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# Math OER Shapeshifting: Changing a Number

Our first foundational topic is Shapeshifting. This is our nickname for when we do something to a single number.

What can we do with a single number?

Consider a situation in which child rolls a die 1,000 times. An even number is rolled 499 times.

We can write this result in fraction format as 4991,000.

This fraction is very close to one-half. In fact, we would expect that if the child rolled the dice many more times the nearness to one-half would increase. The child has "proved" in his or her mind that the chance rolling an even number is one-half. So it makes sense to round the answer to one-half.

We could change the number one-half to decimal format by writing it as 0.5.

Tangentially, when it is spoken aloud the decimal 0.5 has three possible names. An instructor who is lecturing would use the dictation name and say "zero point five" because saying "zero point..." gives a heads-up to listening ears. People thinking to themselves or working together would probably use the casual name and say "point five". There is also a formal name, "five tenths", created from the place value of the final digit.

We could change the number one-half to percent format by writing it as 50%.

Finally, if the number was a measurement we could change the format of its measurement units. For example, a length of 0.5 feet could also be written as 6 inches.

To summarize, we will explore more deeply how and when we "shapeshift" a number by rounding or by changing formats (fraction, decimal, percent, measurement units).

As we study this topic, work on making helpful and organized notes, so you have handy the comments, formulas, and example problems you need.

## Rounding

### What is math?

My scale tells me I weigh 150 pounds.

That's just a number. Numbers are part of math.

Here is one of the geometry formulas we will learn:

Area of a rectangle = length × width

This formula expects us to plug in two measured lengths, and then it provides an area. Formulas are part of math.

We could actually use the formula to calculate an answer. If a rectangular field was 500 feet long and 700 feet wide, the formula claims that the field's area is length × width = 500 ft × 700 ft ≈ 350,000 square feet. Calculations are part of math.

Here is a quick version of the famous barometer story.

The Barometer Story

A physics test asked the question, "How could you measure the height of a tall building, using a barometer?"

One student gave the expected answer: "I could measure the barometric pressures at the top and the bottom of the building, and use the formula about pressure to determine the buildingâ€™s height," but instead of stopping there provided many other answers.

• I could drop the barometer off the roof, time its fall, and use the formulas about falling speed to find the distance it traveled before it shattered on the sidewalk below.
• I could walk down the stairwell while making marks on the wall equal to the barometer's height, and then multiply the number of marks by that height.
• I could tie the barometer to a string, lower it from the roof to the ground, and measure the length of the string.
• I could tie the barometer to a string, lower it from the roof to the ground, swing it like a pendulum, time its frequency, and use the pendulum formulas to find the length of the string.
• I could stand the barometer on the sidewalk, measure its height and the length of its shadow, measure the length of the building's shadow, and use the formula about proportional triangles.
• I could go to the building superintendent and offer him a brand-new barometer if he will tell me the height of the building.

That story talks about numbers and formulas and computations. But it does not actually include any. Thinking about numbers and formulas and computations is also math.

A math class should focus on all four parts of math.

We use lots of numbers! No big deal. Numbers themselves are rarely exciting.

We learn a bunch of famous formulas! We will plug in numbers and use these formulas. But equally important is to learn how to use the formulas. For example, if we were measuring the area of a field that is not perfectly rectangular, how accurate and useful will our answer be?

We do a lot of computation! Correctly doing small problems with right answers helps you feel good about your math ability.

The computations we do are not themselves hard. They can be hard when they are new, just because anything can be hard when it is new! But once they stop being new, they also stop being hard.

Finally, we do a lot of thinking about numbers and formulas and computations.

Most of our thinking about numbers and formulas and computations is planning.

How do you plan a diet to manage your weight? How do you plan a meal to feed a hundred people? How do you plan a household budget, or plan what size house to buy to fit a budget? How do you plan saving for retirement? How do you plan the pricing strategy for a store or restaurant?

These are big problems without right answers. Math can guide us to make good decisions even if there is no one right answer for that situation.

Domain of Science

The Map of Mathematics

### Decimal Accuracy§1.1, §4.1

#### Very Accurate

Decimals can be very accurate.

LEGO bricks fit together so well because the molds that make them are even more precise. Their accuracy is within two thousandths of a millimeter.

How much is two thousandths?

Let's review the place value names.

1. What is the place value of the 8 in the number 85,026,139?

That digit does not mean 8, but 80,000,000. It can say it is "eight ten-millions".

1. What is the place value of the 8 in the number 24.3185?

That digit does not mean 8, but 0.008. It can say it is "eight thousandths".

Thousands vs. Thousandths

2,000       0.20       0.02       0.002

Which of these numbers is two thousands?

Which of these numbers is two thousandths?

1. High-performance gaming optical mice can measure a movement as small as six ten-thousandths of an inch. Write the number "six ten-thousandths" as a decimal.

0.0006

If it helps, work in steps. 0.6 is six tenths. 0.06 is six hundredths. 0.006 is six thousandths. Finally we get to 0.0006 as six ten-thousandths.

1. State the place value of each digit in the number 9876.54321

from left to right: hundreds, tens, ones, tenths, hundredths, thousandths, ten-thousandths, hundred-thousandths

We should also do two problems that check whether zeroes in place values confuse us.

1. Place the following numbers in ascending order (smallest to largest)

20,680       2,680       2,860       2,608

2,608 → 2,680 → 2,860 → 20,680

1. Place the following numbers in ascending order (smallest to largest).

0.57429       0.57491       0.574905       0.57949

0.57429 → 0.574905 → 0.57491 → 0.57949

Here is an example in which so many decimal places really matter.

Cole Paviour invented a famous machine that used a computerized motion-capture of an athlete moving to control the timing of more than two thounsand separate water drop dispensers. This machine needs accuracy to the millionths place.

Imagine that one of Cole Paviour's measurements was 5.714286 seconds.

1. Confirm that the measurement 5.714286 seconds is accurate to the millionths place.

Yes. The 7 is in the tenths, the 1 in the hundredths, the 4 in the thousandths, the 2 in the ten-thousandths, the 8 in the hundred-thousandths, and the 6 in the millionths.

#### Too Accurate?

In the situation above the decimals really were amazingly accurate. However, some decimals look accurate but are not.

The most common source of fake accuracy is division.

Consider three different division stories that use 40 ÷ 7 = 5.714286....

1. I am planning my kid's birthday party. There will be six children attending, so I need 7 party favor bags (including the one for my own kid). I have 40 pieces of candy for in the bags. How many pieces go in each bag?

To fairly distribute 40 pieces of candy into 7 party favor bags, each bag gets 5 pieces. I can see 5 pieces ÷ 7 = 35 pieces. Then I eat the remaining 5 pieces myself.

An alert student might notice that this problem did not follow the usual rules for rounding. The tenths place was 5 or more (it was 7) but we still rounded down. That is okay. Real life can be like that.

It is true that 40 ÷ 7 = 5.714286... which looks like the number from Example 4. But rounding this answer with any decimal places does not make sense. No one will get a part of a candy. We should just drop all the decimal places.

Notice how the decimal places appeared because of division, not careful measurement. They were fake accuracy.

1. A pharmacist has a small container whose label tells her it contains 40 milligrams of medicine. She divides it into 7 doses. How many milligrams are in each dose?

Since the initial value was only accurate to a whole number of milligrams, we could use the same number of decimal places and say 40 milligrams ÷ 7 ≈ 6 milligrams. Making equal piles might not increase our accuracy beyond what we originally knew. We have no reason to believe the medicine's accurate total weight is precisely 40 milligrams. It could be 39.6 milligrams or 40.4 milligrams, or some other amount that whomever sent it to the pharmacist had rounded. The pharmacist did not re-weigh the medicine before dividing it into doses, knowing that for this medicine that would be a waste of her time. So 6 milligrams is an honest answer.

We could also use one more decimal place and say 40 milligrams ÷ 7 ≈ 5.7 milligrams. The pharmacist's scale can measure accurately to the tenth of a milligram. Making equal piles can often increase our accuracy by one decimal place. That is another honest answer.

But if we said 40 milligrams ÷ 7 ≈ 5.71 milligrams we would be exaggerating. The pharmacist's scale is not accurate to the hundredth of a milligram. No matter how carefully we make equal piles we probably cannot increase our accuracy by two decimal places.

Notice how the decimal places appeared because of division, not careful measurement. But adding one more decimal place could be justified. Making about ten piles could increase our accuracy by a tenth. Also, the pharmacist's scale could measure accurately to the tenth of a milligram.

1. A carpenter had a 40 inch board, and cut it into seven equal lengths. How many inches long is each piece?

If we said 40 inches ÷ 7 ≈ 6 inches we probably insult the carpenter. A tape measure and saw should be accurate to the tenth of an inch.

We should use one more decimal place and say say 40 inches ÷ 7 ≈ 5.7 inches. That is an honest answer.

If we said 40 inches ÷ 7 ≈ 5.71 inches we are probably exaggerating. Maybe the original board was not actually shorter than 40 inches, maybe the carpenter was extremely careful and used his tools to the limit of their accuracy, and maybe the extra wood was discarded unstead of letting the final piece be a smidgen too long. In this situation the tools might justify using two more decimal places in our answer. But those are a lot of maybes.

Notice how the decimal places appeared because of division, not careful measurement. But adding one more decimal place is very justified. Making about ten piles could increase our accuracy by a tenth. Also, the carpenter's tools should measure accurately to the tenth of an inch.

In conclusion, a measurement tool can be very accurate. But when division is the source of a decimal answer, the general guideline is to not add more than one more decimal place than in the original values.

A General Rule

The examples above suggest a a General Rule for Accuracy: when division is the source of a decimal answer, use the same number of decimal places as in the original values, or perhaps one more decimal place.

However, the real-life context takes priority. When making party favor bags, do we cut candies into pieces? When sawing a board, how accurate can we cut?

Your turn to do an example.

1. Clara cans 15 pints of salsa. There are 16 pints in a gallon. How much of a gallon did she make?

15 pints ÷ 16 pints per gallon = 0.9375 of a gallon ≈ 0.94 of a gallon

We can reliably divide a gallon into 100 pieces, each would be about 2.5 tablespoons. Clara's salsa production is probably not accurate to the tenth of a tablespoon, so we should stop there.

In the examples above the decimal answer was rounded for you. Now it is your turn to round decimal answers!

First, a carefully worded definition.

Definition

Rounding is ideally reducing the accuracy of an answer to remove fake accuracy, using four steps:

• Pick an appropriate place value

• Make zero the digits to the right (this removes decimal digits)

• Of the digits just made zero, if the largest/leftmost was 5 or more then increase by 1 the digit of the picked place value

• Do not change the digits to the left

No one else defines rounding so carefully. But we can be extra awesome.

Let's unpack this definition.

First, we ideally only round answers with fake accuracy. The exceptions should be few. When someone like Cole Paviour works hard to be genuinely super-accurate it would be a shame to throw away that accuracy for no reason!

Second, we pick an appropriate place value. You were just introduced to the relevant issues. How accurate are the situation's initial values? How accurate are the tools being used? Is the situation one of the rare cases when making equal sized piles can add not merely one but two decimal places of accuracy?

(In general, students worry much more about this decision than math instructors. We want you to learn math, not spend time agonizing over whether a teaspoon is accurate to the hundredth or thousandth of a pint.)

Third, notice that we are making digits zero. It can be tempted to use sloppy thinking and imagine that we are "removing" decimal place digits. But truthfully they are still there. We have only set them to zero, which means we can skip writing them.

Time for many more examples!

1. Round each number to the nearest... ones?      tens?      hundreds?      thousands?
• (a) 4,573.1
• (b) 624.95
• (c) 17,348.9

(a) 4,573 to the ones, 4,570 to the tens, 4,600 to the hundreds, 5,000 to the thousands

(b) 625 to the ones, 620 to the tens, 600 to the hundreds, 1,000 to the thousands

(c) 17,349 to the ones, 17,350 to the tens, 17,300 to the hundreds, 17,000 to the thousands

1. Round each number to the nearest... tenth?      hundredth?      thousandth?      ten-thousandth?
• (a) 1.95283
• (b) 3.85264
• (c) 0.07249
• (d) 25.79013
• (e) 0.6666666...
• (f) 1.27272727...
• (g) 5.749507

(a) 2.0 to the tenth, 1.95 to the hundredth, 1.953 to the thousandth, 1.9528 to the ten-thousandth

(b) 3.9 to the tenth, 3.85 to the hundredth, 3.853 to the thousandth, 3.8526 to the ten-thousandth

(c) 0.1 to the tenth, 0.07 to the hundredth, 0.072 to the thousandth, 0.0725 to the ten-thousandth

(d) 25.8 to the tenth, 25.79 to the hundredth, 25.790 to the thousandth, 25.7901 to the ten-thousandth

(e) 0.7 to the tenth, 0.67 to the hundredth, 0.667 to the thousandth, 0.6667 to the ten-thousandth

(f) 1.3 to the tenths, 1.27 to the hundredths, 1.273 to the thousandths, 1.2727 to the ten-thousandths

(g) 5.7 to the tenth, 5.75 to the hundredth, 5.750 to the thousandth, 5.7495 to the ten-thousandth

In part (d) of the above problem please notice the unexpected thing when rounding to the nearest thousandth. When we are asked to write to the thousandths place, we must write three decimal digits. We write a zero to "fill up" to the thousandths place.

1. Find the rounding error below, where someone tried to round 478.3469 to different place values.

The hundreds digit is less than five, so when rounding to the nearest thousand we get 0 instead of 1,000.

Our definition of rounding warned us to not round in the middle of a problem. If we round too early we introduce error. Let's consider a few examples.

1. In Eugene it rains, on average, about 1.15 inches per week. How many inches of rain is this per year? Round correctly, only rounding the answer.

1.15 inches × 52 weeks = 59.8 inches

1. In Eugene it rains, on average, about 1.15 inches per week. How many inches of rain is this per year? Round incorrectly, doing 1 inch × 52 weeks.

1 inch × 52 weeks = 52 inches

Soon we will have the tools to see that rounding too early in the problem made the answer about 13% too small. That might matter a lot if we were farmers budgeting for our annual water bill.

Here is another example of rounding too early.

1. You do assistant contractor work that earns a little over \$1,000 per month after taxes. You work for 3 months and 1 week, then have time between jobs, then work for 5 months and 1 week. How much do you earn total? Round correctly, only rounding the answer.

(3.25 + 5.25) × \$8,500

1. You do assistant contractor work that earns a little over \$1,000 per month after taxes. You work for 3 months and 1 week, then have time between jobs, then work for 5 months and 1 week. How much do you earn total? Round incorrectly, doing (3 + 5) × \$1,000.

(3 + 5) × \$8,000

Soon we will have the tools to see that rounding too early in the problem made the answer about 6% too small. That could be a big deal if we were planning our personal budget.

As a concluding exercise, work with a partner to invent a word problem. Solve it accurately, doing any rounding at the end. Then round too early, and see how much the answer changes. Try to invent a problem for which rounding early would be utterly disastrous.

Video example problems have three images to click on. You can see a video step-by-step answer, a written step-by-step answer, or only the answer. If you find yet more helpful videos, please let your instructor know so that this website can be updated and improved!

Introduction to Place Value

Finding Place Value

The Organic Chemistry Tutor

Rounding Numbers and Rounding Decimals - The Easy Way!

Round 34,528 to the nearest thousand. Identify the requested place value: 34,527
Also consider the place value immediately to the right: 34,528
Is the 5 as big as 5 or more? Yes, so it makes us increase the 4 before we zero out that 5 and everything to its right: 35,000
35,000

Round 34,528 to the nearest ten. Identify the requested place value: 34,528
Also consider the place value immediately to the right: 34,528
Is the 8 as big as 5 or more? Yes, so it makes us increase the 2 before we zero out that 8 (there is nothing to its right): 35,530
35,530

Round 34,528 to the nearest hundred. Identify the requested place value: 34,528
Also consider the place value immediately to the right: 34,528
Is the 2 as big as 5 or more? No, so leave the 5 unchanged as we zero out that 2 and everything to its right: 34,500
34,500

Round 5.6783 to the nearest one. Identify the requested place value: 5.6783
Also consider the place value immediately to the right: 5.6783
Is the 6 as big as 5 or more? Yes, so it makes us increase the 5 before we zero out that 6 and everything to its right: 6
6

Round 5.6783 to the nearest hundredth. Identify the requested place value: 5.6783
Also consider the place value immediately to the right: 5.6783
Is the 8 as big as 5 or more? Yes, so it makes us increase the 7 before we zero out that 8 and everything to its right: 5.68
5.68

Round 5.6783 to the nearest thousandth. Identify the requested place value: 5.6783
Also consider the place value immediately to the right: 5.6783
Is the 3 as big as 5 or more? No, so leave the 8 unchanged as we zero out that 3 (there is nothing to its right): 5.678
5.678

Round 5.6783 to the nearest tenth. Identify the requested place value: 5.6783
Also consider the place value immediately to the right: 5.6773
Is the 7 as big as 5 or more? Yes, so it makes us increase the 6 before we zero out that 7 and everything to its right: 5.7
5.7

Textbook Exercises for Decimal Accuracy

This list of recommended odd-numbered textbook problems is designed for a hypothetical student with a "typical" math background. If your math foundation is weak, do even more odd-numbered problems. If your math foundation is strong, do fewer.

Section 1.1 (page 9) # 13, 15, 17, 19, 67, 71

Section 4.1 (page 197) # 17, 19, 21, 27, 29, 31, 33, 35, 49, 55, 57, 59, 69, 71

### Estimating§1.3, §4.1

As we have seen, rounding early can drastically change an answer. But it can be a very helpful thing to do as a quick "check" before we attempt a problem.

Definition

Rounding at the very beginning of a problem is called estimating.

For now, use estimating to quickly get a rough idea how large the answer should be. This an artificial but useful reason to estimate.

After we estimate, we could actually solve the problem. Is our answer close to the estimated amount? If not, we might have made a careless mistake (on paper or with the calculator) and can go back to find what went wrong.

1. Estimate the answer to \$47,900 ÷ 12 by rounding both numbers to the nearest ten, before using a calculator to do the problem.

\$47,900 ÷ 10 = \$4,790, which is not very close to the actual calculation of \$47,900 ÷ 12 = \$3,991.67

Many real life situations force us to estimate. For example, if I am planning a road trip the price of gasoline will not be the same at every gas station. To budget for my trip I will pick one value to use as my "typical" estimated price.

For that reason we will (later on) do math in situations that start with estimation and have no precise "right answer" to afterwards compare.

1. Estimate the sum or difference. Round all numbers (you pick how much to round) before mentally adding or subtracting.
• (a) 57 + 34 ≈ 60 + 30 =
• (b) 353 − 92 ≈
• (c) 119 + 75 ≈
• (d) 462 − 183 ≈
• (e) 6,035 + 107 ≈
• (f) 8.15 − 0.006 ≈
• (g) 3,542.55 + 6,276.1 ≈
• (h) 1.675 − 0.9999 ≈

Note: answers will be different if the amount rounded is different.

(a) 60 + 30 = 90

(b) 350 − 90 = 260

(c) 120 + 80 = 200

(d) 460 − 180 = 280

(e) 6,000 + 100 = 6,100

(f) 8.15 − 0.01 = 8.14

(g) 3,500 + 6,300 = 9,800

(h) 1.675 − 1 = 0.675

1. Estimate the product or quotient. Round all numbers (you pick how much to round) before mentally multiplying or dividing.
• (a) 18 × 27 ≈ 20 × 30 =
• (b) 932 ÷ 6 ≈
• (c) 53 × 5 ≈
• (d) 2,084 ÷ 175 ≈
• (e) 868.52 × 7.5 ≈
• (f) 461 ÷ 19.2 ≈
• (g) 0.005 × 0.014 ≈
• (h) 245 ÷ 241 ≈

Note: answers will be different if the amount rounded is different.

(a) 20 × 30 = 600

(b) 930 ÷ 10 = 93

(c) 50 × 5 = 250

(d) 2,000 ÷ 200 = 10

(e) 870 × 10 = 8,700

(f) 460 ÷ 20 = 23

(g) 0.01 × 0.01 = 0.0001

(h) 200 ÷ 200 = 1

Here are more estimation problems.

1. This week the national average price for a certain item was \$3.025, which is an increase of \$0.284 from last week. First estimate last week's price. Then solve to find the actual value of last week's price.

estimate \$3.025 − \$0.284 ≈ \$3.0 − \$0.3 = \$2.70

actual \$3.025 − \$0.284 = \$2.741

1. A square has four equal sides of length 2.48 feet. The perimeter of a square is the sum of its four sides. First estimate the square's perimeter. Then solve to find the actual value of the square's perimeter.

estimate 2.48 × 4 ≈ 2.5 × 4 = 10 feet

actual 2.48 × 4 = 9.92 feet

1. Circle the correct response. If each number in a sum is rounded up, the estimated total
2. will be an overestimate    /    will be an underestimate    /    could be either an overestimate or underestimate

will be an overestimate

1. Circle the correct response. If each number in a sum is rounded down, the estimated total

will be an overestimate    /    will be an underestimate    /    could be either an overestimate or underestimate

will be an underestimate

1. Circle the correct response. If some of the numbers in a sum are rounded up and others are rounded down, the estimated total

will be an overestimate    /    will be an underestimate    /    could be either an overestimate or underestimate

could be either an overestimate or underestimate

Video example problems have three images to click on. You can see a video step-by-step answer, a written step-by-step answer, or only the answer. If you find yet more helpful videos, please let your instructor know so that this website can be updated and improved!

Estimate 4,872 + 1,691 + 777 + 6,124 by first rounding each number to the nearest thousand. ≈ 5,000 + 2,000 + 1,000 + 6,000 = 14,000 14,000

Estimate the sum 23,649 + 54,746 by first rounding to the nearest hundred. ≈ 23,600 + 54,700 = 78,300 78,300

Estimate the difference 54,751 − 23,649 by first rounding to the nearest hundred. ≈ 54,800 − 23,600 = 31,200 31,200

Estimate the product 824 × 489 by first rounding to the nearest hundred. ≈ 800 × 500 = 400,000 400,000

Estimate the product 8.91 × 22.457 by first rounding to the nearest one. ≈ 9 × 22 = 198 198

Estimate the quotient 78.2209 ÷ 16.09 by first rounding to the nearest ten. ≈ 80 ÷ 20 = 4 4

Textbook Exercises for Estimating

This list of recommended odd-numbered textbook problems is designed for a hypothetical student with a "typical" math background. If your math foundation is weak, do even more odd-numbered problems. If your math foundation is strong, do fewer.

Section 1.3 (page 31) # 41, 43, 45, 47, 59, 61, 63

Section 1.6 (page 74) # 35, 37, 39, 41, 43, 45

Congratulations! You are at the end of the first subtopic (Rounding) for our first big topic (Shapeshifting).

You have now thought more carefully than most people about decimals, their accuracy, when to round, and how much to round.

### Ten Exercises

Try these ten exercises on scratch paper. Work in a study group if you can! Notice where your notes need improvement. After you are very happy with your answers, you can use this form to ask me to check your work. Can you get at least 8 out of 10 correct?

1. What number is two hundred twelve thousandths?

2. Which is the largest decimal? 0.243, 0.423, 0.324, 0.0423, 0.4023

3. Which is the smallest decimal? 0.243, 0.423, 0.324, 0.0423, 0.4023

4. Round 17,398.967 to the nearest ten.

5. Round 98,807.728 to the nearest tenth.

6. Round 54,498.66118 to the nearest thousand.

7. Notice that 2 ÷ 11 = 0.181818... Round that decimal to the nearest thousandth.

8. Alicia's rectangular garden bed has two long sides of length 2.74 meters and two short sides of length 1.22 meters. Estimate the sum of all four sides.

9. Ben's rectangular garden bed has two long sides of length 3.048 meters and two short sides of length 0.9144 meters. Estimate the sum of all four sides.

10. Which garden bed estimate will be closer to the actual sum? How can you know without calculating the actual sums?

### Random Exercises

Try these exercises on scratch paper. Work in a study group if you can! Notice where your notes need improvement. Check your work when you are done.

## Fraction Format

### Thinking About Division§1.6, §2.1, §2.5

There are two ways to think about division. Unfortunately, most people are only taught one way and this causes people to get stuck.

#### Dealing Out Cards

The first way to think about division is dealing out cards.

I can model 6 ÷ 3 = 2 by acting out having six cards and dealing them out to three people until I am done.

How does it work?

1. When I model 6 ÷ 3 = 2 by dealing out cards, what question do I ask to find the answer? (An answer we know is 2.)

Division: Dealing Out Cards (Works Great!)

We can nicely model 6 ÷ 3 = 2 by dealing out cards.

6 becomes the number of cards total.

3 becomes the number of piles.

2 becomes the number of cards per pile.

So the question is: How many cards end up in each pile?

Unfortunately, this way of thinking does not help with dividing by fractions.

1. When I try to model 6 ÷ 12 = 12 by dealing out cards, what goes wrong?

Division: Dealing Out Cards (Does Not Work)

We cannot model 6 ÷ 12 = 12 by dealing out cards.

6 becomes the number of cards total.

½ becomes the number of piles. Now we get stuck! How do we deal cards to half a pile? What is "half a pile" anyway?

The question no longer makes sense. I cannot ask: How many cards end up in each pile?

#### Making Piles of a Fixed Size

The second way to think about division is making piles of a fixed size.

I can model 6 ÷ 3 = 2 by acting out having six cards and making piles of size three until I am done.

How does it work?

1. When I model 6 ÷ 3 = 2 by making piles of a fixed size, what question do I ask to find the answer? (An answer we know is 2.)

Division: Piles of Fixed Size (Works Great!)

We can nicely model 6 ÷ 3 = 2 by making piles of a fixed size

6 becomes the number of cards total.

3 becomes the number of cards per pile.

2 becomes the number of piles.

So the question is: How many piles do I make before I run out of cards?

This way of thinking does help us think about dividing by fractions.

1. When I model 6 ÷ 12 = 12 by making piles of a fixed size, what question do I ask to find the answer? (An answer we know is 12.)

Division: Piles of Fixed Size (Works Great!)

We can nicely model 6 ÷ 12 = 12 by making piles of a fixed size

6 becomes the number of cards total.

½ becomes the number of cards per pile. I rip each card in half!

12 becomes the number of piles.

The question still makes sense: How many piles do I make before I run out of cards?

The second way of thinking also allows us to understand why division by zero, which is undefined, is in a few situations treated as if the answer is infinity. If I had some cards and tried to make piles of size zero I can do this easily. I just never stop!

1. Solve each equation by multiplying or dividing, if possible.
• (a) 8 ÷ 4 =
• (b) 8 ÷ ¼ =
• (c) ½ × 2 =
• (d) ½ ÷ 2 =
• (e) 1 ÷ ¼ =
• (f) ½ ÷ ¼ =

(a) We can imagine dealing out 8 cards to 4 people. Each person gets 2 cards.

We can also imagine making 8 cards into piles of 4 cards. We make 2 piles before we run out of cards.

(b) Imagine having eight pieces of paper. We rip them into quarters, while setting those quarters down as "piles". We get 32 piles.

(c) ½ + ½ = 1. This is nothing more than what "one half" means!

(d) We can imagine dealing out ½ of a card to 2 people. Rip! Each person gets ¼ of a card.

We can also imagine making ½ of a card into piles of size 2 cards. We run out to soon. We can only make ¼ of a pile of that size.

(d) ¼

(e) Imagine having one piece of paper. We rip it into quarters, while setting those quarters down as "piles". We get 4 piles.

(f) Imagine having half a piece of paper. We rip it in half again, making quarters, while setting those quarters down as "piles". We get 2 piles.

How about some multiplication and division problems that some students try to memorize as separate rules, instead of merely understanding division well?

1. Solve each equation by multiplying or dividing, if possible.
• (a) 28 × 1 =
• (b) 28 ÷ 1 =
• (c) 72 × 0 =
• (d) 72 ÷ 0 =
• (e) 100 × 100 =
• (f) 100 ÷ 100 =

(a) 28

(b) 28

(c) 0

(d) undefined

(e) 10,000

(f) 1

Video example problems have three images to click on. You can see a video step-by-step answer, a written step-by-step answer, or only the answer. If you find yet more helpful videos, please let your instructor know so that this website can be updated and improved!

The Idea of Division

Ways to Represent Division

Textbook Exercises for Thinking About Division

This list of recommended odd-numbered textbook problems is designed for a hypothetical student with a "typical" math background. If your math foundation is weak, do even more odd-numbered problems. If your math foundation is strong, do fewer.

Section 1.6 (page 76) # 77, 79, 81

Section 2.1 (page 102) # 75, 77, 81, 83

Section 2.5 (page 139) # 55, 59

### Divisibility and Factors§2.3

#### Visualizing Factors

Our next step in thinking carefully about division is to visualize factors.

Definition

Factors are the numbers you multiply to get another number.

That is a slightly sloppy definition, but it is good enough.

We can look at a multiplication equation to find factors.

Factors of 15

3 and 5 are two of the factors of 15 because 3 × 5 = 15

1 and 15 are two more factors of 15 because 1 × 15 = 15

Some students find it helpful to think about factors by imagining all the ways to make a rectangle with blocks or coins.

Note that 1 and the number itself are always factors!

Very soon we will look at shortcuts for finding factors. But we can do some problems with small numbers first to make sure we understand the definition.

1. Find both factors of 5.

The only factors are 1 and 5.

Definition

A prime numbers has no factors other than itself and 1.

1. Find all three factors of 25.

The three factors are 1, 5, and 25.

Notice that the number 5 does appear twice in the equation 5 × 5 = 25. But we do not mention it twice when naming the factors.

1. Find all four factors of 6.

The four factors are 1, 2, 3, and 6.

#### Divisibility Shortcuts

How would we find factors if we do not have multiplication equations handed to us?

We could just think hard, and maybe guess-and check.

Does 6 Work?

Is 6 a factor of 96?

Um...let me think...yes, because 6 × 16 = 96

But some tricks can help us.

Here are divisibility shortcuts you already know:

• A number divisible by 2 is even, and its one's digit is 0, 2, 4, 6, or 8.
• A number divisible by 5 has for its one's digit either 0 or 5.
• A number divisible by 10 has 0 as its one's digit.

Here are divisibility shortcuts that might be new to you:

• A number is divisible by 3 if the sum of its digits is divisible by 3.
• A number is divisible by 6 if it is divisible by both 2 and 3.
• A number is divisible by 4 if the two-digit number formed from its ten's and one's digits is also divisible by 4.
• A number is divisible by 9 if the sum of its digits is divisible by 9.

There are other divisibility tricks, but these are the ones that are easy enough to use to count as shortcuts.

Let's apply our divisibility shortcuts to 13,512.

1. Is 13,512 divisible by... 2?       3?       4?       5?       6?       9?

Divisible by 2? Yes, because the one's place value digit is 2, which is even.

Divisible by 3? Yes, because the sum of digits is 1 + 3 + 5 + 1 + 2 = 12 and three goes into 12.

Divisible by 4? Yes, because two-digit number formed from its ten's and one's digits is 12 and four goes into 12.

Divisible by 5? No, because the one's place value digit is not zero or five.

Divisible by 6? Yes, because it was divisible by both 2 and 3.

Divisible by 9? No, because the sum of digits is 1 + 3 + 5 + 1 + 2 = 12 and nine does not go into 12.

Let's also apply our divisibility shortcuts to 2,016.

1. Is 5,661 divisible by... 2?       3?       4?       5?       6?       9?

Divisible by 2? No, because the one's place value digit is 1, which is odd.

Divisible by 3? Yes, because the sum of digits is 8 + 6 + 6 + 1 = 18 and three goes into 18.

Divisible by 4? No, because two-digit number formed from its ten's and one's digits is 61 and four does not go into 61.

Divisible by 5? No, because the one's place value digit is not zero or five.

Divisible by 6? No, because it was not divisible by both 2 and 3.

Divisible by 9? Yes, because the sum of digits is 8 + 6 + 6 + 1 = 18 and nine goes into 18.

Let's also apply our divisibility shortcuts to 5,025.

1. Is 5,025 divisible by... 2?       3?       4?       5?       6?       9?

Divisible by 2? No, because the one's place value digit is 5, which is odd.

Divisible by 3? Yes, because the sum of digits is 2 + 2 + 5 = 12 and three goes into 12.

Divisible by 4? No, because two-digit number formed from its ten's and one's digits is 25 and four does not go into 25.

Divisible by 5? Yes, because the one's place value digit is a zero or five.

Divisible by 6? No, because it was not divisible by both 2 and 3.

Divisible by 9? No, because the sum of digits is 1 + 3 + 5 + 1 + 2 = 12 and nine does not go into 12.

Let's also apply our divisibility shortcuts to 73,080.

1. Is 73,080 divisible by... 2?       3?       4?       5?       6?       9?

Divisible by 2? Yes, because the one's place value digit is 0, which is even.

Divisible by 3? Yes, because the sum of digits is 7 + 3 + 8 = 18 and three goes into 18.

Divisible by 4? Yes, because two-digit number formed from its ten's and one's digits is 80 and four goes into 80.

Divisible by 5? Yes, because the one's place value digit is a zero or five.

Divisible by 6? Yes, because it was divisible by both 2 and 3.

Divisible by 9? Yes, because the sum of digits is 7 + 3 + 8 = 18 and nine goes into 18.

These divisibility shortcuts let us quickly find some factors.

Let's play the Factor Game! Here is a copy of the game board to use with an in-class demonstration.

Once you have played the Factor Game yourself, try to give advice in the following situation.

1. Xavier and Odette are playing the Factor Game. Xavier is X's. Odette is O's. The board is shown below. It is Xavier's turn.
• (a) What is Xavier's best move? (Hint: Odette does not get to claim any spots in reply.)
• (b) What will Odette's best move be on her turn? (Hint: Xavier can reply, but she still comes out ahead by 14 points.)
• (c) Conidering the 7 spot is already taken, why shouldn't Odette claim the 14 spot as a simpler way to come out ahead by 14 points?

(a) If Xavier picks 15 then Odette cannot reply and he comes out ahead by 15 points.

(b) She picks 28 and Xavier replies with 14.

(c) The next turn Xavier would be able to pick 28 and she would be unable to reply.

Being able to find factors quickly will soon allow us to do fraction arithmetic more easily. But we have a bit more preparation to do.

There are two flavors of thorough factor finding. In some situations we want to find all the factors. In other situations we want to find the prime factors.

Finding all the factors will be useful when reducing a fraction.

Finding prime factors will useful when finding a common denominator for fractions.

#### Finding All the Factors

We find all the factors of a number by making a two-column list. Count 1, 2, 3,... in the first column. List any matching factors in the second column. When the columns get to the same value we can stop.

1. Find all the factors of 66.

This time our first column counts up from 1 to 10. The, the next value for the first column would be 11, which is already listed in the second column. So we can stop.

1       66
2       33
3       22
4       not a factor
5       not a factor
6       11
7       not a factor
8       not a factor
9       not a factor
10      not a factor

So 66 has eight factors: 1, 2, 3, 6, 11, 22, 33, 66

1. Find all the factors of 24.

24 has eight factors: 1, 2, 3, 4, 6, 8, 12, 24

1. Find all the factors of 60.

60 has twelve factors: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

1. Which has more factors, 36 or 40?

36 has nine factors (more): 1, 2, 3, 4, 6, 9, 12, 18, 36

40 has eight factors: 1, 2, 4, 5, 8, 10, 20, 40

1. Of all the numbers we just considered, only 36 had an odd number of factors. What happened to make 36 special?

36 = 6 × 6. When we made those two columns, the bottom number for both columns was the same, 6. So even though factors come in pairs, there is a special case where the "bottom" pair of factors is the same number repeated.

#### Finding Prime Factors

We find prime factors by making a factor tree and noting the "leaves".

Remember factor trees?

Here is one factor tree for 48.

1. Write a different factor tree for 48 that does not start with 6 and 8.

We could start with 2 × 24 or with 3 × 16 or with 4 × 12.

We could also start by splitting 48 into three factors, such as 2 × 2 × 12 or 2 × 3 × 8.

Either circle or "bring down" the leaves of your factor tree so you do not make a careless mistake and forget any of them when writing your answer.

To be polite, list the prime factors in order. Write them as a product (separated by × symbols).

Definition

The ordered list of prime factors is called the prime factorization.

Optionally, you may show off your fluency with exponents by writing the prime factorization as compactly as possible using exponents.

With or Witout Exponents

For example, the prime factorization of 66 is 2 × 2 × 2 × 3.

Or you could be fancy and write the prime factorization of 66 as 23 × 3.

1. Find the prime factorization of 48.

Looking at the factor trees for 48 we see the prime factorization is 2 × 2 × 2 × 2 × 3 = 48.

This can also be written as 24 × 3 = 48.

1. Find the prime factorization of each number.
• (a) 24
• (b) 18
• (c) 51
• (d) 49
• (e) 80
• (f) 280

(a) 24 = 2 × 2 × 2 × 3

(b) 18 = 2 × 3 × 3

(c) 51 = 3 × 17

(d) 49 = 7 × 7

(e) 80 = 2 × 2 × 2 × 2 × 5

(f) 280 = 2 × 2 × 2 × 5 × 7

Video example problems have three images to click on. You can see a video step-by-step answer, a written step-by-step answer, or only the answer. If you find yet more helpful videos, please let your instructor know so that this website can be updated and improved!

Finding Factors

Divisibility Tests for 2, 3, 4, 5, 6, 9, 10

Finding Factors of a Number

Identify the products and factors: 30 = 2 × 3 × 5 The answer to a multiplication problem is the product.
The amounts being multiplied are the factors.
So the product is 30, and the factors are 2, 3, and 5.
The product is 30. The factors are 2, 3, and 5.

Identify the products and factors: 9 × 8 = 72 The answer to a multiplication problem is the product.
The amounts being multiplied are the factors.
So the product is 72, and the factors are 8 and 9.
The product is 72. The factors are 8 and 9.

Determine whether 784 is divisible by 9. The sum of the digits is 7 + 8 + 4 = 19.
Does 9 go into 19? No.
So 9 does not go into our original number either.
No.

Determine whether 5,552 is divisible by 5. The one's place value digit is 5
Is this 0 or 5? Yes.
So 5 goes into our number.
Yes.

Determine whether 2,322 is divisible by 6. The one's place value digit is 2
Is this 0, 2, 4, 6, or 8? Yes.
So 2 goes into our number.
The sum of the digits is 2 + 3 + 2 + 2 = 9.
Does 3 go into 9? Yes.
So 3 goes into our original number also.
Both 2 and 3 work, so 6 also works.
Yes.

Find all the factors of 300. Start counting, and writing the matching factor.
1 × 300. 2 × 150. 3 × 100. 4 × 75. 5 × 60. 6 × 50. 7 doesn't work. 8 doesn't work. 9 doesn't work. 10 × 30. 11 doesn't work. 12 × 25. 13 doesn't work. 14 doesn't work. 15 × 20. 16 doesn't work. 17 doesn't work. 18 doesn't work. 19 doesn't work.
Now we are up to 20, a number that already appeared. We can stop.
1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 25, 30, 50, 60, 75, 100, 150, 300

Find the prime factorization of 18. Make a factor tree. However you do that, the "leaves" will be 2, 3, and 3.
So you can write 2 × 3 × 3 or you can write 2 × 32
You can write 2 × 3 × 3, or you can write 2 × 32

Find the prime factorization of 60. Make a factor tree. However you do that, the "leaves" will be 2, 2, 3, and 5.
So you can write 2 × 2 × 3 × 5 or you can write 22 × 3 × 5
You can write 2 × 2 × 3 × 5, or you can write 22 × 3 × 5

Textbook Exercises for Divisibility and Factors

This list of recommended odd-numbered textbook problems is designed for a hypothetical student with a "typical" math background. If your math foundation is weak, do even more odd-numbered problems. If your math foundation is strong, do fewer.

Section 2.3 (page 117) # 9, 11, 13, 15

### Desired Denominators§2.1, §2.3, §3.1

How can we change a fraction so it has a desired denominator?

That is an excellent question. We will answer it soon. But first it is time for cake!

Let us consider the fraction one-third drawn as a slice of cake.

#### Un-Reducing

The most intuitive thing to do with those big slices of cake is to keep cutting the cake.

1. Change the three-piece cake by making one cut so the cake will have six equal pieces. How many are shaded?

If we make a horizontal cut through the center, the two left-most pieces will be shaded.

We can represent this cake-cutting with numbers. We take 13 and multiply the numerator and denominator by two.

1. Change the three-piece cake by making two cuts so the cake will have nine equal pieces. How many are shaded?

If we make two horizontal cuts, the three left-most pieces will be shaded.

We can again represent this cake-cutting with numbers. We take 13 and multiply the numerator and denominator by three.

Definition

Un-Reducing a fraction represents with numbers the act of slicing the cake more. The numerator and denominator are multiplied by the same value.

Yikes. I am glad our first three definitions were admirable. Because the fourth was slightly sloppy. And this one talks about cake? What kind of word is "un-reducing" anyway? I don't think that is actual math jargon. Sigh. I thought we had standards.

Anyway, notice that there are four ways to write "un-reducing" one-third into two-sixths. Three are appropriate. One is wrong.

Writing Un-reducing

"Un-reduce" the fraction 13 into an equivalent fraction with a denominator of six.

(a) write two little, floating × 2, one each for numerator and denominator

(b) write a single, centered × 2 intended to work on both numerator and denominator

(c) write another fraction: × 22

(d) cross out the one and three and replace them with the larger values

Do not use (b)! This is how we write multiplication, not "un-reducing" numerator and denominator.

A moment ago we asked, "How can we change a fraction so it has a desired denominator?" Now we have one answer to that question.

If the current denominator is a factor of our desired denominator, we can multiply both numerator and denominator by something.

Enough about "un-reducing"! I don't think it is a real word anyway. I have never heard it before, Google doesn't know about it, and it keeps appearing inside quotation marks.

#### Reducing

Can we use a cake diagram to think about dividing both numerator and denominator of a fraction.

Sure. We could make big slices of cake by gluing smaller slices together. Maybe we use very sticky icing?

Here is a whole cake.

We will prepare the whole cake by making two vertical slices and one horizontal slice to show sixths. Shade in the left-most two of the sixths.

(To be extra clear, we will use one color pen for the vertical slices, and another color for the horizontal slice.)

1. Glue cake slices together to change two-sixths into one-third.

We can represent this cake-gluing with numbers. We take 26 and divide the numerator and denominator by two.

Next we erase, and prepare the whole cake by making two vertical slices and two horizontal slice to show ninths. Shade in the left-most three of the ninths.

(To be extra clear, we will use one color pen for the vertical slices, and another color for the horizontal slices.)

1. Glue cake slices together to change three-ninths into one-third.

We can represent this cake-gluing with numbers. We take 39 and divide the numerator and denominator by three.

Here is our definition.

Definition

We reduce a fraction by dividing the numerator and denominator by the same value. That value must be a factor of both the numerator and denominator.

Since reducing is not make-believe jargon, we actually need a second definition.

Definition

A fraction is fully reduced if its numerator and denominator have no factor in common (except 1).

Many books and websites call a fully reduced fraction "simplified". We avoid that word because it has a bad tendency to be used in too many situations without careful definitions. It ends up fuzzy and vague, for many students meaning no more than "make the instructor happy".

Again there are four ways to write reducing two-sixths into one-third. Three are appropriate. One is wrong.

Writing Reducing

Reduce the fraction 26 into an equivalent fraction with a denominator of three.

(a) write two little, floating ÷ 2, one each for numerator and denominator

(b) write a single, centered ÷ 2 intended to work on both numerator and denominator

(c) write another fraction: ÷ 22

(d) cross out the two and three and replace them with the smaller values

Do not use (b)! This is how we write division, not reducing numerator and denominator.

Notice that reducing is a process. Canceling is one possible step in the process. (Reducing and canceling are related, but not the same thing!)

A moment ago we asked, "How can we change a fraction so it has a desired denominator?" Now we have another answer to that question.

If the desired denominator is a factor of our current denominator, we can try to divide both numerator and denominator by something.

Notice the word try. We cannot always reduce. If we wanted thirds, but currently have 56, we are stuck. Our desired denominator of 3 is indeed a factor of our current denominator of 6. But trying to reduce does not work with that numerator. We do not always get what we want.

An important comment is that efficiency is a convenience, not a virtue when reducing fractions.

1. Use three steps (each of dividing the numerator and denominator by 2) to fully reduce the fraction 824

That worked great!

1. Use one step (of dividing the numerator and denominator by 8) to fully reduce the fraction 824

That also worked great!

Some students like to reduce with many steps. It can feel satisfying and safe to chip away at the numbers a little bit at a time.

Other students like to reduce with one step. It can feel satisfying and powerful to find which factor is most efficient.

You can use either method.

To help us talk about the difference, we need one more definition.

Definition

The greatest common factor of two numbers is the...um...biggest factor they both have...

Oops. That's not a definition. That's a tautology. Sorry.

We have not yet tried finding a common denominator for two or more fractions. When that task arrives you are prepared with a solid foundation.

One more example problem as a finale.

1. Fully reduce the fraction 330462

Video example problems have three images to click on. You can see a video step-by-step answer, a written step-by-step answer, or only the answer. If you find yet more helpful videos, please let your instructor know so that this website can be updated and improved!

Equivalent Fractions

Visualizing Equivalent Fractions

Equivalent Fraction Models

Textbook Exercises for Desired Denominators

This list of recommended odd-numbered textbook problems is designed for a hypothetical student with a "typical" math background. If your math foundation is weak, do even more odd-numbered problems. If your math foundation is strong, do fewer.

Section 2.1 (page 99) # 31, 33, 35, 37, 47, 49, 51, 53, 65

Section 2.3 (page 117) # 17, 19, 23, 37, 39, 41

Section 3.1 (page 151) # 17, 19, 23, 37, 45, 47, 19, 51

### Fractions and Decimals§4.5

#### Decimals to Fractions

Remember when at the very top of this page we mentioned that decimals have three names?

Decimal Aliases

When it is spoken aloud the decimal 0.5 has three possible names.

• People thinking to themselves or working together would probably use the casual name and say "point five". We could write .5

• An instructor who is lecturing would use the dictation name and say "zero point five" because saying "zero point..." gives a heads-up to listening ears. We could write 0.5

• There is also a formal name, "five tenths", created from the place value of the final digit. We could write five-tenths in a legal document or other type of formal writing.

1. Write the formal name for 0.08.

8 hundredths

1. Write the dictation name for 45 hundredths.

0.45

To change a decimal into a fraction, we just say its formal name and then reduce if possible. This is one reason why we need to remember place value names!

1. Write 0.5 in fraction format.

5 tenths = 510 = 12

You probably knew that 0.5 is the same as 12. Hopefully you also see how the process works.

1. Write 0.08 in fraction format.

8 hundredths = 8100 = 225

1. Write 0.125 in fraction format.

125 thousandths = 1251,000 = 18

One way to review converting decimals to fractions is to play The Decimals to Fractions Game.

Game Rules: The Decimals to Fractions Game

A moderator provides five numbers.

Students, in teams with one representative at the board, race to change three of the numbers at a time nto a correct decimal-to-fraction equation. The three numbers used become the decimal, numerator, and denominator.

How many solutions can the class find?

Example of the Decimals to Fractions Game

The five numbers are 1, 2, 5, 6, and 30.

One possible solution uses the three numbers 1, 2, and 5 to create the equation 0.5 = 12

A second possible solution uses the three numbers 2, 6, and 30 to create the equation 0.2 = 630

The five numbers are 1, 4, 8, 25, and 125.

The five numbers are 4, 8, 16, 20, and 25.

The five numbers are 3, 6, 15, 20, and 75.

#### Fractions to Decimals

To change a fraction into a decimal, we just treat the fraction as a division problem. We do numerator ÷ denominator = decimal.

1. Write 15 in decimal format.

Divide top ÷ bottom. 1 ÷ 5 = 0.2

1. Write 38 in decimal format.

Divide top ÷ bottom. 3 ÷ 8 = 0.375

If we are not allowed to use a calculator, we can sometimes avoid long division. There is a clever way to change a fraction to a decimal by making the denominator a power of ten

Cleverness

125 = 125 × 44 = 4100 = 4 hundredths = 0.04

25 is a factor of 100. So we can "un-reduce" to make the denominator 100, then say the fraction's formal name.

However, this clever trick is not really a shortcut. In most real life situations, checking whether fraction denominator were factor of 10, 100, etc. slows you down a lot compared to simply using a calculator.

#### Repeating Decimals to Fractions

We cheated above by pretending that all decimals have a formal name.

But some decimals do not!

Definition

Repeating decimals have a never-ending pattern. We write them using a bar over the repeating portion instead of ellipsis to avoid ambiguity.

Why We Need a Bar

Which do we mean by 0.321...

0.3211111...?       0.321212121...       0.321321321...

Note that a repeating decimal has no formal name. So we cannot use the usual method to change it into a fraction.

For the three most common repeating decimals it is best to simply memorize their fraction equivalents.

Memorize These

One-third = ?

Two-thirds = ?

One-sixth = ?

There is a method for changing a repeating decimal into a fraction, but it is tricky.

Knowing how to do this is not required. We have not discussed equations yet. This will not appear on any homework assignment, quiz or test!

The method involves setting the repeating decimal equal to y, multiplying both sides of the equation by a 10 one or more times, and then subtracting the two equations before solving.

It makes more sense when you see it happen.

1. Change 0.222222... into a fraction.

Set the repeating decimal equal to y.
y = 0.222222...
Multiply both sides by 10.
10y = 2.22222...
Subtract the first from the second.
9y = 2
Divide both sides by 9 to get y by itself.
y = 29

Congratulations again! You are at the end of the second subtopic (Fraction Format) for our first big topic (Shapeshifting).

You have now thought more carefully than most people about division, factors, and reducing and "un-reducting" fractions. You know tricks for finding all the factors, and have seen that those tricks are useful when reducing a fraction.

Video example problems have three images to click on. You can see a video step-by-step answer, a written step-by-step answer, or only the answer. If you find yet more helpful videos, please let your instructor know so that this website can be updated and improved!

Common fractions and decimals

Rewriting decimals as fractions

Math Antics

Convert any Fraction to a Decimal

Convert 78 to a decimal Divide 7 ÷ 8 = 0.875 0.875

Convert 1125 to a decimal Divide 11 ÷ 25 = 0.44 0.44

Textbook Exercises for Fractions and Decimals

This list of recommended odd-numbered textbook problems is designed for a hypothetical student with a "typical" math background. If your math foundation is weak, do even more odd-numbered problems. If your math foundation is strong, do fewer.

Section 4.5 (page 238) # 5, 9, 11, 21, 23, 25, 27, 29, 31

### Ten Exercises

Try these ten exercises on scratch paper. Work in a study group if you can! Notice where your notes need improvement. After you are very happy with your answers, you can use this form to ask me to check your work. Can you get at least 8 out of 10 correct?

1. Consider the following six numbers: 315, 1101, 7628, 13025, 110124, and 876. Which are divisible by 3?

2. Consider the following six numbers: 315, 1101, 7628, 13025, 110124, and 876. Which are divisible by 4?

3. How many factors does 126 have?

4. What is the prime factorization of 126?

5. Fully reduce the fraction 16120

6. Change 0.45 to a fraction. Remember to simplify.

7. What is the decimal notation for four-twentieths?

8. Write this decimal, which is less than one: three hundred and eleven ten-thousandths.

9. Which decimal is greater, 0.040199 or 0.040201?

10. Little Bo Peep has a dozen sheep. She shears 1 pound of wool from the first, 1.5 pounds from the second, 2 pounds from the third, and so forth increasing by 0.5 pounds each time until she shears 6.5 pounds of wool from the last sheep. What is the average amount of wool taken per sheep? How do you know how much to round this answer?

### Random Exercises

Try these exercises on scratch paper. Work in a study group if you can! Notice where your notes need improvement. Check your work when you are done.

### Interlude: Am I In the Right Math Class?

Many college math classes include a quick review of the topics we have done so far.

This can cause a problem!

If you have only seen review topics, instead of the "real" topics for your class, how do you know if you are in the right math class?

If you attend an Oregon college, you might be taking a class named Level C Math or Math 20. You can look at this class's practice finals.

If those seem easy, or something you could do with a few hours of review with a tutor or videos, then perhaps you should move up to the next math class.

In Oregon, the earlier classes had state required objectives. For Level B Math, students should be able to:

• use place value to check which whole numbers are biggest or smallest
• use place value to round whole numbers
• change fractions to common denominators to check which are biggest or smallest
• describe common polygons and break apart polygons into pieces of equal area
• find the perimeter of polygons, or if given the perimeter find the length of missing side(s)
• read or draw a bar graph

For Math 10, students should be able to not only do the above list but also:

• list all the factors of a number
• make a factor tree to find a prime factorization
• find the greatest common factor of two numbers
• list a few multiples of a number
• change fractions to decimals to compare them with other decimals
• confirm if an order of operations statement is true
• confirm if a measurement unit conversion is true

As you have seen, we will review all of these skills.

If you are rusty, or have never been taught them properly, do not worry!

If the review has been going well for you, then you are in the right class.

If the review has been very confusing, or your reaction is "It feels impossible to make time this term to feel caught up!", then maybe you should consider moving down to the previous math class. Talk to your instructor about your personal math background and your time management issues this term.

## Ratio Format

### Ratios and Rates§5.1

We are not done yet with thinking about fractions.

So far all of our fractions were proper fractions. They were cake slice numbers. The denominator counted the number of pieces. The numerator counted the shaded pieces.

Cooks or carpenters deal with those fractions daily. But for many people their real-life fractions are not like that. Their fractions do have a sense of "total pieces" and "shaded pieces".

When is a fraction not a fraction? Cue the suspenseful music...

Let's notice something interesting about fractions and division.

We know that we can change fractions to decimals by doing division. In this sense fractions are like division problems waiting to happen.

A Fraction as Division Waiting to Happen

15 = 1 ÷ 5 = 0.2

We often reduce fractions. In this sense fractions are not like division. When we divide one number by another we do not reduce the numbers first.

This Hurts My Brain

105 = 10 ÷ 5 = 2 ÷ 1 = 2

It looks really weird to "reduce" 10 ÷ 5 and change it into 2 ÷ 1 in the middle of a division problem. No one does that!

We talk about "equivalent fractions" but never "equivalent division problems".

So our first clue is that fraction format is a good way to patiently lurk like a vulture, spying on a situation that does not yet involve division but might soon.

This happens frequently when we compare numbers.

A package of 24 energy drinks costs \$35.99. Does anyone care how much one drink costs? Maybe. We lurk, waiting to see if division will happen.

A contractor submits a \$2,800 bid for installing a 120 foot long fence. Does anyone care about how much this costs per foot? Maybe. We lurk, waiting to see if division will happen.

So we found a reason to deal with a pair of numbers as if they were the numerator and denominator in a fraction, even though the numbers have meaning very unlike "total pieces" and "shaded pieces".

This is the job for a ratio.

Definition

A ratio is a comparison of two numbers, usually written as a fraction.

Some ratios do have a sense of "total pieces" and "shaded pieces".

1. Snoopy interviewed 10 dentists and 8 of them approved of a certain breath mint. Write this comparison as a ratio.

Other ratios do not have a sense of "total pieces" and "shaded pieces".

1. The Red Baron can fly 38 miles in 40 minutes. Write this comparison as a ratio.

There are even ratios with a hidden sense of "total pieces" and "shaded pieces".

1. A classroom has 4 left-handed people and 16 right-handed people. Write this comparison as a ratio.

We could look at the ratio and find the whole (20 total people). But the provided numbers do not explicitly tell us the whole.

Notice that, like Snoopy pretending to be the Flying Ace, we can even write a ratio upside down. This does not lose any meaning. It is not confusing.

How different from a normal fraction, in which the numerator and denominator count "total pieces" and "shaded pieces". Those would have their meaning changed if written upside down.

1. A classroom has 4 left-handed people and 16 right-handed people. Write this comparison as a different ratio, "upside down" compared to your previous answer.

Often it does not make sense to reduce a ratio. If I reduced Snoopy's breath mint ratio I would switch from an 810 which involved numbers he actually counted to 45. Neither the 4 nor 5 represent something from the real world. He did not interview 5 dentists, of whom 4 approved. Yes, the fractions have the same value. But it would be silly to reduce this ratio for no reason.

Maybe his friends were also asking dentists about other brands of breath mints. But his friends were lazier, and each only spoke to five dentists. In that situation reducing his ratio might be useful, so it could be more easily compared to the ratios representing his friends' data.

Reducing ratios can be sensible. The point is merely that it is not always sensible, unlike how most math classes have a rule that fraction answers should be fully reduced.

Similarly I would lie if I changed Snoopy's ratio to 80100 this makes it sound like he did a lot more work by interviewing 100 dentists. (Note that many people, especially in advertising, do change ratios in this manner.)

Tangentially, not all ratios are written as fractions. The ratio 810 can also be written as 8 to 10 or as 8 : 10. But in real life, almost no one but the writers of SAT tests and gamblers write ratios that way.

Some ratios include words to label the numbers.

A Ratio with Word Labels

A child has 18 green candies and 9 yellow candies.

We could write this rate as 18 greens9 yellows.

Time for another definition.

Definition

A rate is a kind of ratio in which the two numbers have different labels.

Once again, we could change a rate into an equivalent fraction but typically do not do so to preserve the record of a real life situation.

Video example problems have three images to click on. You can see a video step-by-step answer, a written step-by-step answer, or only the answer. If you find yet more helpful videos, please let your instructor know so that this website can be updated and improved!

Intro to Ratios

Basic Ratios

Part to Whole Ratios

Ratios as Fractions

Ratio Review

Intro to Rates

Write the ratio "85 to 97" in fraction notation. Simply write one number above the other, the fraction bar represents the word "to"
8597
8597

Write the ratio "0.34 to 124" in fraction notation. Simply write one number above the other, the fraction bar represents the word "to"
0.34124
0.34124

Textbook Exercises for Ratios and Rates

This list of recommended odd-numbered textbook problems is designed for a hypothetical student with a "typical" math background. If your math foundation is weak, do even more odd-numbered problems. If your math foundation is strong, do fewer.

Section 5.1 (page 264) # 1, 3, 5, 11, 13, 15, 17, 21, 23, 29

### Unit Rates§5.2

So far we have imagined ratios and rates as division problems waiting to happen. We lurk nearby, patiently waiting to be told when to divide.

When we stop lurking and finally do the division problem, it changes a rate into a different rate whose second value is 1.

This happens so often there is a special name for it.

Definition

A unit rate is a rate whose second value is 1.

1. A child has 18 green candies and 9 yellow candies. Write this as a unit rate.

The unit rate is 2 greens1 yellow

This is subtle. 18 ÷ 9 = 2. But our answer is not the plain number 2.

Because we keep both labels, and we write the comparison as a fraction, we still have a rate.

1. Kim solved a 300 piece jigsaw puzzle in 90 minutes. Write this as a unit rate.

The unit rate is about 33 pieces1 minute

The unit rate can also be written as 33 piecesminute

The unit rate can also be written as 33 pieces per minute

Our answer should retain the same labels.

We often express unit rates in English by omitting the denominator of 1 and using the word "per" to represent the fraction bar between the unit labels.

If you did not click on the button to see the previous problem's answer, please do that. Notice how we can write a unit rate in different ways.

1. A child has 18 green candies and 9 yellow candies. What is the average number of greens per yellow?

The unit rate is about 2 greens1 yellow

The unit rate can also be written as 2 greensyellow

The unit rate can also be written as 2 greens per yellow

You do not need to write the answer in three different ways. Pick your favorite.

Sometimes the word "average" is used when requesting a unit rate.

1. Kim solved a 300 piece jigsaw puzzle in 90 minutes. What was her average rate of pieces solved per minute?

Kim solved about 33 pieces per minute.

The unit rate can also be written as 33 piecesminute

The unit rate can also be written as 33 pieces per minute

Any rate can be changed into a unit rate by treating the rate as a division problem. Doing "top ÷ bottom" simplifies the rate into a unit rate.

Unit rates make comparisons easy. We might prefer either the biggest or the smallest unit rate.

Shopping Example

I am shopping for fancy hand lotion. Which is a better deal, 10 ounces for \$5 or 24 ounces for \$8?

If we find ounces per dollar we want the most ounces per dollar.

• 10 ounces ÷ \$5 = 2 ounces per dollar
• 24 ounces ÷ \$8 = 3 ounces per dollarbest buy

In the first case, imagine someone with \$1 shopping at a gas station. They want the most gas for their \$1.

If we find dollars per ounce we want the least cost per ounce.

• \$5 ÷ 10 ounces = \$0.50 dollars per ounce
• \$8 ÷ 24 ounces ≈ \$0.33 dollars per ouncebest buy

In the second case, imagine someone with who needs 1 ounce of something shopping at the bulk food section of a grocery store. They need an unusual ingredient for a recipe, and want to spend as little as possible for the correct amount.

Video example problems have three images to click on. You can see a video step-by-step answer, a written step-by-step answer, or only the answer. If you find yet more helpful videos, please let your instructor know so that this website can be updated and improved!

Solving Unit Rate Problem

Solving Unit Price Problem

Mathispower4u

Rates and Unit Rates

A twelve pound shankless ham contains sixteen servings. What is the rate in servings per pound? The word "per" means division. So we do servings ÷ pounds = 16 ÷ 12 ≈ 1.3 servings per pound 1.3 servings per pound

A car will travel 464 miles on 14.5 gallons of gasoline. What is the rate in miles per gallon? The word "per" means division. So we do miles ÷ gallon = 464 ÷ 14.5 = 32 miles per gallon 32 miles per gallon

A sixteen ounce bag of salad greens costs \$2.39. Find the unit price in cents per ounce.EA sixteen ounce bag of salad greens costs \$2.39. Find the unit price in cents per ounce. The word "per" means division. So we do cents ÷ ounce = 239 ÷ 16 ≈ 15 cents per ounce 15 cents per ounce

Textbook Exercises for Ratios and Rates

This list of recommended odd-numbered textbook problems is designed for a hypothetical student with a "typical" math background. If your math foundation is weak, do even more odd-numbered problems. If your math foundation is strong, do fewer.

Section 5.2 (page 272) # 1, 3, 5, 7, 11, 13, 17, 19, 21, 29

### Rates with Variables

The interesting applications of rates involve variables.

Comparisons using the words "per", "to", "for" almost always can be written as rates.

1. How much of a snack bar can you eat for a 100 calorie snack? Write "g grams per 100 calories" as a rate.

Write it like a fraction. g grams100 calories

1. A hiking group has 5 adults and some kids. Write "5 adults to k kids" as a rate.

Write it like a fraction. 5 adultsk kids

1. How much money is needed to buy 4 candy bars? Write "d dollars for 4 candy bars" as a rate.

Write it like a fraction. d dollars4 candy bars

Sometimes the words "in", "on", or "from" are also cues to write a rate.

1. You wonder how many feet your pet mouse can run in 10 seconds. Write "f feet in 10 seconds" as a rate.

Write it like a fraction. f feet10 seconds

1. You wonder how many gallons of gas are required for a road trip. Write "430 miles from g gallons" as a rate.

Write it like a fraction. 430 milesg gallons

Congratulations again! You are at the end of the third subtopic (Ratio Format) for our first big topic (Shapeshifting).

You have now thought more carefully than most people about writing one number on top of another, and waiting before doing division. That might not sound like a significant accomplishment, but it will help you in future math topics.

### Ten Exercises

Try these ten exercises on scratch paper. Work in a study group if you can! Notice where your notes need improvement. After you are very happy with your answers, you can use this form to ask me to check your work. Can you get at least 8 out of 10 correct?

1. Recall that a "proper fraction" has the format "part ÷ whole". Invent a situation that is described using a ratio that is not a proper fraction.

2. A contractor submits a \$2,800 bid for installing a 120 foot long fence. What is the cost per foot?

3. A contractor submits a \$2,800 bid for installing a 120 foot long fence. How many feet of fence are installed per dollar?

4. Karl solved a 300 piece jigsaw puzzle in 45 minutes. Write this as a unit rate.

5. A snack bag has 38 rice crackers and 6 wasabi peas. What is the average number of wasabi peas per rice cracker?

6. Invent a situation that is described using a ratio that includes a variable.

7. A sixteen ounce chocolate mocha coffee with whipped cream contains 520 calories, 18 grams of fat, 60 grams of carbohydrates, and 15 grams of protein. Find the unit rate of calories per ounce.

8. The Leaning Tower of Pisa is 184.5 feet tall and leans 13 feet from its base. What is the ratio of the distance it leans to its height?

9. The area is 0.75 square miles. The population is 35,427 people. What is the unit rate of people per square mile?

10. Brand M soup weighs 54 ounces and costs \$4.79. Brand T soup weighs 59 ounces and costs \$5.99. Which is the better buy?

### Random Exercises

Try these exercises on scratch paper. Work in a study group if you can! Notice where your notes need improvement. Check your work when you are done.

## Percent Format

Activation is strengthened by repetition in context.

- Gregory Mulder

### Decimal Point Scoots§4.3

As we transition from fraction format to percent format, let us review what happens when we multiply or divide by powers of ten.

1. Multiply by powers of ten. Use a calculator if you wish.
• (a) 123.456 × 10
• (b) 123.456 × 100
• (c) 123.456 × 1,000

(a) 1,234.56

(b) 12,345.6

(c) 123,456

The decimal point scoots to the right.

How many scoots just happened compared to the zeroes in the power of ten?

• 123.456 × 10 → one zero, one scoot
• 123.456 × 100 → two zeros, two scoots
• 123.456 × 1,000 → three zeros, three scoots

Now let's look at division.

1. Divide by powers of ten. Use a calculator if you wish.
• (a) 123.456 ÷ 10
• (b) 123.456 ÷ 100
• (c) 123.456 ÷ 1,000

(a) 12.3456

(b) 1.23456

(c) 0.123456

The decimal point scoots to the left.

How many scoots happen compared to the zeroes in the power of ten?

• 123.456 ÷ 10 → one zero, one scoot
• 123.456 ÷ 100 → two zeros, two scoots
• 123.456 ÷ 1,000 → three zeros, three scoots

The next step is to multiply and divide by decimal amounts. These are powers of one-tenth.

1. Multiply by powers of ten. Use a calculator if you wish.
• (a) 123.456 × 0.1
• (b) 123.456 × 0.01
• (c) 123.456 × 0.001

(a) 12.3456

(b) 1.23456

(c) 0.123456

The decimal point scoots to the left.

How many scoots just happened compared to the zeroes in the power of one-tenth?

• 123.456 × 0.1 → one zero, one scoot
• 123.456 × 0.01 → two zeros, two scoots
• 123.456 × 0.001 → three zeros, three scoots

So dividing by powers of ten works just like multiplying by powers of one-tenth.

1. Divide by powers of ten. Use a calculator if you wish.
• (a) 123.456 ÷ 0.1
• (b) 123.456 ÷ 0.01
• (c) 123.456 ÷ 0.001

(a) 1,234.56

(b) 12,345.6

(c) 123,456

The decimal point scoots to the right.

How many scoots just happened compared to the zeroes in the power of one-tenth?

• 123.456 ÷ 0.1 → one zero, one scoot
• 123.456 ÷ 0.01 → two zeros, two scoots
• 123.456 ÷ 0.001 → three zeros, three scoots

So multiplying by powers of ten works just like dividing by powers of one-tenth.

1. Practice multiplication for reliability and speed. Do the following without a calculator.
• (a) 314.67 × 100
• (b) 314.67 × 1,000
• (c) 314.67 × 0.1
• (d) 314.67 × 10
• (e) 314.67 × 0.01
• (f) 314.67 × 0.0001
• (g) 314.67 × 10,000
• (h) 314.67 × 0.001
• (i) 314.67 × 1

(a) 31,467

(b) 314,670

(c) 31.467

(d) 3,146.7

(e) 3.1467

(f) 0.031467

(g) 3,146,700

(h) 0.31467

(i) 314.67

1. Practice division for reliability and speed. Do the following without a calculator.
• (a) 59.028 ÷ 10
• (b) 59.028 ÷ 0.1
• (c) 59.028 ÷ 1,000
• (d) 59.028 ÷ 0.01
• (e) 59.028 ÷ 0.001
• (f) 59.028 ÷ 100
• (g) 59.028 ÷ 100,000
• (h) 59.028 ÷ 0.00001
• (i) 59.028 ÷ 1

(a) 5.9028

(b) 590.28

(c) 0.059028

(d) 5,902.8

(e) 59,028

(f) 0.59028

(g) 0.00059028

(h) 5,902,800

(i) 59.028

Video example problems have three images to click on. You can see a video step-by-step answer, a written step-by-step answer, or only the answer. If you find yet more helpful videos, please let your instructor know so that this website can be updated and improved!

Multiplying and dividing decimals by 10, 100, 1000

The Organic Chemistry Tutor

Ratio and Proportion Word Problems

Textbook Exercises for Decimal Point Scoots

This list of recommended odd-numbered textbook problems is designed for a hypothetical student with a "typical" math background. If your math foundation is weak, do even more odd-numbered problems. If your math foundation is strong, do fewer.

Section 4.3 (page 216) # 23, 25, 27

### Four Replacements for %§6.1

Our society is used to using numerical scales to rate how nice things are.

Here is table to complete. We'll just make up answers, rating things.

Rating Scales

Scale of
1 to 5
Scale of
1 to 10
Scale of
1 to 100
Jogging
Banana Runts
Elevator Music

In English, the popularity of product reviews and Auto Club travel guides means a "star" is assumed to be a rating on a scale of 1 to 5.

There is no nice name in English for how something rates on a scale of 1 to 10.

The name for how something rates on a scale of 1 to 100 used to be called per cent, meaning "per 100" since the word "cent" means "100". But over time those words became customarily squished together, and now we say percent.

This idea of "per 100" or "out of 100" is so important that we should be more formal.

Definition

Percent means "out of 100".

Therefore the symbol % can be replaced with "out of 100".

By the way, we know four ways to do "out of 100" with arithmetic.

• ÷ 100
• two decimal point scoots to the left
• writing the number as a fraction with denominator 100
• × 1100

Let's do each of those four operations from the definition of percent.

1. Use each of the four ways to do "out of 100" with arithmetic.
• (a) Rewrite 47% by replacing the % symbol with ÷ 100.
• (b) Rewrite 47% by replacing the % symbol with two decimal point scoots to the left.
• (c) Rewrite 47% by replacing the % symbol with writing the number as a fraction with denominator 100.
• (d) Rewrite 47% by replacing the % symbol with × 1100.

These examples will look incomplete. They are incomplete. Very soon we will see when each is actually useful. For now we are merely warming up.

If you are a visual learner, it might help to picture percentages using a grid of 100 boxes or a circle with 100 tic marks. These can help us draw pictures for "out of 100". The diagrams are especially helpful for thinking carefully about small percentages.

1. Use the grid to represent percentages.
• (a) 47%
• (b) 4.7%
• (c) 0.47%
1. Use the circle to represent percentages.
• (a) 70%
• (b) 7%
• (c) 0.7%

In real life percents often come as a set of values that add up to 100%, and a missing value must be found using subtraction.

1. How much orange juice is sold?

But that type of problem only involves subtraction, so it is too boring to appear again in this class.

Video example problems have three images to click on. You can see a video step-by-step answer, a written step-by-step answer, or only the answer. If you find yet more helpful videos, please let your instructor know so that this website can be updated and improved!

Write 14.7% as a decimal. Use RIP LOP to remind us to go "left out of percent" two scoots, to get 0.147 0.147

Write 0.38 as a percent. Use RIP LOP to remind us to go "right into percent" two scoots, to get 38% 38%

Write 65% as a fraction. Replace the % symbol with "write the number over 100", to get 65100 = 1320 1320

Write 118 as a percent. First do "top ÷ bottom" to change 118 into 1.375
Then use RIP LOP to remind us to go "right into percent" two scoots, to get 137.5%
137.5%

Textbook Exercises for Four Replacements for %

This list of recommended odd-numbered textbook problems is designed for a hypothetical student with a "typical" math background. If your math foundation is weak, do even more odd-numbered problems. If your math foundation is strong, do fewer.

Section 6.1 (Page 316) # 1, 3, 5

### Percents, Decimals, and Fractions§6.1

Remember our four ways to create "out of 100" using arithmetic?

• ÷ 100
• two decimal point scoots to the left
• writing the number as a fraction with denominator 100
• × 1100

Let's explore which help us change the format of numbers. We can change among decimal format, percent format, and fractions.

#### Percent to Decimal

What if we start in percent format and want to switch to decimal format?

We need to replace the % symbol using arithmetic. Which of the four versions of what percent means is the most helpful in this situation?

1. Write 47.1% as a decimal.

The easiest replacement is "two decimal point scoots to the left" to get 0.471

1. Let's do more of these, to practice for reliability and speed. Do the following without a calculator.
• (a) Write 3% as a decimal.
• (b) Write 30% as a decimal.
• (c) Write 103% as a decimal.
• (d) Write 10.3% as a decimal.

(a) 0.03

(b) 0.3

(c) 1.03

(d) 0.103

#### Decimal to Percent

What if we start in decimal format and want to switch to percent format?

Now we need to create the % symbol by "un-doing" one of those four options. Which of the four versions of what percent means is the most helpful in this situation?

1. Write 0.75 as a percent.

If we stick a % symbol on the number we also need to "undo" it to be fair. After all, we cannot change the number for no reason! The easiest un-doing is "two decimal point scoots to the right" to get 75%.

1. Let's do more of these, to practice for reliability and speed. Do the following without a calculator.
• (a) Write 0.04 as a percent.
• (b) Write 2.3 as a percent.
• (c) Write 0.375 as a percent.
• (d) Write 0.66... as a percent.

(a) 4%

(b) 230%

(c) 37.5%

(d) 66.6...%

#### RIP LOP

Definition

The acronym RIP LOP summarizes moving between decimal format and percent format. It stands for Right Into Percent, Left Out of Percent.

Notice that RIP LOP does not tell us "two places of decimal scoots". But we can remember that we always scoot twice.

Percents are always about 100, and 100 always has two zeroes, and we saw earlier that multiplying or dividing by 100 always causes two scoots.

1. Write 0.76 as a percent.

Two decimal point scoots to the right gives us 76%.

1. Write 3.3% as a decimal.

Two decimal point scoots to the left gives us 0.033.

#### Percent to Fraction

What if we start in percent format and want to switch to fraction format?

We again need to replace the % symbol using arithmetic. Which of the four versions of what percent means is the most helpful in this situation?

1. Write 42% as a fraction.

The easiest replacement is "writing the number as a fraction with denominator 100" to get 42100 which reduces to 2150

Let's do another example.

1. Write 80% as a fraction.

80% = 80100 = 810 = 45

In the previous two example we dealt with a whole number in percent format. What about changing a fraction in percent format into a fraction not in percent format?

A different one the four versions of what percent means is the most helpful in this situation!

1. Write 2 35 % as a fraction.

This time we will replace the % symbol with × 1100

2 35 % = 135 % = 135 × 1100 = 13500

We have not talked about fraction multiplication yet. If this example does not make sense to you, please file it away in the back of your mind until we do discuss fraction multiplication.

#### Fraction to Percent

What if we start in fraction format and want to switch to percent format?

A few fractions are too easy. We can simply look at 51100 and see that it is "51 out of 100" or 51%.

In general it is quickest to use division to change the fraction into decimal format, and then use RIP LOP.

1. Write 14 as a percent.

14 = 1 ÷ 4 = 0.25 = 25%

1. Write 78 as a percent.

78 = 7 ÷ 8 = 0.875 = 87.5%

Notice that our percent format answer may include a repeating decimal.

1. Write 19 as a percent.

19 = 1 ÷ 9 = 0.111... = 11.111...%

Congratulations! You are at the end of the third subtopic (Percent Format) for our first big topic (Shapeshifting).

You have now thought more carefully than most people about the relationship between decimals, percentages, and fractions. You word problems arrive you are prepared with a solid foundation.

Video example problems have three images to click on. You can see a video step-by-step answer, a written step-by-step answer, or only the answer. If you find yet more helpful videos, please let your instructor know so that this website can be updated and improved!

Multiplying a Decimal by a Power of 10

Dividing a Decimal by a Power of 10

Dividing a Decimal by a Power of 10: Pattern

Multiplying Decimals by 10, 100, and 1000 (worksheet)

Dividing Decimals by 10, 100, and 1000 (worksheet)

morgankenneth12

Percents, Fractions, Decimals Review

Mathispower4u

Introduction to Percent

Relating Fractions, Decimals, and Percents - Example 1

Relating Fractions, Decimals, and Percents - Example 2

Textbook Exercises for Percents, Decimals, and Fractions

This list of recommended odd-numbered textbook problems is designed for a hypothetical student with a "typical" math background. If your math foundation is weak, do even more odd-numbered problems. If your math foundation is strong, do fewer.

Section 6.1 (Page 316) # 11, 13, 15, 17, 25, 31, 33, 35, 37, 41, 43, 45, 49, 59, 61, 63, 65

### Ten Exercises

Try these ten exercises on scratch paper. Work in a study group if you can! Notice where your notes need improvement. After you are very happy with your answers, you can use this form to ask me to check your work. Can you get at least 8 out of 10 correct?

1. An easy way to divide a number by 1,000 is to scoot the decimal point ___ places to the _____.

2. An easy way to multiply a number by 0.01 is to scoot the decimal point ___ places to the _____.

3. Write 5.5% as a decimal.

4. Write 0.8911 as a percent.

5.Write 0.8% as a fraction.

6. Write 25 38 % as a fraction.

7. Write the fraction 58 as a percent.

8. Write the fraction 1750 as a percent.

9. After we change two-thirds into percent format, do its digits repeat endlessly?

10. Use what you have memorized to write 33 13 percent as a decimal and a fraction.

### Random Exercises

Try these exercises on scratch paper. Work in a study group if you can! Notice where your notes need improvement. Check your work when you are done.

## Measurement Units

We begin our topic of measurement unit conversions with a definition.

Definition

A measurement's unit is the word that labels the measurement.

Unit = Word Label for Measurements

I am 63 inches tall. The measurement unit is inches.

By convention we use plurals of such words when talking about units.

A large box is 1 yard tall. The measurement unit is yards.

Note that unit rates have two units.

A Rate Comparing Two Units

A speed limit is 30 miles per hour. The measurement units are miles and hours.

There are measurements that use more than two units, but we will not use them in this class. As one example, thinking carefully about a 20-watt light bulb requires considering six units!

Watt?

A watt is the rate of power of moving a weight of one kilogram at a speed of one meter per second when resisted by a deceleration of one meter per second per second.

### Units and Their History(none)

Time to take a short break from mathematics to talk about this history of measurement units. (A thank you to Wikipedia for providing much of the following history lesson.)

Where did we get our measurement units? The older system of units from Europe is now named Standard, American, or Imperial units. The newer system is named SI units, but most people still call it by its older name, the metric system.

#### Length

We'll start by talking about units for measuring length.

Many European countries used lengths named the "inch", "foot", and "yard".

European historical records from the twelfth century show much more care in measuring the inch than the foot or yard. Apparently a town would measure an inch by taking the average value of thumb width of a few men, which produced a result that had little regional variation. Carpentry and other crafts needed a standardized inch more than a foot or yard.

Having twelve inches in one foot was a remnant of the Roman Empire. So a foot was always twelve inches. This was much longer than nearly everyone's feet, but could be estimated by stepping heel-toe heel-toe in boots.

For many centuries a "yard" meant many things in Europe. The common theme was that a yard was about a stride length. In different places a yard was either three feet, average waist circumference, or two cubits (elbow to elbow when touching fingertips). In places that based yards on feet, a yard was always three feet.

The original mile was five thousand Roman "paces". People stopped using these "paces" but kept the mile. The result is that one mile is 5,280 feet. The details involve furlongs, and you can read more here.

The metric system was created in France, beginning in 1791. In 1793 the French Academy of Sciences adopted as the country's official unit of length a meter, defined as one ten-millionth of the distance from the Equator to the North Pole through Paris.

How long was this meter? It is the same meter we use today. The expedition sent to measure this distance was only off by one-fiftieth of a percent. This was remarkable accuracy considering the technology available then! But it does mean the meter we use is not quite the originally intended length, so people felt free to adjust the definition in later years.

In 1889 the first General Conference on Weights and Measures established a new definition. This group built an International Prototype Metre: a bar composed of an alloy of ninety percent platinum and ten percent iridium, marked with two lines whose distance apart was measured at the melting point of ice. The purpose was to keep the same length but have a physical example that could be easily copied by anyone visiting Paris.

Many of the rulers we use have one side with inches and feet, and the other side with centimeters and millimeters. Can you look at such a ruler and see that one inch is about two-and-a-half centimeters? Interest in redefining measurement units for length resurfaced in the late 1950s. Since the meter had a proptype but the inch did not, in 1958 the inch was redefined as exactly 2.54 centimeters.

In 1960 the meter was redefined again, this time as 1,650,763.73 wavelengths of the orange-red emission line in the electromagnetic spectrum of the krypton-86 atom in a vacuum. This definition again aimed to keep the length unchanged while creating a physical example that could be duplicated almost exactly by anyone, anywhere in the world, with common laboratory equipment.

In 1983 the meter was redefined for the last time. Now it is the distance traveled by light in free space in 1299,792,458 of a second. This final definition again kept the length unchanged, and finally created a physical example that could be duplicated exactly by anyone, anywhere in the world, with common laboratory equipment. Much easier and less expensive than visiting Paris to measure a metal bar!

A meter is broken up into 100 centimeters or 1,000 millimeters. (To help remember this think about 100 years in one century and 1,000 years in one millenium.)

#### Mass

The Roman Empire used weights from which we get the words "pounds" and "ounces". After the Roman Empire, an ounce saw various definitions across Europe, usually 450 or 480 grains of barleycorn because that grain is very uniform. The Roman pound was 12 ounces. But soon after the Roman Empire ended, most European merchants switched to a sixteen ounce pound.

For the metric system, a gram was defined as the mass of one cubic centimeter of water. This is a small amount! It is useful in the laboratory, but not as practical for commerce. So 1,000 grams (a kilogram) is used more in normal daily life.

(A question to ponder: Why wasn't one cubic meter of water used as the standard mass? That would be a logical choice, since a meter is the standard length.)

As an aside, scientists use the word mass to denote how much physical stuff an object is made up of. This is different than weight because it is independent of an object's location. The weight of an object changes proportionally to the local gravity.

1. The moon's gravity is one-sixth as strong as the Earth's gravity. If I weigh 162 pounds on Earth, how much do I weigh on the moon?

162 pounds ÷ 6 = 27 pounds

(My mother used to have a t-shirt that said, "I'm not overweight, I'm just on the wrong planet" because she had a thyroid condition that meant she would always be overweight.)

#### Volume

Volume was not used much until the eighteenth century. A standard size bag was much more difficult to measure, let alone construct, than a standard length rod/cord or standard weight of barleycorn grains.

For most of history, volume measurements were not used. When they were used, they were merchants' jargon for the volume taken up by a certain weight of a certain trade good. For example, a gallon was originally the volume of eight pounds of wheat.

In modern times the volumes of the American/Standard/Imperial system have been standardized.

• 8 ounces = 1 cup
• 2 cups = 1 pint
• 2 pints = 1 quart
• 4 quarts = 1 gallon

Bakers and canners might have the advantage of already knowing these conversion rates by heart.

Is there an easy way to remember the parts of a gallon? You betcha.

Here is our gallon.

If we chop it into fourths we get quarts. The name quarts is like "quarters".

Each quart is two pints.

Each pint is two cups. Think about two short one-cup milk cartons stacked on top of each other to become the size of a one-pint carton of whipping cream.

Some students prefer to combine all of those diagrams into a single, more complicated picture.

Pubs in England used to give credit to regular customers. People would only get paid once per month, and in the meanwhile the bartender would mark p's and q's (for pints and quarts) below their name on the wall behind the bar. At the end of the month this record would show what they owed to bar.

The expression "Mind your p's and q's" meant "Don't drink too much before payday!"

Yet nowadays we usually say that phrase to young children!

Be aware that there is a different unit of volume called a "dry quart" which is not quite the same amount as liquid quarts. We will not use in Math 20. But if you are part of the Willamette Valley grass seed business and shop for bushels of grain the dry quart is important! (It is almost the same amount as a liquid quart, so for grocery shopping no one notices the difference.)

Since a gram was defined as the mass of one cubic centimeter of water, that same size cube was used to define volume in the metric system. A milliliter was defined as the volume of one cubic centimeter. Note that some professions use "cubic centimeter" or cc instead of milliliter.

A milliliter is pretty small! So the liter (1,000 milliliters) is the size more appropriate for most practice uses.

(A question to ponder: Why wasn't a cubic meter used as the standard volume? That would be a logical choice, since a meter is the standard length.)

Video example problems have three images to click on. You can see a video step-by-step answer, a written step-by-step answer, or only the answer. If you find yet more helpful videos, please let your instructor know so that this website can be updated and improved!

Math Antics

Intro to the Metric System

Textbook Exercises for Units and Their History

This list of recommended odd-numbered textbook problems is designed for a hypothetical student with a "typical" math background. If your math foundation is weak, do even more odd-numbered problems. If your math foundation is strong, do fewer.

(Our textbook has no exercises for this topic.)

### One Step Conversions§7.1, §7.3

What happens when we convert between measurement units?

These problems involve either multiplying or dividing.

Consider two examples that use the fact that there are 12 inches in 1 foot.

1. 7.5 feet is how many inches?

7.5 feet × 12 = 90 inches

In the above example we split apart each foot into twelve inches. You can imagine breaking, distributing, or shattering a bunch of one-foot rulers.

We will say a foot is multiple inches, in the same way that 15 is multiple 3s. In both cases the bigger thing is several copies of the smaller thing.

Feet are multiple inches. Yards are multiple feet or inches. Miles are multiple yards, or feet.

Pints are multiple ounces. Quarts are multuiple pints or ounces. Gallons are multiple quarts, pints, or ounces.

Centimeters are multiple millimeters. Meters are multiple centimeters or millimeters. Kilometers are multiple meters, centimeters or millimeters.

1. 30 inches is how many feet?

30 inches ÷ 12 = 2.5 feet

In the above example we grouped together a couple sets of twelve inches to each make one foot. You can imagine fusing, bundling, or gluing inches into one-foot rulers.

We will say an inch is a factor of a foot, in the same way that 3 is a factor of 15. In both cases several copies of the smaller thing make the larger thing.

Yards are a factor of miles. Feet are a factor of yards or miles. Inches are a factor of feet, yards, or miles.

Quarts are a factor of gallons. Pints are a factor of quarts or gallons. Ounces are a factor of pints, quarts or gallons

Meters are a factor of kilometers. Centimeters are a factor of meters or kilometers. Millimeters are a factor of centimeters, meters or kilometers.

So we can see the following guidelines:

• When converting from a multiple measurement unit (for example feet → inches) we multiply.
• When converting from a factor measurement unit (for example inches → feet) we divide.

Either direction happens in one step.

Definition

A one-step unit conversion can happen when we switch a measurement's unit and we know the rate that compares the old and new units.

1. Your turn, to practice one-step unit conversions for reliability and speed.
• (a) How many inches are in 5.75 feet?
• (b) How many feet are in 198 inches?
• (c) How many cups are in 12 gallons?
• (d) How many gallons are in 40 cups?
• (e) How many ounces are in 152 pounds?
• (f) How many pounds are in 152 ounces?

(a) We are going to inches, which are smaller than feet, so we multiply to get 69 inches

(b) We are going to feet, which are bigger than inches, so we divide to get 16.5 feet

(c) We are going to cups, which are smaller than gallons, so we multiply to get 192 cups

(d) We are going to gallons, which are bigger than cups, so we divide to get 2.5 gallons

(a) We are going to ounces, which are smaller than pounds, so we multiply to get 2,432 ounces

(b) We are going to pounds, which are bigger than ounces, so we divide to get 9.5 pounds

Video example problems have three images to click on. You can see a video step-by-step answer, a written step-by-step answer, or only the answer. If you find yet more helpful videos, please let your instructor know so that this website can be updated and improved!

How many yards is 10.5 feet? There are 3 feet in 1 yard. We are bundling groups of feet into fewer yards, so we use ÷3.
10.5 ÷ 3 = 3.5 yards
3.5 yards

Textbook Exercises for One Step Conversions

This list of recommended odd-numbered textbook problems is designed for a hypothetical student with a "typical" math background. If your math foundation is weak, do even more odd-numbered problems. If your math foundation is strong, do fewer.

Section 7.1 (Page 371) # 1, 5, 7, 9, 11, 31

Section 7.3 (Page 388) # 1, 5, 11, 31, 39, 41

We have already seem some examples of metric prefixes:

• a kilogram is 1 gram × 1,000
• a milliliter is 1 liter ÷ 1,000
• a millimeter is 1 meter ÷ 1,000
• a centimeter is 1 meter ÷ 100

These examples are part of a general pattern.

Here are the metric prefixes used in these math lectures:

• kilo × 1,000
• (unit)
• centi ÷ 100
• milli ÷ 1,000
• micro ÷ 1,000,000

In this list (unit) is a placeholder for the plain unit, whether meters, grams, or liters.

These are prefixes are part of a longer list.

 kilo hecto deca (unit) deci centi milli • • micro × 1,000 × 100 × 10 × 1 ÷ 10 ÷ 100 ÷ 1,000 • • ÷ 1,000,000

Strangely, there are no prefixes for the two spots between milli and micro (for ten-thousandths and hundred-thousandths of a unit).

Even though we do not use hecto-, deca-, or deci- it is still useful to be familiar with this longer list. Why? Because of a shortcut!

But for the shortcut we do not write the whole list. Just the initials are sufficient.

SI Prefix Initials

k      h      d      unit      d      c      m      •      •      micro

Many math students memorize a cute saying (try making your own!) and remember the acronym for the first seven spots. For example, "King Henry does usually drink chocolate milk" or "Killer hobos dance under dazzling crystal mobiles."

Writing the acronym at the top of your homework and your test scratch paper is a great idea when doing SI-to-SI unit conversions. There is a shortcut!

SI One-Step Conversion Shortcut

To convert between metric prefixes, first write the list of all SI prefix initials.

Then count decimal point scoots: for every "place" you move right or left along the list of initials, move the decimal point the same way.

1. How many millimeters is 2.8 centimeters?
1. A calculator is 16 centimeters long. Change this to meters and millimeters.
1. A backpack weighs 6,480 grams. Change this to kilograms.
1. A shampoo bottle contains 450 cubic centimeters. Change this to milliliters and liters.

Video example problems have three images to click on. You can see a video step-by-step answer, a written step-by-step answer, or only the answer. If you find yet more helpful videos, please let your instructor know so that this website can be updated and improved!

Mathispower4u

Metric Unit Conversion

How many meters is 6 kilometers? On the list of SI prefixes, move 3 scoots right from "kilo" to "plain units", to get 6,000 m 6,000 m

How many centimeters is 8.7 millimeters? On the list of SI prefixes, move 1 scoot left from "milli" to "centi", to get 0.87 cm 0.87 cm

Something is 0.5 cm wide. Express this width in meters and millimeters. For meters, on the list of SI prefixes, move 2 scoots left from "centi" to "plain units", to get 0.005 m
For millimeters, on the list of SI prefixes, move 1 scoot right from "centi" to "milli", to get 5 mm
0.005 m and 5 mm

Marvin L. Bittinger is 1.8542 meters tall. Express this height in centimeters and millimeters. For centimeters, on the list of SI prefixes, move 2 scoots right from "plain units" to "centi", to get 185.42 cm
For millimeters, on the list of SI prefixes, move 3 scoots right from "plain units" to "milli", to get 1,854.2 mm
185.42 cm and 1,854.2 mm

How many liters is 3,080 milliliters? On the list of SI prefixes, move 3 scoots left from "milli" to "plain units", to get 3.080 L 3.080 L

How many milliliters is 0.25 liters? On the list of SI prefixes, move 3 scoots right from "plain units" to "milli", to get 250 mL 250 mL

How many grams is 3.8 kilograms? On the list of SI prefixes, move 3 scoots right from "kilo" to "plain units", to get 3,800 g 3,800 g

How many grams is 2,200 milligrams? On the list of SI prefixes, move 3 scoots left from "milli" to "plain units", to get 2.200 g 2.200 g

How many micrograms is 0.37 milligrams? On the list of SI prefixes, move 3 scoots right from "milli" to "micro", to get 370 mcg 370 mcg

Textbook Exercises for More about SI Prefixes

This list of recommended odd-numbered textbook problems is designed for a hypothetical student with a "typical" math background. If your math foundation is weak, do even more odd-numbered problems. If your math foundation is strong, do fewer.

Section 7.1 (Page 371) # 15, 19, 21, 23, 27, 39

Section 7.3 (Page 388) # 15, 17, 21, 25, 29

### Exponents and Square Roots§1.7, §2.3, §4.6, §7.2

#### Exponents

Consider this very simple square.

We know the area of a rectangle (which includes a square) is length × width. So the area of that square is 1 foot × 1 foot = 1 square foot.

The answer is labeled with square feet because it is counting how many squares it takes to cover the area.

If we changed each side from feet to inches, how should we write the area?

Now the area is 12 inches × 12 inches = 144 square inches.

The answer is labeled with square inches because it is counting how many squares it takes to cover the area, using smaller squares this time.

Sometimes square inches are written abbreviated as sq. in. or as in2.

Similar abbreviations are used with other square units. Square feet can be sq. ft. or ft2. Square centimeters can be sq. cm. or cm2.

Later in the class, when we deal with shapes that are not rectangles, it will help to think of answers written in "square units" not as multiplication problem answers but as counting squares.

Usually we think of an exponent as its own number. It floats above its "base" and tells you how many times to multiply.

4 3 = 4 × 4 × 4 = 64

But for measurement units it can sometimes help to think of an exponent of 2 as the second half of a two-part number. It is saying "instead of a line of the base length, we are drawing a big square and then counting little squares".

(4 inches)2 = 4 inches × 4 inches = 16 square inches

1. What is the difference between these two amounts?

22 ft              2 ft2

The first measures a distance, a length covered by 4 rulers each one foot long.

The second measures an area, a flat space covered by 2 squares each one square foot.

#### Square Roots

We actually just talked about square roots! You might not have noticed because we did not make that clear. Let's add clarity with a definition.

Definition

The square root of a number is the amount that, when multiplied by itself, gets to that number.

For example, the square root of 25 is 5 because 5 × 5 = 25.

We usually write square roots with a symbol.

But do not forget the definition just because you are looking at a symbol!

The picture we used earlier of a square foot broken up into inches...

...shows us that the square root of 144 is 12.

1. Use a calculator to find the square root of the following numbers: 4, 6, 8, and 16.

Why are only some of those answers whole numbers?

Definition

When we multiply a whole number by itself, we call the answer a perfect square.

Perfect squares have nice square roots. When 144 or less, they are easy to find with mental math if you have memorized the multiplication table up to 12s.

1. Draw the two perfect squares with area 4 and 16 from the previous problem.

Video example problems have three images to click on. You can see a video step-by-step answer, a written step-by-step answer, or only the answer. If you find yet more helpful videos, please let your instructor know so that this website can be updated and improved!

BrainSTEM

Metric Conversion Trick!! Part 1

Metric Conversion Trick!! Part 2

Write as multiplication, then multiply: 32 = 32 = 3 × 3 = 9 9

Twelve square feet is how many square inches? One square foot is 12 inches × 12 inches = 144 square inches.
So twelve square feet is twelve of that: 1,728 sq. in.
1,728 sq. in.

Three square centimeters is how many square meters? One square meter is 100 cm × 100 cm = 10,000 square cm.
For three square cm we do 3 ÷ 10,000 = 0.0003 sq. m
0.0003 sq. m

What is the square root of 225? A square with sides 15 would have area 15 × 15 = 225.
15

What is the square root of 87? (Round to the thousandths place.) Let our calculator do the work, to find the answer of 9.327 9.327

Textbook Exercises for Exponents and Square Roots

This list of recommended odd-numbered textbook problems is designed for a hypothetical student with a "typical" math background. If your math foundation is weak, do even more odd-numbered problems. If your math foundation is strong, do fewer.

Section 1.7 (Page 84) # 11, 13, 15, 19, 21

Section 2.3 (Page 119) # 81, 83

Section 4.6 (Page 247) # 1, 3, 5, 7, 29, 31

Section 7.2 (Page 381) # 1, 3, 11, 15, 29

### Ten Exercises

Try these ten exercises on scratch paper. Work in a study group if you can! Notice where your notes need improvement. After you are very happy with your answers, you can use this form to ask me to check your work. Can you get at least 8 out of 10 correct?

1. 1 foot is how many yards?

2. 7.1 miles is how many feet?

3. 5,280 yards is how many miles?

4. An index card is 0.27 mm thick. What is that thickness in centimeters?

5. An index card is 0.27 mm thick. What is that thickness in meters?

6. A football field is 4,844 cm wide. What is that width in kilometers?

7. 80 ounces is how many pounds?

8. 5,200 grams is how many kilograms?

9. One gallon is how many fluid ounces?

10. 806 liters is how many milliliters?

### Random Exercises

Try these exercises on scratch paper. Work in a study group if you can! Notice where your notes need improvement. Check your work when you are done.

### Interlude: Why is three-fifths one number?

Mathematicians have some funny beliefs.

We believe that 4 is a number. We believe that 4,000 is a number. We believe that 1.5 is a number. We believe that 0.6 is a number. We believe that 35 is a number. We believe that 60% is a number.

So far so good, right?

But we do not call 2+15 a number. That is called an expression. We say it must be simplified into 35 before we consider it a number.

This is funny!

Yeah, it is true that we can "do something" to 2+15.

But we can also "do something" to 35. We can change it into 0.6 or 60%.

So why don't mathematicians call 35 an expression? What makes it one number instead of two numbers?

The answer is that in real life rates are funny creatures and we don't trust them.

If I told you that drove my car 3 miles down main street, or that it took me 5 minutes, you have a pretty realistic idea about how accurate those numbers are. Surely it was not exactly 3 miles or 5 minutes. But the information can be talked about in a meaningful way.

If I told you that drove my car down main street at a rate of 3 miles per 5 minutes, you have zero useful information. I probably was not driving that speed the entire time, because of other vehicles and traffic lights. Perhaps I was never actually going that speed, but was always slower or faster, and am reporting an average. You do not know how long my trip was. Was it 5 minutes or an hour? An average speed for only 5 minutes is not anything anyone cares about. An average speed for an hour might be something the city planners do care about as they adjust traffic light timing during rush hour. Also, did I really measure either of the 3 miles or the 5 minutes, or did I simply read 36 miles per hour on my spedometer and reduce 3660 for some reason? If I did reduce, why did I do that?

In other words, rates have issues. Their social media status is always "it's complicated". They can be very useful. Sometimes they are needed to understand the world. But they tend to make false claims, or to eagerly support unreliable conclusions.

Most mathematicians do not spend enough time doing carpentry or in the kitchen baking. They usually see 35 as a rate, perhaps as 3 miles5 minutes. So they call it one number instead of two numbers because is a single thing (a certain rate) and they are used to thinking especially carefully about that kind of thing.

On the other hand, no one gives you 2+15 as a rate. In real life something like that is always the last step in solving a bigger math problem. So that is called an expression instead of a single number.

If you want to play around with rates (division) being epecially unreliable, try the error propagation playground.