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Math 20
 Shapeshifting: Changing a Number 

What can we do with a single number?

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

We can write ths number we get from the data as the fraction 4991,000.

This fraction is very close to one-half. In fact, we would expect that if many more rolls happened, the nearness to one-half would increase. The kid 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 from fraction format to percent format by writing it as 50%.

We could change the number 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" to give 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.

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.

So we begin this class by exploring more deeply how and when we "shapeshift" a number by rounding or by changing formats (fraction, decimal, percent, measurement units).

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Rounding

How to Round Decimals

Definition

Rounding a decimal is reducing the accuracy of a decimal by removing the digits of tiny place value, and perhaps incrementing by 1 the smallest remaining place value digit.

Consider rounding to the hundredths place. Any digits of place value to the left of hundredths stay the same. (So the whole number digits and tenths digit are unchanged.) Any place value digits to the right of the hundredths digit are removed. The hundredths place digit might go up by 1. This happens if the digit to its right, the biggest of those removed, was 5 or more.

Example 8

Round 1.95283 to the nearest... tenth?       hundredth?       thousandth?       ten-thousandth?

Example 9

Round 5.749507 to the nearest... tenth?       hundredth?       thousandth?       ten-thousandth?

(Notice the unexpected thing when rounding to the nearest thousandth. Still write three decimal digits to "fill up" to the thousandths place.)

videos

Bittinger Chapter Tests, 11th Edition

Chapter 1 Test, Problem 20: Round 34,528 to the nearest thousand.

Chapter 1 Test, Problem 21: Round 34,528 to the nearest ten.

Chapter 1 Test, Problem 22: Round 34,528 to the nearest hundred.

Chapter 4 Test, Problem 14: Round 5.6783 to the nearest one.

Chapter 4 Test, Problem 15: Round 5.6783 to the nearest hundredth.

Chapter 4 Test, Problem 16: Round 5.6783 to the nearest thousandth.

Chapter 4 Test, Problem 17: Round 5.6783 to the nearest tenth.

How Much to Round Decimals

When we do math (especially with a calculator) we often get answers that have long decimals. For example, the problem 40 ÷ 7 = 5.7142857... has a long decimal merely because we did division.

With how many decimal places should we write our answers?

This question is actually more about philosophy than mathematics. Let us consider a few cases. In all these cases we will show that we changed the answer by using the about equal sign, which is written like .

We follow three guidelines:

Example 10
Criteria One: Usefulness of the Answer

40 ÷ 7 = 5.7142857...

Perhaps I have a 40 inch board, and I need to cut it into seven equal lengths.

When doing carpentry with a tape measure I can only measure to tenths or hundredths of an inch. It does not make sense in this situation to use more decimal places.

40 ÷ 7 ≈ 5.71

With the carpentry example, our tools limited our accuracy. This happens often in real life, but not often in Math 20 word problems.

Example 11
Criteria Two: Indivisible Units

40 ÷ 7 = 5.7142857...

Perhaps I have a six guests coming to my kid's birthday party, so I need to put 40 pieces of candy into 7 bags of party favors.

I do not want to cut pieces of candy into pieces. To avoid having kids argue I should just drop the decimal entirely.

40 ÷ 7 ≈ 5

In the bags of candy example, our answers needed to be whole numbers. This is very rare in Math 20 word problems.

Example 12
Criteria Three: Communicating Measurement Accuracy

40 ÷ 7 = 5.7142857...

Imagine that a pharmacist has about 40.0 mg of a medicine and divides it into 7 doses.

This measured value was not notably precise: 40.0, not 40.0268 or 40.00035. We should not add more than one decimal place. We do not want to pretend we have more accuracy than we do.

40.0 ÷ 7 ≈ 5.7 or 40.0 ÷ 7 ≈ 5.71

With the pharmacist example, our tools were not what limited our accuracy. A chemist's scale can measure much more accurately than one-tenth of a milligram. Instead, our initial accuracy was what limited us. We cannot gain accuracy just by doing division!

In Math 20 word problems this is the most common guideline for rounding.

Your turn to do an example.

Example 13

With a yardstick I measure that a cardboard box is 15/16 of a yard tall. Write this fraction as a decimal and round appropriately.

Do Not Round Decimals in the Middle of a Problem

Normally we do not round in the middle of a problem. By waiting until the end, we avoid introducing error. Let's consider a few examples.

Example 14

In Eugene it rains, on average, about 1.15 inches per week.

1.15 inches × 52 weeks = 59.8 inches annually

If we rounded, the difference is noticeable!

1 inch × 52 weeks = 52 inches annually

That difference of 7.8 inches would matter a lot if you were a farmer, or planning the storm drain system.

Example 15

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.

(3.25 + 5.25) × $1000.20 = $8,501.70 ≈ $8,502

If we rounded first, the difference is significant!

(3 + 5) × $1000 = $8,000

Example 16

You are a realtor who earns a 0.06 commission. You sell a house for $210,000.

0.06 × $210,000 = $12,600

If we rounded only the commission:

0.1 × $210,000 = $21,000

If we rounded only the house value:

0.06 × $200,000 = $12,000

Your turn. Work with a partner to invent a word problem. Solve it accurately, doing any rounding at the end. Then round first, and see how much the answer changes.

Estimating

Definition

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

As we have seen, this can change the answer a lot. But it can be a very helpful thing to do as a quick "check" after we finish a problem. Is our answer close to the estimated amount? If not, we probably made a careless calculator mistake and can go back to find what went wrong.

Example 17

Estimate the answer to 47,900 ÷ 12 before using a calculator to do the problem.

videos

Chapter 1 Test, Problem 23: Estimate the sum 23,649 + 54,746 by first rounding to the nearest hundred.

Chapter 1 Test, Problem 24: Estimate the difference 54,751 − 23,649 by first rounding to the nearest hundred.

Chapter 1 Test, Problem 25: Estimate the product 824 × 489 by first rounding to the nearest hundred.

Chapter 4 Test, Problem 39: Estimate the product 8.91 × 22.457 by first rounding to the nearest one.

Chapter 4 Test, Problem 40: Estimate the quotient 78.2209 ÷ 16.09 by first rounding to the nearest ten.

Fraction Format

Understanding Division

It is time to talk conceptually about division.

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.

If I want to model 6 ÷ 3 = 2 then I act out having six cards, and dealing them out to three people until I am done.

How does it work? What question do I ask to find the answer? (An answer we know is 2.)

Example 1

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.

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

Example 2

Deal out cards to model 6 ÷ 1/2 = 12.

We get stuck! How do we deal cards to half a pile? What is "half a pile" anyway?

Making Piles of a Fixed Size

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

If I want to model 6 ÷ 3 = 2 then I act out having six cards, and dealing them out them out into piles of size 3 until I am done.

How does it work? What question do I ask to find the answer? (An answer we know is 2.)

Example 3

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.

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

Example 4

Model 6 ÷ 1/2 = 12 by making piles of a fixed size.

6 becomes the number of cards total.

1/2 becomes the number of cards per pile. (We must rip cards in half!)

12 becomes the number of piles.

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!

Your turn to work out two more examples of fraction division.

Example 5

Model 4 ÷ 1/2 = ? by making piles of a fixed size.

Example 6

Model 1/2 ÷ 1/4 = ? by making piles of a fixed size.

videos

Kahn Academy

The Idea of Division

Factors

Definition

Factors are the numbers you multiply to get another number.

Example 7

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

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

factors example

Note that 1 and the number itself are always factors!

The traditional way to memorize common factors is to use flash cards. But it is more fun to play the Factor Game! Here is a copy of the game board to use with an in-class demonstration.

factor game

videos

Kahn Academy

Finding Factors

Divisibility Rules

Some divisibility rules you already know:

Other divisibility rules may be new to you:

Let's apply the divisibility rules to 13,512.

Example 1

Is 13,512 divisible by... 2?       3?       4?       5?       6?       9?      10?

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.

Next let's apply the divisibility rules to 2,016.

Example 2

Is 2,016 divisible by... 2?       3?       4?       5?       6?       9?      10?

videos

Kahn Academy

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

Bittinger Chapter Tests, 11th Edition

Chapter 2 Test, Problem 6: Determine whether 1,784 is divisible by 8.

Chapter 2 Test, Problem 7: Determine whether 784 is divisible by 9.

Chapter 2 Test, Problem 8: Determine whether 5,552 is divisible by 5.

Chapter 2 Test, Problem 9: Determine whether 2,322 is divisible by 6.

The rule for six works because 2 × 3 = 6.

There are yet more divisibility rules, but these are the ones that are easy to use.

Finding Factors

There are two tasks involving factor finding. In some situations we want to find all the factors. In other situations we want to find the prime factorization.

(In this class the most common use for finding all the factors is considering how to reduce a fraction, and the most common use for finding the prime factorization is to create a common denominator for fractions.)

The first task is Finding All the Factors. We find all the factors of a number by making a two-column list. Count up in the first column. List any matching factors in the second column. When the columns get to the same value we can stop.

Example 3

Find all the factors of 66.

Our first column counts up from 1 to 10.

166
233
322
4not a factor
5not a factor
611
7not a factor
8not a factor
9not a factor
10not a factor

The next value for the first column would be 11, which is already listed in the second column. So we can stop.

Example 4

Find all the factors of 24.

videos

Kahn Academy

Finding Factors of a Number

Bittinger Chapter Tests, 11th Edition

Chapter 2 Test, Problem 1: Find all the factors of 300.

The second task is Finding the Prime Factorization. We do this by making a factor tree and noting the "leaves".

Remember factor trees?

Here is one factor tree for 48.

rounding example

Example 5

Try writing a different factor tree for 48 that does not start with 6 and 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).

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

Example 6

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

Example 7

Find the prime factorization of 66.

videos

Chapter 2 Test, Problem 4: Find the prime factorization of 18.

Chapter 2 Test, Problem 5: Find the prime factorization of 60.

Desired Denominators

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

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.

Example 1

Make one cut so the cake has six pieces. How many are shaded?

cake cutting example

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

Example 2

Make two cuts so the cake has nine pieces. How many are shaded?

cake cutting example

We can also 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.

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

Example 3

"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.

Reducing

We could also work backwards. We could make big slices of cake by gluing smaller slices together.

Example 4

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

cake cutting example

Example 5

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

cake cutting example

Maybe we use very sticky icing?

Definition

Reducing a fraction represents with numbers the act of gluing cake slices together. The numerator and denominator are divded by the same value.

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

Example 6

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!)

Ratios and Rates

How much are fractions like division?

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

Example 1

1/5 = 1 ÷ 5 = 0.2

Fractions typically belong in their most reduced form. In this sense they are not like division. When we divide one number by another we do not reduce the numbers first. We talk about "equivalent fractions" but never "equivalent division problems".

Example 2

10/5 = 10 ÷ 5 = 2 ÷ 1 = 2

Sometimes it helps to deal with a pair of numbers as if they were the numerator and denominator in a fraction. This can be true even when the pair of numbers comes from a real situation that gives each number an individual meaning (neither is inherently "numerator" or "denominator"). That is the job for a ratio.

Definition

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

Example 3

If I interviewed 10 dentists and 8 of them approved of a certain breath mint, I could write the ratio 8/10.

Often it does not make sense to reduce a ratio. If I reduced the above ratio I would get 4/5. As a fraction (or as a decimal) this has the same value. But it no longer represents something from the real world. I did not interview 5 dentists, of whom 4 approved!

Similarly I would lie if I changed the ratio to 80/100 this makes it sound like I did a lot more work by interviewing 100 dentists. (Note that many people, especially in advertising, do change ratios in this manner.)

One big difference between normal fractions and ratios is that normal fractions always have the numerator count special parts, and the denominator count total parts.

Example 4

3 special parts/8 total parts.

The pie is cut into 8 pieces and I get 3 of them.

In contrast, a ratio might be a comparison of two partial amounts. The whole can be missing!

Example 5

4 left-handed people/16 right-handed people.

We could look at this ratio and find the whole (20 total people). But the rate does not explicitly tell us the whole.

We can even write a ratio upside down without losing any meaning or becoming confusing.

Example 6

16 right-handed people/4 left-handed people.

Normal fractions, in which the denominator is always the total number of parts, cannot be flipped like that without their meaning changing.

Not all ratios are written as fractions. The ratio 8/10 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.

videos

Kahn Academy

Intro to Ratios

Ratios as Fractions

Bittinger Chapter Tests, 11th Edition

Chapter 5 Test, Problem 1: Write the ratio "85 to 97" in fraction notation. Do not simplify.

Chapter 5 Test, Problem 2: Write the ratio "0.34 to 124" in fraction notation. Do not simplify.

Rates

Did you notice that some ratios include words to label the numbers?

Definition

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

Example 7

If I interviewed 10 dentists and 8 of them approved of a certain breath mint, I could write the rate 8 approvals/10 dentists.

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.

videos

Kahn Academy

Intro to Rates

Unit Rates

Definition

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

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.

More formally, we are reducing the original rate by dividing the top number and the bottom number by the bottom number. This changes the bottom to 1, which we then ignore. So overall we only see "top ÷ bottom".

Example 8

My favorite candy costs $14 for a two-pound box. What is the price per pound?

videos

Kahn Academy

Solving Unit Rate Problem

Solving Unit Price Problem

Bittinger Chapter Tests, 11th Edition

Chapter 5 Test, Problem 6: A twelve pound shankless ham contains sixteen servings. What is the rate in servings per pound?

Chapter 5 Test, Problem 7: A car will travel 464 miles on 14.5 gallons of gasoline in highway driving. What is the rate in miles per gallon?

Chapter 5 Test, Problem 8: A sixteen ounce bag of salad greens costs $2.39. Find the unit price in cents per ounce.

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

Example 9

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.

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

Percent Format

Percent as a Rating Scale

Instead of diving into percents suddenly, let's approach the topic slowly. First we'll think about how 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.

Example 1

As a group, let's rate the following on a scale of:

Desert1 to 51 to 101 to 100
Carrot Cake   
Banana Runts   
Dark Chocolate   

In general there is no nice name in English for how something rates on a scale of 1 to 5, or a scale of 1 to 10. (There are some exceptions. The popularity of Auto Club travel guides means most people know a "4 star hotel" is rated on a scale of 1 to 5.)

There is a nice name in English for how something rates on a scale of 1 to 100. This 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.

Since this is a math class, we should be more formal.

Definition

Percent means "out of 100". We know four ways to do "out of 100" with arithmetic:

We could also use a grid of 100 boxes, or a circle with 100 tic marks, to draw pictures for "out of 100".

circle with 100 tic marks square with 100 boxes

Often percents come as a set of values that add up to 100%, and a missing value must be found.

Example 2

How much orange juice is sold?

percent-juices.png

But this almost never happens in our textbook problems.

Decimal Format

Decimal Point Scoots

When we multiply or divide by powers of ten it only scoots the decimal point.

Example 8

Multiply 123.456 by 10

Multiply 123.456 by 100

Multiply 123.456 by 1,000

Example 9

Divide 123.456 by 10

Divide 123.456 by 100

Divide 123.456 by 1,000

When we multiply or divide by powers of one-tenth it also scoots the decimal point, but in the other direction.

Example 10

Multiply 123.456 by 0.1

Multiply 123.456 by 0.01

Multiply 123.456 by 0.001

Example 11

Divide 123.456 by 0.1

Divide 123.456 by 0.01

Divide 123.456 by 0.001

videos

Kahn Academy

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)

Fractions and Decimals

Fractions to Decimals

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

Example 18

1/5 = 1 ÷ 5 = ?

Example 19

3/8 = 3 ÷ 8 = ?

There is a shortcut for changing a fraction to a decimal if the denominator is a factor of 10, 100, etc.

If we "un-reduce" to make an equivalent fraction whose denominator is 10, 100, etc. then we can simply read the new fraction.

Example 20

1/25 = 1/25 × 4/4 =

Normal Decimals to Fractions

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

Example 21

Write 0.08 as a fraction.

Remember that there are four ways to write reducing eight hundredths into two twenty-fifths. Three are appropriate. One is wrong.

Example 22

Write 0.08 as a fraction.

(a) show reducing by writing two little, floating ÷ 4, one each for numerator and denominator

(b) show reducing by writing a single, centered ÷ 4 intended to work on both numerator and denominator

(c) show reducing with a fraction: ÷ 4/4

(d) show reducing by crossing out the eight and twenty-five and replacing them with the reduced values

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

Example 23

Write 0.375 as a fraction.

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 24

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 = 1/2.

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

factor game

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.

An .svg image with both the instructions and the outline is here.

Repeating Decimals to Fractions

Definition

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

Example 25

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.

Example 26

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 for Math 20 students. This tricky method 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.

Uhg!

It makes more sense when you see it happen.

Example 27

Change 0.222222... into a fraction.

Percents, Decimals, and Fractions

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

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?

Example 2

Write 47.1% as a decimal.

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

Let's do more examples of changing from percent format to decimal format.

Example 3

Write 3% as a decimal.

Write 30% as a decimal.

Write 103% as a decimal.

Write 10.3% as a decimal.

videos

Chapter 6 Test, Problem 1: Write 14.7% as a decimal.

Decimal to Percent

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

Example 4

Write 0.75 as a percent.

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?

Let's do more examples of changing from decimal format to percent format.

Example 5

Write 0.04 as a percent.

Write 2.3 as a percent.

Write 0.375 as a percent.

Write 0.66... as a percent.

videos

Chapter 6 Test, Problem 2: Write 0.38 as a percent.

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. (Because percents are always about 100, and 100 always has two zeroes.)

Example 6

Write 0.76 as a percent.

Write 3.3% as a decimal.

Percent to Fraction

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

Example 7

Write 42% as a fraction.

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?

Let's do another example of changing from percent format to decimal format.

Example 8

Write 80% as a fraction.

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

Example 9

Write 2 3/5 % as a fraction.

videos

Chapter 6 Test, Problem 4: Write 65% as a fraction.

Fraction to Percent

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

Example 10

Write 1/4 as a percent.

We again 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?

The previous example worked so nicely because the denominator was a factor of 100. In general, a fraction may not be so friendly. Switching from fraction format to decimal format is usually a helpful intermediate step.

Example 11

Write 1/8 as a percent.

(Since 8 is not a factor of 100, using an intermediate step of decimal format works nicely.)

videos

Chapter 6 Test, Problem 3: Write 11/8 as a percent.

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

Example 12

Write 1/9 as a percent.

Let's look at the homework for §6.1 (pages 342-344) and §6.2 (pages 349-352) for any scary problems. (The problems involving mixed numbers often seem worse than they really are.)

Example 13

(Students pick one or more problems from §6.1 and §6.2.)

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.

Example 1

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

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

Example 2

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

Note that some measurements have two units.

Example 3

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 in Math 20 we will not encounter anything this complicated. As one example, thinking carefully about a 20-watt light bulb requires considering six units!

Example 4

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.

History

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 as always three feet.

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 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 1/299,792,458 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.

Example 5

The moon's gravity is one-sixth as strong as the Earth's gravity. If I weigh 152 pounds on Earth, how much do I weigh on the moon?

(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.

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

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.)

One Step Conversions

Lots of math problems from earlier in the class used measurement units, but only as labels to keep track of and append to the final answer. Our new task is to "convert" between measurement units.

Sometimes this is easy and can be done with the tools we already have learned. We will examine that situation first.

Definition

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

This problem is a one-step unit conversion.

Example 6

How many inches is 7 feet?

We know the rate "12 inches to 1 foot" so this is a one-step conversion. (In a moment we'll actually do this problem.)

Here is a nice memory aid. When converting units from larger to smaller units (i.e., miles to feet, or gallons to quarts) then multiply. When converting units from smaller to larger, divide.

Unfortunately, life is not always so simple. This problem is not a one-step unit conversion.

Example 7

How many inches is 7 miles?

Most people do have memorized the rate of inches to miles, so this conversion will require more than one step.

We can use a proportion to do a one-step unit conversion. As a class you can figure out how to use those old tools (making and solving a proportion) for a new trick (a one-step unit conversion).

Example 6 (again)

How many inches is 7 feet?

But we cannot use a proportion when the conversion requires more than one step. Why not? Let's talk about this while looking at an example.

Example 7 (again)

How many inches is 7 miles?

Is it clear why the old math we know is not enough? We need a new technique.

In real life many unit conversions require more than one step. However, almost all the homework problems in the textbook are only one-step unit conversions. If Math 20 is your last math class, you can get by with using proportions for unit conversion problems. The second midterm and final exam will only have a few problems requiring more than one step. But to be better prepared for real world problems and math classes beyond Math 20, we need a new technique.

Conversion Rates

Turn to page 468, the beginning of §8.1. Notice there is a purple box that has all of the section's conversion rates.

Almost every section in Chapter 8 begins this way. The Chapter provides a new purple box of unit conversion rates to use in the Unit Analysis technique. The process of Unit Analysis does not change. Just look up the rates you need in the proper purple box. (Well, aside from §8.3 which inexplicably on page 481 has a blue table instead of a purple box.)

You will eventually be asked to memorize some of these unit conversion rates. But for now only worry about memorize the parts of a gallon, which is described next.

Parts of a Gallon

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

Here is our gallon.

parts of a gallon

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

parts of a gallon

Each quart is two pints.

parts of a gallon

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.

parts of a gallon

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

parts of a gallon

Why do people say "Mind your p's and q's"?

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. So "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!)

More About SI Prefixes

We have already seem some examples of metric prefixes:

These examples are part of a general pattern.

Here are the metric prefixes used in these math lectures:

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

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 memorize this longer list. Why? Because of a shortcut!

We can use Unit Analysis to convert between metric prefixes. We would use one unit conversion rate to remove the prefix and a second unit conversion rate to put on the desired one.

Example 12

How many millimeters is 2.8 centimeters? (Use Unit Analysis.)

But it is quicker to use a shortcut.

Shortcut

To convert between metric prefixes, first write the list of all SI prefixes. Then count decimal point scoots: for every "place" you move right or left along the list of prefixes, move the decimal point the same way.

Example 13

How many millimeters is 2.8 centimeters? (Use the shortcut.)

Notice that we do not actually need to write the whole list. Just the initials are sufficient.

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. Then you can use the shortcut instead of Unit Analysis. But remember that the shortcut is only for doing a metric-to-metric unit conversion.

Let's do more examples.

Example 14 (§8.2 # 39)

A calculator is 18 centimeters long. Change this to meters and millimeters.

Example 15 (§8.4 # 33)

A backpack weighs 8,492 grams. Change this to kilograms.

Example 16 (§8.5 # 35)

A shampoo bottle contains 355 cubic centimeters. Change this to milliliters and liters.

videos

Chapter 8 Test, Problem 3: How many meters is 6 kilometers?

Chapter 8 Test, Problem 4: How many centimeters is 8.7 millimeters?

Chapter 8 Test, Problem 7: Something is 0.5 cm wide. Express this width in meters and millimeters.

Chapter 8 Test, Problem 8: Marvin L. Bittinger is 1.8542 meters tall. Express this height in centimeters and millimeters.

Chapter 8 Test, Problem 9: How many liters is 3,080 milliliters?

Chapter 8 Test, Problem 10: How many milliliters is 0.25 liters?

Chapter 8 Test, Problem 13: How many grams is 3.8 kilograms?

Chapter 8 Test, Problem 15: How many grams is 2,200 milligrams?

Chapter 8 Test, Problem 21: How many micrograms is 0.37 milligrams?