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# Math Lecture NotesArithmetic Review Problems

The first few days of our Math 20 class are for reviewing foundational topics about arithmetic and fractions.

The homework problems below hope to provide an interesting, thought-provoking, and useful review of arithmetic. (The fraction review problems are here.)

As always, this is merely a "best of their kind" collection of problems. If your math background is strong you might feel there are too many. If your math background is rusty or weak you might need to do additional problems from the textbook to get enough review and practice.

Try to work these problems without using a calculator. Calculators will not be allowed on the midterm that covers these review topics.

Use the floating buttons in the bottom right corner of the screen to show or hide the answers. Before you look at the answers you should have tried to work the problems "forwards". Do not examine the answers first and then try to work "backwards". For these foundational review topics it is better to wait and ask questions during class.

When you have done enough review, try the arithmetic homework.

The arithmetic and fraction review topics are explained in the first four chapters of our textbook. However, our review days do not have lecture or homework straight from the textbook.

The first four textbook chapters are a valuable "unofficial" resource. Its chapters contain another way to explain each topic. Its example problems can help when practicing on homework problems.

But the textbook is not required until we start chapter five. This policy allows students who recently used the textbook for Math 10 to see the material in a different way, without repeating the same textbook homework problems. It also postpones any potential issues from a student having the wrong textbook or the bookstore running out of textbooks.

Most of the problems below have the corresponding textbook section(s) provided in this font: §1.5.

## Understanding Division

The more deeply we understand division, the more deeply we can understand fraction and ratios.

The online notes for this topic are here.

1. Solve each equation by multiplying or dividing, if possible. §1.5

• (a) 28 × 1 = 28
• (b) 28 ÷ 1 = 28
• (c) 72 × 0 = 0
• (d) 72 ÷ 0 = undefined
• (e) 100 × 100 = 10,000
• (f) 100 ÷ 100 = 1

2. In the previous problem, three of the equations can be rewritten using a fraction on the left-hand side. Do that. §1.5

3. Draw a picture to show dealing 52 cards into four equal piles. (How many cards are in each pile?) Be able to explain how your picture represents division.

5. Draw another picture to show dealing 52 cards into piles of four cards. (How many piles do you draw?) Be able to explain how your picture represents division.

## Factors

Practicing our mental math helps us think about factors quickly.

The online notes for this topic are here.

1. Find all twelve factors of 60. §2.1 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

2. Xavier and Odette are playing the Factor Game. The board is shown below. It is Xavier's turn. (Xavier is X's. Odette is O's.)

• (a) What is his best move? (Hint: Odette does not get to claim any spots in reply.) If he picks 15 then Odette cannot reply and he comes out ahead by 15 points.
• (b) What will Odette's best move be on her turn? (Hint: Xavier can reply, but she still comes out ahead by 14 points.) She picks 28 and Xavier replies with 14.
• (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? The next turn Xavier would be able to pick 28 and she would be unable to reply.

## Place Value

Place value names are needed to discuss rounding. They are also used when changing decimals into fractions.

The online notes for this topic are here.

1. State the place value of each digit in the number 9876.54321. §1.1, §4.1 from left to right: hundreds, tens, ones, tenths, hundredths, thousandths, ten-thousandths, hundred-thousandths

2. Place the following numbers in ascending order (smallest to largest). §1.6 2,608 → 2,680 → 2,860 → 20,680

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

3. Place the following numbers in ascending order (smallest to largest). §4.1 0.57429 → 0.574905 → 0.57491 → 0.57949

0.57429       0.57491       0.574905       0.57949

4. Do the following nine multiplication problems without a calculator to review the shortcut for multiplying by powers of ten. §4.3

• (a) 314.67 × 100 = 31,467
• (b) 314.67 × 1,000 = 314,670
• (c) 314.67 × 0.1 = 31.467
• (d) 314.67 × 10 = 3146.7
• (e) 314.67 × 0.01 = 3.1467
• (f) 314.67 × 0.0001 = 0.031467
• (g) 314.67 × 10,000 = 3,146,700
• (h) 314.67 × 0.001 = 0.31467
• (i) 314.67 × 1 = 314.67

5. Do the following nine division problems without a calculator to review the shortcut for dividing by powers of ten. §4.4

• (a) 59.028 ÷ 10 = 5.9028
• (b) 59.028 ÷ 0.1 = 590.28
• (c) 59.028 ÷ 1,000 = 0.059028
• (d) 59.028 ÷ 0.01 = 5,902.8
• (e) 59.028 ÷ 0.001= 59,028
• (f) 59.028 ÷ 100 = 0.59028
• (g) 59.028 ÷ 100,000 = 0.00059028
• (h) 59.028 ÷ 0.00001 = 5,902,800
• (i) 59.028 ÷ 1 = 59.028

## Rounding

The rules for rounding are used when writing the answers to most word problems.

The online notes for this topic are here.

1. Round each number four times: to the nearest ones, tens, hundreds, and thousands. §1.6 4,573.1 → ones 4,573; tens 4,570; hundreds 4,600; thousands 5,000
624.95 → ones 625; tens 620; hundreds 600; thousands 1,000
17,348.9 → ones 17,349; tens 17,350; hundreds 17,300; thousands 17,000

4,573.1       624.95       17,348.9

2. Round each number four times: to the nearest tenths, hundredths, thousandths, and ten-thousandths. §4.1 3.85264 → tenths 3.9; hundredths 3.85; thousandths 3.853; ten-thousandths 3.8526
0.07249 → tenths 0.1; hundredths 0.07; thousandths 0.072; ten-thousandths 0.0725
25.79053 → tenths 25.8; hundredths 25.79; thousandths 25.791; ten-thousandths 25.7905
0.6666666... → tenths 0.7; hundredths 0.67; thousandths 0.667; ten-thousandths 0.6667

3.85264       0.07249       25.79013       0.6666666...

3. Find the rounding error below, where someone tried to round 478.3469 to different place values. §1.6, §4.1 The hundreds digit is less than five, so when rounding to the nearest thousand we get 0 (not 1,000).

4. Round 1.27272727... to the nearest tenth, hundredth, and thousandth. §4.1 tenths 1.3; hundredths 1.27; thousandths 1.273

## Estimating

Estimating is helpful for getting a rough sense of the size of a reasonable answer, which can help noticing a calculator mistake.

The online notes for this topic are here.

1. Estimate the sum or difference. Round all numbers (you pick how much to round) before mentally adding or subtracting. §1.6, §4.6 Answers will be different if the amount rounded is different.

• (a) 57 + 34 ≈ 60 + 30 = 90
• (b) 353 − 92 ≈ 350 − 90 = 260
• (c) 119 + 75 ≈ 120 + 80 = 200
• (d) 462 − 183 ≈ 460 − 180 = 280
• (e) 6,035 + 107 ≈ 6,000 + 100 = 6,100
• (f) 8.15 − 0.006 ≈ 8.15 − 0.01 = 8.14
• (g) 3,542.55 + 6,276.1 ≈ 3,500 + 6,300 = 9,800
• (h) 1.675 − 0.9999 ≈ 1.675 − 1 = 0.675

2. Estimate the product or quotient. Round all numbers (you pick how much to round) before mentally multiplying or dividing. §1.6, §4.6

• (a) 18 × 27 ≈ 20 × 30 = 600
• (b) 932 ÷ 6 ≈ 930 ÷ 10 = 93
• (c) 53 × 5 ≈ 50 × 5 = 250
• (d) 2,084 ÷ 175 ≈ 2,000 ÷ 200 = 10
• (e) 868.52 × 7.5 ≈ 870 × 10 = 8,700
• (f) 461 ÷ 19.2 ≈ 460 ÷ 20 = 23
• (g) 0.005 × 0.014 ≈ 0.01 × 0.01 = 0.0001
• (h) 245 ÷ 241 ≈ 200 ÷ 200 = 1

3. 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. §4.2, §4.6 estimate \$3.025 − \$0.284 ≈ \$3.0 − \$0.3 = \$2.70, and actual \$3.025 − \$0.284 = \$2.741

4. 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. §4.2, §4.6 estimate 2.48 × 4 ≈ 2.5 × 4 = 10 feet, and actual 2.48 × 4 = 9.92 feet

5. Circle the correct response. If each number in a list is rounded up, the total estimate

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

6. Circle the correct response. If each number in a list is rounded down, the total estimate

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

7. Circle the correct response. If some of the numbers in a list are rounded up and others are rounded down, the total estimate

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

## The Perils of PEMDAS

The acronym PEMDAS can help with order of operaton problems. Make sure you understand it correctly.

The online notes for this topic are here.

1. Find the value of each expression. §1.9, §4.4

• (a) 24 ÷ 4 × 3 = 6 × 3 = 18
(all multiplication and division, so work left to right)
• (b) 100 − 30 + 20 − 10 = 70 + 20 − 10 = 90 − 10 = 80
(all addition and subtraction, so work left to right)
• (c) 50 + 30 ÷ 10 × 2 − 6 = 50 + 3 × 2 − 6 = 50 + 6 − 6 = 50
(start by doing the central multiplication and division portion working left to right)

2. Rewrite each of the three expressions to replace any fraction bar with one or two pairs of parenthesis. (Do not solve for an answer. Later in the class all expressions of this complexity will be calculator work.) §3.7

(5 × 7 − 12) ÷ (9 + 3)
128 + (1 + 6) ÷ 3.5
(0.4 × 12) ÷ (11 ÷ 2) + 2.7 ÷ (8 × 3)

## Terms

Breaking up a long expression into terms is the most powerful way to deal with order of operations.

The online notes for this topic are here.

1. Find the number of terms in each expression.

• (a) 5 × 4 + 3 × 6 ÷ 12 = (5 × 4) + (3 × 6 ÷ 12) = two terms
• (b) 2 × (7 + 1) − 42 + 5 = [2 × (7 + 1)] − (42) + 5 = three terms
• (c) 24 ÷ 3 + 8 × 5 − 22 = (24 ÷ 3) + (8 × 5) − (22) = three terms

2. Use terms to find the value of each of those three expressions. §1.9, §4.4

• (a) 5 × 4 + 3 × 6 ÷ 12 = 5 × 4 + 3 × 6 ÷ 12 = 20 + 18 ÷ 12 = 20 + 1.5 = 21.5
• (b) 2 × (7 + 1) − 42 + 5 = [2 × 8] − (16) + 5 = 16 − 16 + 5 = 5
• (c) 24 ÷ 3 + 8 × 5 − 22 + 9 ÷ 3 = (8) + (40) − (4) + (3) = 47

3. Use terms to find the value of each of these long expressions. (Hint: focus first on the part inside square brackets as its own problem.) §1.9, §4.4

• (a) 2 × [5 + 2 × (9 − 4)] ÷ 3 + 6 × (12 − 7) ÷ 3 = the bracketed portion is 5 + 2 × 5 = 5 + 10 = 15
the entire expression is then (2 × 15 ÷ 3) + (6 × 5 ÷ 3) = (30 ÷ 3) + (30 ÷ 3) = 10 + 10 = 20
• (b) [(11 + 33) − 5] ÷ 11 + 6 − 4 × [(1 + 2) − 2] + 15 = the first bracketed portion is (11 + 27) − 5 = 38 − 5 = 33
the second bracketed portion is 3 − 2 = 1
the entire expression is then (33 ÷ 11) + 6 − (4 × 1) + 15 = 3 + 6 − 4 + 15 = 9 − 4 + 15 = 5 + 15 = 20
• (c) 5 × 32 + 2 × [20 ÷ (11 − 7)] − 45 + 40 ÷ 22 = the bracketed portion is 20 ÷ 4 = 5
the entire expression is then (5 × 9) + (2 × 5) − 45 + (40 ÷ 4) = 45 + 10 − 45 + 10 = 20
• (d) 3 × [4 + 3 × (10 − 8)] + 60 ÷ 10 − 10 ÷ (0.75 − 0.25) + 4 = the bracketed portion is 4 + (3 × 2) = 4 + 6 = 10
the entire expression is then (3 × 10) + (60 ÷ 10) − (10 ÷ 0.5) + 4 = 30 + 6 − 20 + 4 = 20

## Divisibility Rules

Knowing all the simple divisibility rules allows us to find important factors quickly.

The online notes for this topic are here.

1. Is 5,025 divisible by... 2?       3?       4?       5?       6?       9?      10? (note that the sum of digits is 12, which is a multiple of 3 but not a multiple of 9)
2 no, 3 yes, 4 no (25 is not a multiple of 4), 5 yes, 6 no (it is not true that both 2 and 3 work), 9 no, 10 no

2. Is 73,080 divisible by... 2?       3?       4?       5?       6?       9?      10? (note that the sum of digits is 18, which is a multiple of both 3 and 9)
2 yes, 3 yes, 4 yes (80 is a multiple of 4), 5 yes, 6 yes (both 2 and 3 work), 9 yes, 10 yes

## Finding Factors

Notice that finding all the factors of a number and finding the prime factorization of that number are a very different tasks, using very different techniques.

The online notes for this topic are here.

1. Which has more factors, 36 or 40? §2.1 find all the factors of 36 → 1 × 36, 2 × 18, 3 × 12, 4 × 9, 6 × itself → nine factors
find all the factors of 40 → 1 × 40, 2 × 20, 4 × 10, 5 × 8 → eight factors
so 36 has more factors

2. Find the prime factorization of each number. §2.1

• (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

## One-Step Equations

Isolate the variable by doing the same thing to both sides of the equation.

For most of Math 20 the only equations to solve by isolating a variable are made by cross-multiplying. One side of the equation has the format "y times a number" so we divide both sides by that number to isolate y.

For these problems you may use a calculator.

The online notes for this topic are here.

1. Solve the following equations. Show your work using the vertical format. §1.7

• (a) 5 + y = 55 subtract 5 from both sides to find y = 50
• (b) y + 12 = 34 subtract 12 from both sides to find y = 22
• (c) 2.5 + y = 8.6 subtract 2.5 from both sides to find y = 6.1
• (d) y − 0.01 = 1.2 add 0.01 to both sides to find y = 1.21

2. Solve the following equations. Show your work using the vertical format. §1.7

• (a) 6 × y = 180 divide both sides by 6 to find y = 30
• (b) y ÷ 16 = 128 multiply both sides by 16 to find y = 2,048
• (c) y × 0.1 = 100 divide both sides by 0.1 (recall the shortcut is one decimal point scoot to the right) to find y = 1,000
• (d) y ÷ 4.575 = 2 multiply both sides by 4.575 to find y = 9.15

3. Twelve times a number equals 27015. What is the number? §1.7 our equation is 12 × y = 270 ÷ 15
divide both sides by 12 to find y = (270 ÷ 15) ÷ 12 = 18 ÷ 12 = 1.5