Algebra Foundations for Normal People by Gabriel Beguerie - HTML preview

Chapter 13

Worked example (no skipped steps)

13.1 The sum of a number and 9 is 20. Find the number.

Let the number be 𝑥. Then:

𝑥 + 9 = 20

Subtract 9:

𝑥 = 11

Answer: 11.

Worked example (no skipped steps)

13.2 Three times a number is 27. Find the number.

Let the number be 𝑥. Then:

3 𝑥 = 27

Divide by 3:

𝑥 = 9

Answer: 9.

Worked example (no skipped steps)

13.3 Twice a number plus 4 is 18. Find the number.

Let the number be 𝑥. Then:

2 𝑥 + 4 = 18

Subtract 4:

2 𝑥 = 14

Divide by 2:

𝑥 = 7

Answer: 7.

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Worked example (no skipped steps)

13.4 A number decreased by 7 gives 12. Find the number.

Let the number be 𝑥. Then:

𝑥 − 7 = 12

Add 7:

𝑥 = 19

Answer: 19.

Worked example (no skipped steps)

13.5 A number divided by 4 equals 7. Find the number.

Let the number be 𝑥. Then:

𝑥 = 7

4

Multiply by 4:

𝑥 = 28

Answer: 28.

Worked example (no skipped steps)

13.6 Five more than a number is 17. Find the number.

Let the number be 𝑥. Then:

𝑥 + 5 = 17

Subtract 5:

𝑥 = 12

Answer: 12.

Worked example (no skipped steps)

13.7 A taxi charges a base fee of $4 plus $3 per mile. If the total bill was $25, how many miles were traveled?

Let the miles be 𝑚. Then:

4 + 3 𝑚 = 25

Subtract 4:

3 𝑚 = 21

Divide by 3:

𝑚 = 7

Answer: 7 miles.

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Step-by-step solutions

Worked example (no skipped steps)

13.8 A phone plan costs $30 plus $5 per extra GB. If the total bill is $55, how many extra GB were used?

Let the extra GB be 𝑔. Then:

30 + 5 𝑔 = 55

Subtract 30:

5 𝑔 = 25

Divide by 5:

𝑔 = 5

Answer: 5 GB.

Worked example (no skipped steps)

13.9 A movie ticket costs $12 and popcorn costs $5. If one popcorn and some tickets cost $41

total, how many tickets were bought?

Let the number of tickets be 𝑡. Then:

12 𝑡 + 5 = 41

Subtract 5:

12 𝑡 = 36

Divide by 12:

𝑡 = 3

Answer: 3 tickets.

Worked example (no skipped steps)

13.10 A gym charges $20 to join and $15 per month. If the total paid is $110, how many months is that?

Let the months be 𝑚. Then:

20 + 15 𝑚 = 110

Subtract 20:

15 𝑚 = 90

Divide by 15:

𝑚 = 6

Answer: 6 months.

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Worked example (no skipped steps)

13.11 A rectangle has length 8 and perimeter 26. What is the width?

Let the width be 𝑤. Use

𝑃 = 2 𝑙 + 2 𝑤

Substitute:

26 = 2(8) + 2 𝑤

Simplify:

26 = 16 + 2 𝑤

Subtract 16:

10 = 2 𝑤

Divide by 2:

𝑤 = 5

Answer: the width is 5.

Worked example (no skipped steps)

13.12 A rectangle has width 5 and perimeter 34. What is the length?

Let the length be 𝑙. Then:

34 = 2 𝑙 + 2(5)

Simplify:

34 = 2 𝑙 + 10

Subtract 10:

24 = 2 𝑙

Divide by 2:

𝑙 = 12

Answer: the length is 12.

Worked example (no skipped steps)

13.13 The perimeter of a square is 44. What is the side length?

Let the side length be 𝑠. Then:

4 𝑠 = 44

Divide by 4:

𝑠 = 11

Answer: the side length is 11.

192

Step-by-step solutions

Worked example (no skipped steps)

13.14 A student scored 68, 74, and 82 on three tests. What score is needed on the fourth test to average 75?

Let the fourth score be 𝑥. Then:

68 + 74 + 82 + 𝑥 = 75

4

Add the known scores:

224 + 𝑥

So:

224 + 𝑥 = 75

4

Multiply by 4:

224 + 𝑥 = 300

Subtract 224:

𝑥 = 76

Answer: 76.

Worked example (no skipped steps)

13.15 Two quiz scores are 70 and 82. What score is needed on the third quiz to average 80?

Let the third score be 𝑥. Then:

70 + 82 + 𝑥 = 80

3

Add:

152 + 𝑥 = 80

3

Multiply by 3:

152 + 𝑥 = 240

Subtract 152:

𝑥 = 88

Answer: 88.

Worked example (no skipped steps)

13.16 Three numbers have average 12. Their sum is what?

Average means

sum = 12

3

Multiply by 3:

sum = 36

Answer: 36.

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Worked example (no skipped steps)

13.17 Two consecutive integers have sum 31. Find them.

Let the first integer be 𝑛. Then the next is 𝑛 + 1.

So:

𝑛 + ( 𝑛 + 1) = 31

Combine:

2 𝑛 + 1 = 31

Subtract 1:

2 𝑛 = 30

Divide by 2:

𝑛 = 15

So the integers are 15 and 16.

Worked example (no skipped steps)

13.18 Two consecutive even integers have sum 42. Find them.

Let the first even integer be 𝑛. Then the next even integer is 𝑛 + 2.

So:

𝑛 + ( 𝑛 + 2) = 42

Combine:

2 𝑛 + 2 = 42

Subtract 2:

2 𝑛 = 40

Divide by 2:

𝑛 = 20

So the integers are 20 and 22.

Worked example (no skipped steps)

13.19 A rope 36 meters long is cut into two pieces. One piece is 8 meters longer than the other. Find both lengths.

Let the shorter piece be 𝑥. Then the longer piece is 𝑥 + 8.

So:

𝑥 + ( 𝑥 + 8) = 36

Combine:

2 𝑥 + 8 = 36

Subtract 8:

2 𝑥 = 28

Divide by 2:

𝑥 = 14

So the pieces are 14 and 22.

194

Step-by-step solutions

Worked example (no skipped steps)

13.20 One number is 6 more than another. Their sum is 30. Find both numbers.

Let the smaller number be 𝑥. Then the larger number is 𝑥 + 6.

So:

𝑥 + ( 𝑥 + 6) = 30

Combine:

2 𝑥 + 6 = 30

Subtract 6:

2 𝑥 = 24

Divide by 2:

𝑥 = 12

So the numbers are 12 and 18.

Worked example (no skipped steps)

13.21 A shirt is discounted 20% from $45. What is the sale price?

A 20% discount leaves 80%:

0 . 80 · 45 = 36

Answer: $36.

Worked example (no skipped steps)

13.22 A price of $80 increases by 15%. What is the new price?

Multiply by:

1 + 0 . 15 = 1 . 15

Then:

1 . 15 · 80 = 92

Answer: $92.

Worked example (no skipped steps)

13.23 A car travels at 60 miles per hour for 2 . 5 hours. How far does it go?

Use

𝑑 = 𝑟𝑡

Substitute:

𝑑 = 60(2 . 5) = 150

Answer: 150 miles.

Chapter 14

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Worked example (no skipped steps)

13.24 A water tank has 12 liters at first and then gains 4 liters per minute. After how many minutes will it hold 40 liters?

Let the minutes be 𝑚. Then:

12 + 4 𝑚 = 40

Subtract 12:

4 𝑚 = 28

Divide by 4:

𝑚 = 7

Answer: 7 minutes.