Lesson 8.1.3

Lesson 8.1.3Lesson 8.1.3

Percent Increasesand Decreases

Percent Increasesand Decreases

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Lesson

1.1.1

California Standards:Number Sense 1.3Convert fractions to decimals and percents and use these representations in estimations, computations, and applications.

Number Sense 1.6Calculate the percentage of increases and decreases of a quantity.

What it means for you:You’ll see how to use percents to show how much a quantity has gone up or down by.

Percent Increases and DecreasesPercent Increases and DecreasesLesson

8.1.3

Key words:• percent• increase• decrease• compare

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Lesson

1.1.1

When a number goes up or down, you can use percents to describe how much it has changed by.


8.1.3

This can come in useful in real-life situations like comparing price rises or working out sale discounts.

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Lesson

1.1.1

You Can Increase a Number by a Given Percent


8.1.3

You can increase a number by a certain percent of itself.

So, say if you want to increase a number by 10%, you have to work out what 10% is, then add this to the original number.

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Percent Increases and DecreasesPercent Increases and Decreases

Example 1

Lesson

8.1.3

Increase 50 by 20%.

Solution

50 + 10 = 60

Solution follows…

First work out 20% of 50:

This is the amount you need to increase 50 by:

So, 50 increased by 20% is 60.

20% of 50 = × 50 = 1010020

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Example 2

Lesson

8.1.3

A photograph with a length of 14 cm is enlarged. This increases its length by 8%. What is the final length of the enlarged photograph?

Solution

14 cm + 1.12 cm = 15.12 cm

Solution follows…

First work out 8% of 14 cm:

This is the amount you need to increase 14 cm by:

The length of the enlarged photograph is 15.12 cm.

8% of 14 cm = × 14 cm = 0.08 × 14 cm = 1.12 cm 100

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5. Sarah goes out for lunch. Her bill comes to $15. She wants to leave an extra 17% as a tip for the server. How much should Sarah leave in total?

Lesson

1.1.1

Guided Practice


8.1.3

In Exercises 1–4, find the total after the increase.

110

Solution follows…

1. 100 is increased by 10% 2. 20 is increased by 5%21

334.95 48.2

$17.55

3. 165 is increased by 103% 4. 40 is increased by 20.5%

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Lesson

1.1.1

You Can Describe an Increase as a Percent


8.1.3

When a number goes up, you can give the increase as a percent of the original number.

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Example 3

Lesson

8.1.3

A loaf of bread has 24 slices. As a special buy, a larger loaf is sold, which contains 27 slices. What is the percent increase in the number of slices?

Solution27 – 24 = 3

Solution follows…

First find the increase in the number of slices:

x × 24 = 3100

x% of 24 is 3

The number of slices has increased by 12.5%.

Call x the percent increase and write an equation:

x × 24 = 300

x = 12.5

Multiply both sides by 100

Divide both sides by 24

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6. Reynaldo has 140 marbles. He buys 63 more. By what percent has he increased the size of his marble collection?

Lesson

1.1.1

Guided Practice


8.1.3

45%

Solution follows…

20%

7. A company increases its number of staff from 1665 to 1998. What is this as a percent increase?

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Lesson

1.1.1

You Can Decrease a Number by a Given Percent Too


8.1.3

You can also decrease a number by a percent of itself.

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Example 4

Lesson

8.1.3

Decrease 80 by 15%.

Solution

80 – 12 = 68

Solution follows…

First work out 15% of 80:

This is the amount you decrease 80:

So, 80 decreased by 15% is 68.

15% of 80 = × 80 = 0.15 × 80 = 1210015

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8. 100 is decreased by 15%



11. 10 is decreased by 3.9%

12. Tandi has saved $152. She spends 25% of her savings on a shirt. How much does Tandi have left?

Lesson

1.1.1

Guided Practice


8.1.3

In Exercises 8–11, find the total after the decrease.

100 × 0.15 = 15, 100 – 15 = 85

Solution follows…

40 × 0.35 = 14, 40 – 14 = 26

37 × 0.08 = 2.96, 37 – 2.96 = 34.04

10 × 0.039 = 0.39, 10 – 0.39 = 9.61

152 × 0.25 = 38152 – 38 =$114

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Lesson

1.1.1

You Can Describe a Decrease as a Percent


8.1.3

When a number goes down, you can use a percent to describe how much it has changed by.

The decrease is described as a percent of the original number.

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Example 5

Lesson

8.1.3

A river is 12.8 feet deep on January 1. By September 1, the depth has fallen to 9.6 feet. Find the percent decrease in the river depth.

Solution

Solution follows…

First find the amount that the depth is decreased by:

12.8 feet – 9.6 feet = 3.2 feet

x% of 12.8 feet is 3.2 feet

The river depth has decreased by 25%.

Call x the percent decrease and write an equation.

x × 12.8 feet = 320 feet Multiply both sides by 100

Divide both sides by 12.8 feetx = 25

× 12.8 feet = 3.2 feet100

x

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13. 90 is reduced to 81

14. 4 is reduced to 3.5

15. Jon is selling buttons for a fund-raiser. He starts with 280 buttons and sells all but 21. What percent of his stock has Jon sold?

Lesson

1.1.1

Guided Practice


8.1.3

Find the percent decreases in Exercises 13–14.

10%

Solution follows…

12.5%

92.5%

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Lesson

1.1.1

Use Percents to Compare Changes


8.1.3

You can use percent increases and decreases to compare how much two numbers have changed relative to each other.

For example:

Snowman 1 and Snowman 2 have both lost the same amount in height as they’ve melted — 1 ft.

But the percent decrease is greater for Snowman 1 — 1 ft is a bigger change relative to 6 ft than to 7 ft.

Snowman 2:7 foot 6 foot14% decrease

Snowman 1:6 foot 5 foot17% decrease

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Example 6

Lesson

8.1.3

In a store, a bagel is 40¢ and a loaf of bread is $1.60. The store raises the price of both items by 5¢. Which has the larger percent increase in cost?

Solution

Solution follows…

The price of both items is increased by 5¢.So the percent increase in the cost of a bagel is:

The bagel shows the larger percent increase in cost.

And the percent increase in the cost of a loaf is:

You need to have the original value and the increase in the same units. $1.60 has been converted to 160¢ here.

(5¢ × 100) ÷ 40¢ = 12.5, so a 12.5% increase.

(5¢ × 100) ÷ 160¢ = 3.125, so a 3.125% increase.

× 40¢ = 5¢ 100

x

× 160¢ = 5¢ 100

x

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16. Cindy has 250 baseball cards. Jim has 200 baseball cards. Both buy 50 extra cards. Whose collection increased by the larger percent?

Lesson

1.1.1

Guided Practice


8.1.3

Solution follows…

Cindy: (50 × 100) ÷ 250 = 20% increaseJim: (50 × 100) ÷ 200 = 25% increaseSo, Jim’s collection increased by the larger percent.

17. Ava and Ian have a contest to see whose sunflower will increase in height by the greatest percent. Ava’s starts 10 cm high and grows to 100 cm. Ian’s starts 20 cm high and grows to 110 cm. Who won?

Ava: (90 × 100) ÷ 10 = 900% increaseIan: (110 × 100) ÷ 20 = 550% increaseSo, Ava won.

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In Exercises 1–6, find the amount after the percent change.

1. Increase 200 by 25%

2. Decrease 200 by 75%

3. Increase 49 by 7%

4. Decrease 82 by 56%

5. Increase 50 by 142.6%

6. Decrease 80 by 33.2%


Independent Practice

Solution follows…

Lesson

8.1.3

250

50

52.43

36.08

121.3

53.44

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8. Kiona’s brother Otis is 115.5 cm tall. The last time he was measured, his height was 110 cm. Find the percent increase in his height.



Solution follows…

Lesson

8.1.3

7. At store A, apples used to cost $1.50 a pound. Then the price rose by 6%. What is the new cost of a pound of apples? $1.59

5%

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Solution follows…

Lesson

8.1.3

9. Last year, School C had 120 6th grade students. This year they have 5% fewer 6th graders. How many fewer students is this?

10. Mr. Hill’s house rental costs $900 a month. He moves to a house with a rental of $828 a month. Find the percent decrease in his rental.

11. Duena collects comic books. 10 years ago, Comic A was worth $70 and Comic B was worth $40. Now Comic A is worth $84 and Comic B is worth $49. Which has shown the greater percent increase in value?

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8%

Comic B

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Lesson

8.1.3

Round UpRound Up


Percent increases and decreases tell you how big a change in a number is when you compare it to the original amount.

It’s useful to be able to work them out in real-life situations, especially when you’re thinking about tips and discounts — and you’ll learn more about them in Section 8.2.

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Lesson 8.1.3