1Estimate answers to percent problems involving 1% and 10%.

2Estimate answers to percent problems involving 50%, 25%, 5%, and 15%.

3Estimate answers to percent problems involving 200%.

4Use estimation to solve percent application problems.

Estimation with Percent

Estimation can be used to find approximations when exact answers aren’t necessary. For example, when dining at a restaurant, it’s helpful to be able to estimate the amount of the tip.When shopping, the ability to estimate a discount or the sale price of an item also comes in handy. In this section, we will discuss some estimation methods that can be used to make quick calculations involving percents.

1 Estimate answers to percent problems involving 1% and 10%.

There is an easy way to find 1% of a number that does not require any calculations. First, recall that 1% 1001 0.01.Thus, to find 1% of a number, we multiply it by 0.01, and a quick way to multiply the number by 0.01 is to move its decimal point two places to the left.

Self Check 1

What is 1% of 519.3? Find the exact answer and an estimate using front-end rounding.

Now Try Problem 11

Finding 1% of a Number

To find 1% of a number, move the decimal point in the number two places to the left.

What is 1% of 423.1? Find the exact answer and an estimate using front-end rounding.

Strategy To find the exact answer, we will move the decimal point in 423.1 two places to the left. To find an estimate, we will move the decimal point in an approximation of 423.1 two places to the left.

WHY We move the decimal point two places to the left because 1% of a number means 0.01 of (times) the number.

Solution

Exact answer:

1% of 423.1 4.231 Move the decimal point in 423.1 two places to the left.

Estimate: Recall from Chapter 1 that with front-end rounding, a number is rounded to its largest place value so that all but its first digit is zero.To estimate 1% of 423.1, we can front-end round 423.1 to 400 and find 1% of 400. If we move the understood decimal point in 400 two places to the left, we get 4. Thus,

1% of 423.1 4 Because 1% of 400 4.

Success Tip To quickly find 2% of a number, find 1% of the number by moving the decimal point two places to the left, and then double (multiply by 2) the result. In Example 1, we found that 1% of 423.1 is 4.231. Thus, 2% of 423.1 is 2 4.231 8.462. A similar approach can be used to find 3% of a number, 4% of a number, and so on.

There is also an easy way to find 10% of a number that doesn’t require any

10 1

calculations. First, recall that 10% 100 10 . Thus, to find 10% of a number, we multiply the number by 0.1, and a quick way to multiply the number by 0.1 is to move its decimal point one place to the left.

Finding 10% of a Number

To find 10% of a number, move the decimal point in the number one place to the left.

What is 10% of 6,872 feet? Find the exact answer and an estimate using front-end rounding.

Strategy To find the exact answer, we will move the decimal point in 6,872 one place to the left. To find an estimate, we will move the decimal point in an approximation of 6,872 one place to the left.

WHY We move the decimal point one place to the left because 10% of a number means 0.10 of (times) the number.

Solution

Exact answer:

Estimate: To estimate 10% of 6,872 feet, we can front-end round 6,872 to 7,000 and find 10% of 7,000 feet . If we move the understood decimal point in 7,000 one place to the left, we get 700. Thus,

10% of 6,872 feet 700 feet Because 10% of 7,000 700.

Caution! In Examples 1 and 2, front-end rounding was used to find estimates of answers to percent problems. Since there are other ways to approximate (round) the numbers involved in a percent problem, the answers to estimation problems may vary.

The rule for finding 10% of a number can be extended to help us quickly find multiples of 10% of a number.

Finding 20%, 30%, 40%, . . . of a Number

To find 20% of a number, find 10% of the number by moving the decimal point one place to the left, and then double (multiply by 2) the result. A similar approach can be used to find 30% of a number, 40% of a number, and so on.

EXAMPLE 3 Estimate the answer: What is 20% of 416?

Strategy We will estimate 10% of 416, and double (multiply by 2) the result.

WHY 20% of a number is twice as much as 10% of a number.

Solution Since 10% of 416 is 41.6 (or about 42), it follows that 20% of 416 is about 2 42, which is 84.Thus,

20% of 416 84 Because 10% of 416 41.6 42 and 2 42 84.

Self Check 2

What is 10% of 3,536 pounds? Find the exact answer and an estimate using front-end rounding.

Now Try Problem 15

Self Check 3

Estimate the answer: What is 20% of 129?

Now Try Problem 19

Self Check 4

Estimate the answer: What is 50% of 14,272,549?

Now Try Problem 23

EXAMPLE 4 Estimate the answer: What is 50% of 2,595,603?

Strategy We will divide an approximation of 2,595,603 by 2.

WHY To find 50% of a number, we divide the number by 2.

Solution To estimate 50% of 2,595,603, we will find 50% of 2,600,000. We use 2,600,000 as an approximation because it is close to 2,595,603, because it is even, and, therefore, divisible by 2, and because it ends with many zeros.

50% of 2,595,603 1,300,000 Because 50% of 2,600,000 2,600,000 1,300,000

2

There is also an easy way to find 25% of a number. First, find 50% of the number by dividing the number by 2. Then, since 25% is one-half of 50%, divide that result by 2. Or, to save time, simply divide the original number by 4.

Finding 25% of a Number

To find 25% of a number, divide the number by 4.

Self Check 5

Estimate the answer: What is 25% of 27.16?

Now Try Problem 27

EXAMPLE 5 Estimate the answer: What is 25% of 43.02?

Strategy We will divide an approximation of 43.02 by 4.

WHY To find 25% of a number, divide the number by 4.

Solution To estimate 25% of 43.02, we will find 25% of 44. We use 44 as an approximation because it is close to 43.02 and because it is divisible by 4.

25% of 43.02 11 Because 25% of 44 444 11.

There is a quick way to find 5% of a number. First, find 10% of the number by moving the decimal point in the number one place to the left. Then, since 5% is onehalf of 10%, divide that result by 2.

Finding 5% of a Number

We will use 1,066 as an approximation of this result because it is close to 1,065.6 and because it is even, and, therefore, divisible by 2. Next, we divide the approximation by 2 to estimate 5% of 10,656.

1,066

2

533 Divide the approximation of 10% of 10,656 by 2.

Thus, 5% of 10,656 533. A 5% reduction in electricity usage by the average U.S. household is about 533 kilowatt-hours.

We can use the shortcuts for finding 10% and 5% of a number to find 15% of a number.

Self Check 6

Estimate the answer: What is 5% of 24,198?

Now Try Problems 31

Finding 15% of a Number

To find 15% of a number, find the sum of 10% of the number and 5% of the number.

results.

WHY To find 15% of a number, find the sum of 10% of the number and 5% of the number.

Solution To simplify the calculations, we will estimate the cost of the $77.55 dinner to be $80. Then, to estimate the tip, we find 10% of $80 and 5% of $80, and add.

Self Check 7

TIPPING Estimate the 15% tip on a $29.55 breakfast bill.

Now Try Problems 35 and 75

Self Check 8

Estimate the answer: What is 200% of 12.437?

Now Try Problem 43

Self Check 9

STUDENT DRIVERS Of the 1,550 students attending a high school, 26% of them drive to school. Estimate the number of students that drive to school.

Now Try Problem 85

EXAMPLE 8 Estimate the answer: What is 200% of 5.673?

Strategy We will multiply an approximation of 5.673 by 2.

WHY To find 200% of a number, multiply the number by 2.

Solution To estimate 200% of 5.673, we will find 200% of 6. We use 6 as an approximation because it is close to 5.673 and it makes the multiplication by 2 easy.

200% of 5.673 12 Because 200% of 6 2 6 12.

4 Use estimation to solve percent application problems.

In the previous examples of this section, we were given the percent (1%, 10%, 50%, 25%, 5%, 15%, or 200%), we approximated the base, and then we estimated the amount. Sometimes we must approximate the percent, as well, to estimate an answer.

Of the 350 children attending an elementary school, 24% of them are enrolled in the instrumental music program. Estimate the number of children taking instrumental music.

Strategy We will use the rule from this section for finding 25% of a number.

WHY 24% is approximately 25%, and there is a quick way to find 25% of a number.

Solution 24% of the 350 children in the school are taking instrumental music. To estimate 24% of 350, we will find 25% of 360. We use 360 as an approximation because it is close to 350 and it is divisible by 4.

24% of 350 90 Because 25% of 360 360 90.

4

There are approximately 90 children in the school taking instrumental music.

ANSWERS TO SELF CHECKS

1. 5.193, 5 2. 353.6 lb, 400 lb 3. 26 4. 7,000,000 5. 7 6. 1,210 7. $4.50 8. 24 9. 400 students

S E C T I O N 6.4 STUDY SET

VOCABULARY

Fill in the blanks.

CONCEPTS

5.To find 20% of a number, find 10% of the number by moving the decimal point one place to the left, and

8.To find 5% of a number, find 10% of the number by moving the decimal point in the number one place to

10. To find 200% of a number, multiply the number by

.

GUIDED PRACTICE

What is 1% of the given number? Find the exact answer and an estimate using front-end rounding. See Example 1.

What is 10% of the given number? Find the exact answer and an estimate using front-end rounding. See Example 2.

15.4,059 pounds

16.7,435 hours

17.691.4 minutes

18.881.2 kilometers

Estimate each answer. (Answers may vary.) See Example 3.

19.What is 20% of 346?

20.What is 20% of 409?

21.What is 20% of 67?

22.What is 20% of 32?

Estimate each answer. (Answers may vary.) See Example 4.

23.What is 50% of 4,195,898?

24.What is 50% of 6,802,117?

25.What is 50% of 397,020?

26.What is 50% of 793,288?

Estimate each answer. (Answers may vary.) See Example 5.

27.What is 25% of 15.49?

28.What is 25% of 7.02?

29.What is 25% of 49.33?

30.What is 25% of 39.74?

Estimate each answer. (Answers may vary because of the approximation used.) See Example 6.

31.What is 5% of 16,359?

32.What is 5% of 44,191?

33.What is 5% of 394.182?

34.What is 5% of 176.001?

Estimate a 15% tip on each dollar amount. (Answers may vary.)

See Example 7.

Estimate each answer. (Answers may vary.) See Example 8.

43.What is 200% of 4.212?

44.What is 200% of 5.189?

45.What is 200% of 35.77?

46.What is 200% of 80.32?

TRY IT YOURSELF

Find the exact answer using methods from this section.

47.What is 2% of 600?

48.What is 3% of 700?

49.What is 30% of 18?

50.What is 40% of 45?

Estimate each answer. (Answers may vary.)

51.What is 300% of 59.2?

52.What is 400% of 203.77?

53.What is 5% of 4,605?

54.What is 5% of 8,401?

77.DINING OUT A couple went out to eat at a restaurant. The food they ordered cost $28.55 and the drinks they ordered cost $19.75. Estimate a 15% tip on the total bill.

78.SPLITTING THE TIP The total bill for three businessmen who went out to eat at a Chinese restaurant was $121.10. If they split the tip equally, estimate each person’s share.

79.FIRE DAMAGE An insurance company paid 25% of the $118,000 it cost to rebuild a home that was destroyed by fire. How much did the insurance company pay?

80.SAFETY INSPECTIONS Of the 2,513 vehicles inspected at a safety checkpoint, 10% had code violations. How many cars had code violations?

81.WEIGHTLIFTING A 158-pound weightlifter can bench press 200% of his body weight. How many pounds can he bench press?

82.TESTING On a 60-question true/false test, 5% of a student’s answers were wrong. How many questions did she miss?

83.TRAFFIC STUDIES According to an electronic traffic monitor, 30% of the 690 motorists who passed it were speeding. How many of these motorists were speeding?

84.SELLING A HOME A homeowner has been told she will get back 50% of her $6,125 investment if she paints her home before selling it. How much will she get back if she paints her home?

Approximate the percent and then estimate each answer. (Answers may vary.)

85.NO-SHOWS The attendance at a seminar was only 24% of what the organizers had anticipated. If 875 people were expected, how many actually attended the seminar?

86.HONOR ROLL Of the 900 students in a school, 16% were on the principal’s honor roll. How many students were on the honor roll?

87.INTERNET SURVEYS The illustration shows an online survey question. How many people voted yes?

Online Survey

Live Vote Results

With the high gasoline prices, are you considering buying a more fuel-efficient vehicle?