Chapter 6 — Percent of a Number

Standard: 7.CE.2 (d) — The student will solve problems, including those in context, involving proportional relationships.

By the end of this chapter you will be able to:

• Find benchmark percentages — 1%, 5%, 10%, 25%, 50%, 75%, and 100% — of a whole number using mental math (7.CE.2d)
• Estimate the percentage of a whole number by rounding to a friendly benchmark, and tell whether the estimate is high or low (7.CE.2d)
• Determine the exact percentage of a whole number using decimal multiplication, a fraction, a proportion, or a combination of benchmarks (7.CE.2d)
• Estimate and determine percentages of whole numbers in contextual problems such as tips, tax, discounts, and survey results (7.CE.2d)

Lessons: 6.1 Benchmark Percentages · 6.2 Estimating a Percent of a Number · 6.3 Finding an Exact Percent of a Number · 6.4 Percent in Context

Chapter 5 built the machinery of ratio tables, proportions, and unit conversion. This chapter puts one particular proportional relationship to work: the relationship between a percent and an amount. Everything you learned about proportions still applies, and we will use it — but the goal here is speed and judgment. By the end you should be able to find a tip in your head at a restaurant table, and know without checking whether your answer is roughly right.

Lesson 6.1 — Benchmark Percentages

Percent means "per hundred"

A percent is a ratio that compares a number to 100. The symbol $\%$ means "per hundred," so $35\%$ means 35 out of every 100. That single idea is the whole foundation of this chapter.

Because a percent is a ratio out of 100, every percent can be written three equivalent ways:

$$35\% = \frac{35}{100} = 0.35$$

Each small square in the grid is one hundredth of the whole, which is exactly $1\%$. Shade 35 of them and you have shaded $35\%$ of the grid. When someone asks for "35% of 80," they are asking: if the whole 80 were split into 100 equal pieces, how much would 35 of those pieces be worth?

The whole matters

A percent by itself is not an amount. $50\%$ of 8 is 4, and $50\%$ of 800 is 400. The percent tells you what fraction of the whole to take; the whole (also called the base) tells you what you are taking it from. Always name the whole before you start computing.

The benchmark percentages

A benchmark percentage is a percent whose value you can find in your head, without writing anything down. These seven are worth memorizing as actions, not as facts:

Why the decimal point shifts

Finding $10\%$ means dividing by 10, and finding $1\%$ means dividing by 100. Dividing by 10 shifts the decimal point one place to the left; dividing by 100 shifts it two places. This is not a trick to memorize — it is what our place-value system does, and you saw it in Chapter 1 with powers of ten.

So $10\%$ of $640$ is $64$, and $1\%$ of $640$ is $6.4$. Every whole number has an invisible decimal point sitting after its last digit, which is why $640$ becomes $64.0$ and then $6.40$.

Watch the whole, not the digits. Students sometimes learn "move the decimal" and then apply it to the percent instead of the base. The decimal point that moves belongs to the whole. To find $10\%$ of 74, move the point in 74.

Combining benchmarks

The reason these seven benchmarks are enough for most mental math is that other percents can be built from them by adding, subtracting, or halving.

• $35\% = 25\% + 10\%$
• $15\% = 10\% + 5\%$
• $20\% = 10\% + 10\%$, or double $10\%$
• $60\% = 50\% + 10\%$
• $2\% = 1\% + 1\%$

The bar shows $35\%$ of 80. The $25\%$ piece is worth 20 and the $10\%$ piece is worth 8, so the two together are worth 28. Checking with multiplication: $0.35 \times 80 = 28$. The benchmark route and the multiplication route must always agree, and if they do not, one of them contains an error.

Percents over 100 and under 1

Nothing in the definition stops a percent from being larger than 100 or smaller than 1.

• $200\%$ of a number is twice the number, because $200\% = \frac{200}{100} = 2$. So $200\%$ of 45 is $90$.
• $150\%$ of a number is the whole number plus another half of it: $150\%$ of 48 is $48 + 24 = 72$.
• $0.5\%$ is half of $1\%$. Since $1\%$ of 400 is 4, $0.5\%$ of 400 is $2$.

A percent greater than $100\%$ always produces an amount larger than the whole, and a percent less than $100\%$ always produces an amount smaller than the whole. That is a useful check on every answer you write in this chapter.

A double number line for percents

Because percent and amount are proportional, they can be drawn on a double number line — the same tool you used for rates in Chapter 5. The top line carries the percent, always running from $0\%$ to $100\%$, and the bottom line carries the amount, running from 0 to the whole.

For a class of 60 students, $100\%$ lines up with 60, so each $10\%$ step is 6 students. Reading straight down from any percent gives the matching amount: $30\%$ is 18 students, $70\%$ is 42 students.

Worked examples

Example 1 — Halves and quarters

Find $50\%$ of 48 and $25\%$ of 48.

Half of 48 is 24. A quarter is half of that half.

$$50\% \text{ of } 48 = 48 \div 2 = 24 \qquad 25\% \text{ of } 48 = 24 \div 2 = 12$$

Answer: $50\%$ of 48 is $24$; $25\%$ of 48 is $12$

Example 2 — Tenths and hundredths

Find $10\%$ of 350 and $1\%$ of 350.

Shift the decimal point one place left, then two places left.

$$10\% \text{ of } 350 = 35 \qquad 1\% \text{ of } 350 = 3.5$$

Answer: $35$ and $3.5$

Example 3 — Building 5% from 10%

Find $5\%$ of 240.

First $10\%$, then halve it, because $5\%$ is half of $10\%$.

$$10\% \text{ of } 240 = 24 \qquad 5\% \text{ of } 240 = 24 \div 2 = 12$$

Answer: $12$

Example 4 — Building 75% from two benchmarks

Find $75\%$ of 64.

$$50\% \text{ of } 64 = 32 \qquad 25\% \text{ of } 64 = 16$$ $$75\% \text{ of } 64 = 32 + 16 = 48$$

Check: $0.75 \times 64 = 48$.

Answer: $48$

Example 5 — 100% and 200%

Find $100\%$ of 87 and $200\%$ of 87.

$100\%$ is the whole itself. $200\%$ is two wholes.

$$100\% \text{ of } 87 = 87 \qquad 200\% \text{ of } 87 = 2 \times 87 = 174$$

Answer: $87$ and $174$

Guided practice

1. Find $50\%$ of 86.
2. Find $25\%$ of 120.
3. Find $10\%$ of 74.
4. Find $1\%$ of 74.
5. Find $5\%$ of 60. (Find $10\%$ first, then halve it.)
6. Find $75\%$ of 200. (Add $50\%$ of 200 and $25\%$ of 200.)

Independent practice

7. Find each: a) $50\%$ of 250 b) $25\%$ of 88 c) $10\%$ of 4{,}500 d) $1\%$ of 620

8. Find each: a) $5\%$ of 140 b) $75\%$ of 48 c) $100\%$ of 39 d) $200\%$ of 45

9. Find $10\%$ of 96, then use that result to find $20\%$ of 96.

10. Which is greater, $25\%$ of 160 or $10\%$ of 380? Show both amounts.

11. Explain how to find $10\%$ of 730 without dividing on paper, and give the answer.

12. Complete the benchmark table for a whole of 600.

    Percent	$50\%$	$25\%$	$10\%$	$5\%$	$1\%$	$75\%$
    Amount of 600

13. Application. A restaurant bill is $40. Find a $10\%$ tip and a $20\%$ tip on that bill.

14. Reasoning. Explain why $5\%$ of a number is always exactly half of $10\%$ of that same number.

Exit ticket 6.1

15. Find $50\%$ of 92.
16. Find $10\%$ of 830.
17. Find $1\%$ of 830.
18. Describe two different ways to find $25\%$ of 48, then give the answer.

Lesson 6.2 — Estimating a Percent of a Number

Why estimate at all

You will not always need an exact answer, and you will not always have paper. Standing in a store, "about $12 off" is enough to decide. More importantly, an estimate is how you catch mistakes: if you estimate $20$ and your calculator says $200$, you know to look again before you trust it.

To estimate a percent of a number means to replace the problem with a nearby, friendlier problem you can do in your head. You make two decisions:

1. Round the percent to a nearby benchmark. $48\%$ becomes $50\%$; $26\%$ becomes $25\%$; $9\%$ becomes $10\%$; $73\%$ becomes $75\%$.
2. Round the whole to a friendly number. Friendly means easy to halve, quarter, or divide by 10 — usually a multiple of 10, 20, or 100.

Then compute the friendly problem exactly. That result is your estimate.

Naming the strategy

An estimate is only defensible if you say what you rounded to. "About 40" means nothing on its own; "$50\%$ of 80 is 40, so it's about 40" is a complete answer that another person can check. Throughout this lesson, state the benchmark you used.

$$48\% \text{ of } 82 \approx 50\% \text{ of } 80 = 40$$

The symbol $\approx$ is read "is approximately equal to." Use it whenever you replace a problem with a nearby one, and save the equals sign for statements that are exactly true.

High or low?

You can often tell which side of the exact answer your estimate falls on, just by looking at how you rounded.

• Rounding the percent up makes the estimate larger than the exact answer.
• Rounding the percent down makes the estimate smaller.
• Rounding the whole up makes the estimate larger.
• Rounding the whole down makes the estimate smaller.

For $48\%$ of 250, only the percent was rounded, and it was rounded up from $48\%$ to $50\%$. So the estimate of 125 must be a little high — and it is, since the exact value is 120.

When you round the percent one way and the whole the other way, the two effects push in opposite directions and you cannot tell the direction of the error from the rounding alone. In that situation, say so rather than guessing.

Worked examples

Example 1 — Rounding to 50%

About how much is $48\%$ of 82?

$48\%$ is close to $50\%$, and 82 is close to 80.

$$48\% \text{ of } 82 \approx 50\% \text{ of } 80 = 40$$

Answer: about $40$

Example 2 — Rounding to 25%

About how much is $26\%$ of 119?

$26\%$ is close to $25\%$, and 119 is close to 120, which divides evenly by 4.

$$26\% \text{ of } 119 \approx 25\% \text{ of } 120 = 30$$

Answer: about $30$

Example 3 — Rounding to 10%

About how much is $9\%$ of 71?

$$9\% \text{ of } 71 \approx 10\% \text{ of } 70 = 7$$

Because the percent was rounded up and the whole was rounded down, the two errors partly cancel. The exact value is $6.39$, close to the estimate.

Answer: about $7$

Example 4 — Rounding to 75%

About how much is $73\%$ of 41?

$$73\% \text{ of } 41 \approx 75\% \text{ of } 40 = 30$$

Answer: about $30$

Example 5 — Choosing among options

Which is closest to $19\%$ of 246: 25, 50, 75, or 100?

Round to $20\%$ of 250. Since $10\%$ of 250 is 25, doubling gives $20\%$ of $250 = 50$.

Answer: $50$

Guided practice

19. Estimate $52\%$ of 78 by finding $50\%$ of 80.
20. Estimate $24\%$ of 41 by finding $25\%$ of 40.
21. Estimate $11\%$ of 89 by finding $10\%$ of 90.
22. Estimate $76\%$ of 61 by finding $75\%$ of 60.
23. Estimate $9\%$ of 412 by finding $10\%$ of 400.
24. Estimate $48\%$ of 305 by finding $50\%$ of 300.

Independent practice

25. Estimate each using the benchmark and friendly whole shown in parentheses. a) $49\%$ of 62 ($50\%$ of 60) b) $26\%$ of 79 ($25\%$ of 80) c) $9\%$ of 51 ($10\%$ of 50) d) $74\%$ of 81 ($75\%$ of 80)
26. Which is closest to $31\%$ of 297: 30, 60, 90, or 120? Name the benchmark you used.
27. Which is closest to $6\%$ of 396: 8, 16, 24, or 40? Name the benchmark you used.
28. Estimate $48\%$ of 398 by finding $50\%$ of 400. Is your estimate higher or lower than the exact answer? Explain how you know without computing the exact answer.
29. Estimate $35\%$ of 61 by combining two benchmarks of 60.
30. Application. A jacket costs $59. About how many dollars is $20\%$ of that price? Use $60 as the friendly whole.
31. Application. A school has 412 students, and about $25\%$ of them ride the bus. About how many students ride the bus? Name the friendly whole you used.
32. Reasoning. Marisol estimates $9\%$ of 240 as 24, because $10\%$ of 240 is 24. Is her estimate higher or lower than the exact answer? Explain.

Exit ticket 6.2

33. Estimate $51\%$ of 78 by finding $50\%$ of 80.
34. Estimate $24\%$ of 121 by finding $25\%$ of 120.
35. Which is closest to $11\%$ of 508: 5, 50, 100, or 250?
36. Explain why rounding the whole up, with the percent left alone, makes an estimate greater than the exact answer.

Lesson 6.3 — Finding an Exact Percent of a Number

Four routes to the same answer

When an exact answer is required, you have four reliable methods. They are not competing rules to choose between at random — each is best suited to a particular kind of problem.

Method 1 — Multiply by the decimal. Rewrite the percent as a decimal by dividing by 100, then multiply.

$$18\% \text{ of } 250 = 0.18 \times 250 = 45$$

Method 2 — Multiply by the fraction. Rewrite the percent as a fraction over 100 and multiply. This is often fastest when the fraction simplifies.

$$18\% \text{ of } 250 = \frac{18}{100} \times 250 = \frac{9}{50} \times 250 = 9 \times 5 = 45$$

Method 3 — Set up a proportion. From Chapter 5, a proportion is an equation stating that two ratios are equal. The percent proportion pairs the part-to-whole ratio with the percent-to-100 ratio:

$$\frac{\text{part}}{\text{whole}} = \frac{\text{percent}}{100}$$

For $18\%$ of 250, that gives $\dfrac{x}{250} = \dfrac{18}{100}$. Solve it with the cross-product reasoning from Chapter 5: $100x = 18 \times 250 = 4500$, so $x = 45$.

Method 4 — Combine benchmarks. Break the percent into benchmark pieces, find each piece, and add.

$$18\% \text{ of } 250 = 10\% + 5\% + 1\% + 1\% + 1\% = 25 + 12.5 + 2.5 + 2.5 + 2.5 = 45$$

All four give 45, because they are all expressions of the same proportional relationship. Method 1 is the workhorse. Method 4 is the one to use when you have no pencil.

Choosing well

Writing the decimal correctly

Converting a percent to a decimal means dividing by 100, which shifts the decimal point two places left. Keep track of where the point started.

The single most common error in this lesson is writing $8\%$ as $0.8$. That is $80\%$ — ten times too big. Estimating first protects you: $8\%$ of 250 must be a small piece of 250, so 20 is plausible and 200 is not.

Worked examples

Example 1 — Decimal multiplication

Find $18\%$ of 250.

$$18\% = 0.18$$ $$0.18 \times 250 = 45$$

Estimate check: $20\%$ of 250 would be 50, and $18\%$ is a bit less than that.

Answer: $45$

Example 2 — Benchmark combination

Find $35\%$ of 80.

$$25\% \text{ of } 80 = 20 \qquad 10\% \text{ of } 80 = 8$$ $$35\% \text{ of } 80 = 20 + 8 = 28$$

Check: $0.35 \times 80 = 28$.

Answer: $28$

Example 3 — Using a proportion

Find $42\%$ of 150.

Let $x$ be the part.

$$\frac{x}{150} = \frac{42}{100}$$ $$100x = 42 \times 150 = 6300$$ $$x = \frac{6300}{100} = 63$$

Answer: $63$

Example 4 — A small percent

Find $6\%$ of 45.

$$6\% = 0.06$$ $$0.06 \times 45 = 2.7$$

Check with benchmarks: $1\%$ of 45 is $0.45$, and $6 \times 0.45 = 2.7$.

Answer: $2.7$

Example 5 — Beyond 100% and below 1%

Find $120\%$ of 65 and $0.5\%$ of 400.

$$120\% = 1.2 \qquad 1.2 \times 65 = 78$$ $$0.5\% = 0.005 \qquad 0.005 \times 400 = 2$$

The first answer is greater than 65, as it must be for a percent above $100\%$. The second is tiny, as it must be for a percent below $1\%$.

Answer: $78$ and $2$

Guided practice

37. Find $20\%$ of 45.
38. Find $15\%$ of 60 by adding $10\%$ of 60 and $5\%$ of 60.
39. Find $8\%$ of 250. (Write the decimal carefully.)
40. Find $45\%$ of 80.
41. Find $35\%$ of 140 by adding $25\%$ of 140 and $10\%$ of 140.
42. Find $110\%$ of 90.

Independent practice

43. Find each: a) $30\%$ of 70 b) $12\%$ of 300 c) $65\%$ of 40 d) $4\%$ of 225
44. Find each: a) $55\%$ of 120 b) $85\%$ of 60 c) $150\%$ of 48 d) $2\%$ of 850
45. Find $65\%$ of 240 by combining benchmarks. Show which benchmarks you used.
46. Find $18\%$ of 950 by multiplying by a decimal.
47. Write a proportion for "$24\%$ of 175" and solve it. Show the cross products.
48. Find $0.5\%$ of 600.
49. Application. A store has 640 T-shirts in stock, and $45\%$ of them are blue. How many blue T-shirts are in stock?
50. Reasoning. Explain why finding $25\%$ of 84 gives the same result as dividing 84 by 4. Give the answer as part of your explanation.

Exit ticket 6.3

51. Find $40\%$ of 65.
52. Find $15\%$ of 220.
53. Find $130\%$ of 50.
54. Find $35\%$ of 60 two different ways and show that both give the same answer.

Lesson 6.4 — Percent in Context

The four questions that cover most of it

Almost every percent problem you meet outside a classroom is one of four familiar situations. In all four, the mathematics is identical — find a percent of a whole — and the only new work is reading carefully enough to identify which number is the whole.

• A tip (also called a gratuity) is an amount added for service, calculated as a percent of the bill.
• Sales tax is an amount added by law, calculated as a percent of the purchase price.
• A discount is an amount subtracted from a price. The sale price is the original price minus the discount.
• A survey or count reports a percent of a group; the whole is the size of the group.

A reliable procedure

1. Name the whole. Which number is the bill, the price, the group size?
2. Name the percent.
3. Estimate with a benchmark, so you know roughly what to expect.
4. Compute exactly.
5. Answer the question that was asked. A discount problem may ask for the amount off, or for the sale price. Those are different numbers.

Step 5 is where careful students still lose points. Read the last sentence of the problem twice.

Working with money

Money answers are written with two decimal places, because prices are exact to the cent: write $9.00, not $9, and $8.64, not $8.64000. When a computation produces a third decimal place, the problem will tell you to round to the nearest cent. If it does not tell you, the numbers were chosen so that the answer lands exactly on the cent.

The benchmark tip

The most useful mental-math skill in this whole chapter is the restaurant tip, and it runs entirely on the benchmarks from Lesson 6.1.

• A $10\%$ tip: shift the decimal point one place left. On a $40 bill, that is $4.00.
• A $20\%$ tip: find $10\%$ and double it. On a $40 bill, $8.00.
• A $15\%$ tip: find $10\%$, halve it to get $5\%$, and add. On a $40 bill, $\$4.00 + \$2.00 = \$6.00$.

Worked examples

Example 1 — A tip

Find a $20\%$ tip on a $45 restaurant bill.

The whole is $45. Find $10\%$, then double.

$$10\% \text{ of } 45 = 4.5 \qquad 20\% \text{ of } 45 = 2 \times 4.5 = 9$$

Answer: $9.00

Example 2 — Sales tax and a total

A bicycle helmet costs $120. Sales tax is $6\%$. Find the tax and the total cost.

$$6\% \text{ of } 120 = 0.06 \times 120 = 7.2$$ $$\text{total} = 120 + 7.20 = 127.20$$

Answer: tax $7.20; total $127.20

Example 3 — A discount and a sale price

A jacket regularly costs $85 and is marked $30\%$ off. Find the discount and the sale price.

$$30\% \text{ of } 85 = 0.30 \times 85 = 25.5$$ $$\text{sale price} = 85 - 25.50 = 59.50$$

Answer: discount $25.50; sale price $59.50

Example 4 — A survey

In a school of 350 students, $12\%$ walk to school. How many students walk?

The whole is 350 students.

$$12\% \text{ of } 350 = 0.12 \times 350 = 42$$

An answer of 42 students is sensible: it is a small part of 350, and it is a whole number of people, which it must be.

Answer: 42 students

Example 5 — Estimate, then compute

A dinner bill is $62 and you want to leave a $15\%$ tip. Estimate first, then find the exact tip.

Estimate with a friendly $60: $10\%$ of 60 is 6, half of that is 3, so about $9.

Exact: $0.15 \times 62 = 9.3$.

Answer: about $9; exactly $9.30

Guided practice

55. Find a $10\%$ tip on a $36 bill.
56. Find a $20\%$ tip on a $55 bill.
57. A furniture order costs $240. Find the $5\%$ sales tax.
58. Shoes that cost $64 are marked $25\%$ off. Find the amount of the discount.
59. In a group of 250 students, $40\%$ chose soccer. How many students chose soccer?

Independent practice

60. Find a $15\%$ tip on an $80 bill. (Use $10\%$ plus $5\%$.)
61. A tablet costs $150 and sales tax is $7\%$. Find the tax.
62. A coat regularly costs $120 and is marked $35\%$ off. Find the discount and the sale price.
63. A basketball team attempted 40 free throws and made $65\%$ of them. How many did they make?
64. A phone plan includes 1{,}250 megabytes of data, and Jordan has used $8\%$ of it. How many megabytes has Jordan used?
65. Application. A restaurant bill is $48. Find an $18\%$ tip, then find the total the customer pays.
66. Application. A theater has 350 seats, and $22\%$ of them are in the balcony. First estimate the number of balcony seats using $20\%$ of 350, then find the exact number.
67. Reasoning. Explain why taking $20\%$ off a $60 price gives the same sale price as finding $80\%$ of $60. Show both computations.

Exit ticket 6.4

68. Find a $20\%$ tip on a $65 bill.
69. A chair costs $250 and sales tax is $6\%$. Find the tax.
70. A bike listed at $140 is marked $30\%$ off. Find the amount of the discount.
71. Describe a benchmark strategy for finding a $15\%$ tip on a $40 bill, and give the tip.

Chapter 6 Review

Vocabulary. percent · whole (base) · benchmark percentage · estimate · double number line · discount · sale price · sales tax · tip

Part A — Benchmark percentages (7.CE.2d)

72. Find $50\%$ of 148.
73. Find $25\%$ of 320.
74. Find $10\%$ of 76.
75. Find $1\%$ of 76.
76. Find $5\%$ of 180.
77. Find $75\%$ of 88.

Part B — Estimating a percent of a number (7.CE.2d)

78. Estimate $49\%$ of 122 by finding $50\%$ of 120.
79. Estimate $26\%$ of 79 by finding $25\%$ of 80.
80. Estimate $9\%$ of 305 by finding $10\%$ of 300.
81. Estimate $74\%$ of 41 by finding $75\%$ of 40.
82. Which is closest to $34\%$ of 298: 30, 60, 100, or 150? Name the benchmark you used.
83. Is $10\%$ of 480 a high estimate or a low estimate for $9\%$ of 480? Give both the estimate and the exact value.

Part C — Determining an exact percent (7.CE.2d)

84. Find $30\%$ of 150.
85. Find $12\%$ of 425.
86. Find $65\%$ of 80.
87. Find $4\%$ of 375.
88. Find $140\%$ of 60.
89. Find $0.5\%$ of 800.

Part D — Percent in context (7.CE.2d)

90. Find an $18\%$ tip on a $60 restaurant bill.
91. A backpack costs $45 and sales tax is $6\%$. Find the tax and the total cost.
92. A tent listed at $95 is marked $40\%$ off. Find the discount and the sale price.

Part E — Mixed application and reasoning

93. Find $35\%$ of 260 by combining benchmarks. Show each benchmark amount and the sum.
94. Estimate $48\%$ of 250 using a benchmark, then compute the exact value. Explain why the estimate falls on the high side.
95. Explain why moving the decimal point gives $10\%$ and $1\%$ of a number. Use 640 in your explanation.

Standards coverage check — Chapter 6

Bullet d of 7.CE.2 asks students to "estimate and determine the percentage of a given whole number, including but not limited to the use of benchmark percentages." That single bullet contains three component demands, tracked separately below.

Percents greater than $100\%$ appear in items 8d, 42, 44c, 53, and 88; percents less than $1\%$ appear in items 48 and 89. The whole is a whole number throughout, as the standard specifies.

Scope note. Parts a–c of 7.CE.2 — ratio tables, writing and solving proportions, and unit conversion with a given conversion factor — are covered in Chapter 5. This chapter uses proportions as one of four solution methods in Lesson 6.3 but does not re-teach them. Percent increase and decrease, percent error, and finding the whole from a known part are not part of 7.CE.2d and appear in later standards, so they are deliberately absent here.

Answer keys for every set in this chapter are in Appendix A.