Chapter 3 — Percent Applications: Discounts, Markups, Tax, and Tip

Standard: 8.CE.1 — The student will estimate and apply proportional reasoning and computational procedures to solve contextual problems.

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

• Estimate a percent of a price using benchmark percentages, and judge whether the estimate is high or low (8.CE.1a, b, c)
• Compute one discount and the resulting sale price (8.CE.1a)
• Compute one markup and the resulting selling price (8.CE.1a)
• Compute sales tax and the resulting total (8.CE.1b)
• Compute a tip and the resulting total (8.CE.1b)
• Compute a percent increase or a percent decrease from an original amount and a new amount (8.CE.1c)

Lessons: 3.1 Estimating and Finding a Percent of a Price · 3.2 Discounts and Sale Prices · 3.3 Markups and Selling Prices · 3.4 Sales Tax and the Total · 3.5 Tip and the Total · 3.6 Percent Increase and Percent Decrease

In Grade 7 you learned to find a percent of a number, and you practiced it on tips, tax, and discounts. This chapter does not re-teach that skill — it assumes it. What is new here is the second step. A percent of a price is almost never the answer to a real question. Nobody wants to know that the discount is $12.00; they want to know what the jacket costs. Nobody wants to know that the tax is $2.12; they want to know what to hand the cashier. Every problem in this chapter has two moves: find the percent of the price, then do something with it.

Money note. Every dollar amount in this chapter is rounded to the nearest cent, rounding a half cent up. So $8.965 becomes $8.97 and $0.954 becomes $0.95. Round once, at the end, and say what you rounded.

Numbering note. Item numbers run straight through the chapter, from 1 in Lesson 3.1 to 132 at the end of the review. They do not restart at each lesson.

Lesson 3.1 — Estimating and Finding a Percent of a Price

The one skill everything else is built on

A percent is a ratio out of 100, so $r\%$ of an amount means $\dfrac{r}{100}$ of that amount. To find it, write the percent as a decimal and multiply:

$$r\% \text{ of } A = \frac{r}{100} \times A$$

So $20\%$ of $\$45.00$ is $0.20 \times 45.00 = 9.00$, or $9.00.

The picture worth carrying in your head is a bar cut into ten equal blocks. Each block is $10\%$ of the price.

Two blocks are $20\%$, and each block is worth $4.50, so $20\%$ is worth $9.00. The bar and the multiplication agree, as they must.

Benchmarks are for your head, not your paper

A benchmark percentage is one you can find mentally. These four do almost all the work in this chapter:

Other percents are built by combining benchmarks:

• $15\% = 10\% + 5\%$
• $20\% = 10\% + 10\%$
• $30\% = 10\% + 10\% + 10\%$
• $6\% = 5\% + 1\%$

So $15\%$ of $\$52.00$ is $\$5.20 + \$2.60 = \$7.80$, which you can check against $0.15 \times 52.00 = 7.80$.

The decimal point that moves belongs to the price. To find $10\%$ of $\$74.00$, move the point in $74.00$, not in the $10$. This is the most common slip in mental percent work.

Estimate first, then compute

The standard asks you to estimate as well as solve, and estimating is not busywork. An estimate is a cheap check on an expensive computation. Round the price to a friendly number, use a benchmark, and keep the arithmetic in your head.

To estimate $15\%$ of $\$79.40$: round to $\$80.00$. Then $10\%$ is $8.00 and $5\%$ is $4.00, so the estimate is about $12.00. The exact value is $0.15 \times 79.40 = 11.91$, or $11.91. The estimate landed 9 cents high, because we rounded the price up.

That last sentence is the habit to build. If you round the price up, your estimate comes out high; if you round the price down, your estimate comes out low. Knowing the direction turns an estimate into a real check.

Notice how far the tip row is from its estimate: rounding $\$47.30$ up to $\$50.00$ moved the price by $2.70, and $20\%$ of a price that is too big is a tip that is too big. An estimate tells you the neighborhood, not the address.

Rounding money

A percent of a price often produces more than two decimal places. Sales tax of $5.3\%$ on $\$18.00$ is $0.053 \times 18 = 0.954$ dollars, which is 95.4 cents. There is no such coin, so round to the nearest cent: $0.95.

The rule used everywhere in this chapter: look at the third decimal place. If it is 5 or more, round the cents up; otherwise leave the cents alone. And round once, at the very end of the problem — rounding partway through and then continuing can throw the final answer off by a cent.

Worked examples

Example 1 — A benchmark in your head

Find $10\%$ and $5\%$ of $\$62.00$.

Move the decimal point one place left for $10\%$, then halve that for $5\%$.

$$10\% \text{ of } 62.00 = 6.20 \qquad 5\% \text{ of } 62.00 = 3.10$$

Answer: $6.20 and $3.10

Example 2 — Combining benchmarks

Find $30\%$ of $\$24.50$.

$10\%$ of $\$24.50$ is $2.45, and $30\%$ is three of those.

$$3 \times 2.45 = 7.35$$

Check by multiplying: $0.30 \times 24.50 = 7.35$.

Answer: $7.35

Example 3 — A percent with a decimal in it

Find $5.3\%$ of $\$40.00$.

Write the percent as a decimal. Since $5.3\% = \dfrac{5.3}{100} = 0.053$,

$$0.053 \times 40.00 = 2.12$$

Answer: $2.12

Example 4 — Rounding to the nearest cent

Find $18\%$ of $\$63.50$.

$$0.18 \times 63.50 = 11.43$$

This one comes out exact, with no third decimal place to round. Compare it with $18\%$ of $\$19.99$, which is $3.5982$ and must be rounded to $3.60.

Answer: $11.43

Example 5 — Estimate, then compute, then compare

Estimate $20\%$ of $\$41.20$, then find it exactly, and say whether the estimate was high or low.

Round $\$41.20$ down to $\$40.00$. Then $10\%$ is $4.00, so $20\%$ is about $8.00.

$$0.20 \times 41.20 = 8.24$$

The price was rounded down, so the estimate is low — by 24 cents.

Answer: estimate $8.00; exact $8.24; the estimate is low

Guided practice

1. Find $10\%$ of $\$46.00$.
2. Find $5\%$ of $\$46.00$.
3. Find $1\%$ of $\$84.00$.
4. Find $25\%$ of $\$36.80$.
5. Estimate $20\%$ of $\$38.75$ by rounding the price to $\$40.00$.
6. Find $15\%$ of $\$52.00$ exactly.

Independent practice

7. Find each amount. a) $20\%$ of $\$65.00$ b) $30\%$ of $\$24.50$ c) $6\%$ of $\$120.00$ d) $2.5\%$ of $\$80.00$
8. Find $5.3\%$ of $\$40.00$.
9. Find $18\%$ of $\$63.50$.
10. Round each to the nearest cent. a) $\$7.3125$ b) $\$12.995$ c) $\$0.4449$
11. Estimate $15\%$ of $\$79.40$ using a benchmark, then compute it exactly and say how far off the estimate was.
12. Reasoning. Explain why $1\%$ of a price can be found by moving the decimal point two places to the left.
13. Application. A jacket is priced at $\$74.00$. Find $30\%$ of that price.
14. Error analysis. To find $8\%$ of $\$45.00$, a student computes $8 \times 45 = 360$ and writes $360.00. Explain the error and give the correct amount.

Exit ticket 3.1

15. Find $10\%$ of $\$27.40$.
16. Find $5\%$ of $\$62.00$.
17. Find $12\%$ of $\$95.00$.
18. Estimate $20\%$ of $\$41.20$, then say whether your estimate is high or low and why.

Lesson 3.2 — Discounts and Sale Prices

What a discount is

A discount is an amount subtracted from a price. It is usually announced as a percent of the original price: "$25\%$ off." The percent by itself is not the discount; the discount amount is the dollars you save, and the sale price is what you actually pay.

$$\text{discount amount} = \text{discount rate} \times \text{original price}$$ $$\text{sale price} = \text{original price} - \text{discount amount}$$

Read the question carefully before you answer. "How much do you save?" wants the discount amount. "What does it cost?" wants the sale price. Handing back the discount amount when the sale price was asked for is the single most common mistake in this lesson.

Two routes to the sale price

There is a second route that skips the subtraction. If $25\%$ comes off, then $75\%$ stays on, and $100\% - 25\% = 75\%$ of the price is the sale price.

Both routes on a $\$48.00$ item at $25\%$ off:

$$\textbf{Route 1: } 0.25 \times 48.00 = 12.00, \quad 48.00 - 12.00 = 36.00$$ $$\textbf{Route 2: } 100\% - 25\% = 75\%, \quad 0.75 \times 48.00 = 36.00$$

The number $0.75$ is the complement multiplier for a $25\%$ discount. For any discount rate $r$, the sale price is

$$\text{sale price} = (1 - r) \times \text{original price}.$$

Route 2 is faster and gives fewer chances to slip, but Route 1 is the one that answers "how much did I save?" Know both.

One discount per problem. Grade 8 problems apply exactly one discount to a price. You will never be asked to take $20\%$ off and then another $10\%$ off the result, and you will never combine a discount with a markup on the same item. If a problem seems to ask for two, re-read it.

Estimating a sale price

Round the price, take the benchmark, subtract. A $\$39.80$ lamp at $25\%$ off: round to $\$40.00$, take $25\%$ to get $10.00, and the sale price is about $30.00. The exact discount is $0.25 \times 39.80 = 9.95$, so the exact sale price is $29.85 — a nickel below the estimate, because the price was rounded up.

When the cents get awkward

A $\$149.99$ tent at $40\%$ off has a discount of $0.40 \times 149.99 = 59.996$ dollars, which rounds to $60.00. Subtracting gives $149.99 - 60.00 = 89.99$. Check with the complement route: $0.60 \times 149.99 = 89.994$, which also rounds to $89.99. The two routes agree, which is the point of checking.

Worked examples

Example 1 — Discount amount and sale price

A $\$60.00$ backpack is $20\%$ off. Find the discount amount and the sale price.

$$0.20 \times 60.00 = 12.00 \qquad 60.00 - 12.00 = 48.00$$

Answer: discount $12.00; sale price $48.00

Example 2 — Using the complement multiplier

A $\$32.00$ game is $25\%$ off. Find the sale price without finding the discount.

Since $100\% - 25\% = 75\%$,

$$0.75 \times 32.00 = 24.00$$

Answer: $24.00

Example 3 — A rate that is not a benchmark

A $\$56.40$ tent is $15\%$ off. Find the discount amount and the sale price.

$$0.15 \times 56.40 = 8.46 \qquad 56.40 - 8.46 = 47.94$$

Check: $0.85 \times 56.40 = 47.94$.

Answer: discount $8.46; sale price $47.94

Example 4 — Rounding inside a discount

A $\$99.00$ chair is $45\%$ off. Find the sale price.

$$0.45 \times 99.00 = 44.55 \qquad 99.00 - 44.55 = 54.45$$

Answer: $54.45

Example 5 — Estimate, then compute

A $\$61.75$ sweater is $20\%$ off. Estimate the sale price, then find it exactly.

Round to $\$60.00$: $20\%$ of that is $12.00, so the sale price is about $48.00.

$$0.20 \times 61.75 = 12.35 \qquad 61.75 - 12.35 = 49.40$$

The price was rounded down, so the estimate is low.

Answer: estimate about $48.00; exact sale price $49.40

Guided practice

19. A $\$60.00$ backpack is $20\%$ off. Find the discount amount.
20. Find the sale price of that same backpack.
21. A $\$45.00$ pair of shoes is $10\%$ off. Find the sale price.
22. A $\$32.00$ game is $25\%$ off. Find the sale price using the complement multiplier.
23. A $\$120.00$ bike is $15\%$ off. Find the discount amount and the sale price.
24. Estimate the sale price of a $\$39.80$ lamp at $25\%$ off by rounding the price to $\$40.00$.

Independent practice

25. Find the discount amount and the sale price. a) $\$80.00$ at $35\%$ off b) $\$24.50$ at $20\%$ off c) $\$18.60$ at $50\%$ off d) $\$149.99$ at $40\%$ off
26. A $\$72.00$ coat is $30\%$ off. Find the sale price.
27. A $\$56.40$ tent is $15\%$ off. Find the discount amount and the sale price.
28. A $\$99.00$ chair is $45\%$ off. Find the sale price.
29. Reasoning. Explain why multiplying the original price by $0.80$ gives the sale price directly when an item is $20\%$ off.
30. Application. A store advertises $60\%$ off a $\$85.00$ coat. How much does a shopper save, and what does the coat cost?
31. Error analysis. Asked for the sale price of a $\$48.00$ item at $30\%$ off, a student answers $14.40. Explain what the student found and give the sale price.
32. Estimate, then check. A $\$61.75$ sweater is $20\%$ off. Estimate the sale price, then compute it exactly, and say whether the estimate was high or low.

Exit ticket 3.2

33. A $\$40.00$ item is $25\%$ off. Find the sale price.
34. A $\$65.00$ item is $10\%$ off. Find the discount amount.
35. A $\$36.80$ item is $50\%$ off. Find the sale price.
36. Explain why subtracting the discount and multiplying by the complement give the same sale price.

Lesson 3.3 — Markups and Selling Prices

What a markup is

A store buys an item at some cost and sells it for more. The extra is a markup, and it pays for rent, wages, and profit. Like a discount, it is announced as a percent — but of the cost, and it is added instead of subtracted.

$$\text{markup amount} = \text{markup rate} \times \text{cost}$$ $$\text{selling price} = \text{cost} + \text{markup amount}$$

The bar for a markup is longer than the original, because a markup pushes past $100\%$. That single picture is the difference between this lesson and the last one.

The multiplier for a markup

If $25\%$ is added on, the selling price is $100\% + 25\% = 125\%$ of the cost:

$$\textbf{Route 1: } 0.25 \times 40.00 = 10.00, \quad 40.00 + 10.00 = 50.00$$ $$\textbf{Route 2: } 1.25 \times 40.00 = 50.00$$

In general, for a markup rate $r$,

$$\text{selling price} = (1 + r) \times \text{cost}.$$

Compare the two multipliers side by side and the pattern is hard to forget: a $35\%$ discount multiplies by $1 - 0.35 = 0.65$, and a $35\%$ markup multiplies by $1 + 0.35 = 1.35$. A discount multiplier is always less than 1; a markup multiplier is always greater than 1. If you ever compute a sale price larger than the original, or a selling price smaller than the cost, you have used the wrong one.

Markups can be bigger than 100%

Nothing stops a markup rate from exceeding $100\%$, and in food service it usually does. A cake with $\$8.40$ of ingredients marked up $125\%$ has a markup amount of $1.25 \times 8.40 = 10.50$ dollars and sells for $8.40 + 10.50 = 18.90$, or $18.90. Equivalently, $2.25 \times 8.40 = 18.90$. A markup over $100\%$ means the markup amount is larger than the cost itself — which is exactly what "more than doubling the price" means.

One markup per problem. As with discounts, Grade 8 problems apply exactly one markup. No stacking a second markup on a marked-up price, and no mixing a markup and a discount on the same item.

Worked examples

Example 1 — Markup amount and selling price

An item costs the store $\$40.00$ and is marked up $25\%$. Find both.

$$0.25 \times 40.00 = 10.00 \qquad 40.00 + 10.00 = 50.00$$

Answer: markup $10.00; selling price $50.00

Example 2 — Using the multiplier

An item costs $\$62.00$ and is marked up $10\%$. Find the selling price.

$$1.10 \times 62.00 = 68.20$$

Answer: $68.20

Example 3 — Cents in the markup

An item costs $\$36.80$ and is marked up $35\%$. Find the markup amount and the selling price.

$$0.35 \times 36.80 = 12.88 \qquad 36.80 + 12.88 = 49.68$$

Answer: markup $12.88; selling price $49.68

Example 4 — Rounding to the nearest cent

An item costs $\$14.60$ and is marked up $55\%$. Find the selling price.

$$0.55 \times 14.60 = 8.03$$

The exact product is $8.030$, so no rounding is needed here.

$$14.60 + 8.03 = 22.63$$

Answer: $22.63

Example 5 — Estimate, then compute

An item costs $\$78.40$ and is marked up $25\%$. Estimate the selling price, then find it exactly.

Round to $\$80.00$: $25\%$ of that is $20.00, so the price is about $100.00.

$$0.25 \times 78.40 = 19.60 \qquad 78.40 + 19.60 = 98.00$$

Answer: estimate about $100.00; exact selling price $98.00

Guided practice

37. An item costs $\$40.00$ and is marked up $25\%$. Find the markup amount.
38. Find the selling price of that same item.
39. An item costs $\$18.00$ and is marked up $50\%$. Find the selling price.
40. An item costs $\$62.00$ and is marked up $10\%$. Find the selling price.
41. An item costs $\$85.00$ and is marked up $40\%$. Find the markup amount and the selling price.
42. Estimate the selling price of an item costing $\$29.60$ with a $30\%$ markup, by rounding the cost to $\$30.00$.

Independent practice

43. Find the markup amount and the selling price. a) cost $\$24.00$, markup $15\%$ b) cost $\$150.00$, markup $20\%$ c) cost $\$9.50$, markup $60\%$ d) cost $\$47.25$, markup $40\%$
44. An item costs $\$36.80$ and is marked up $35\%$. Find the selling price.
45. An item costs $\$112.00$ and is marked up $45\%$. Find the markup amount and the selling price.
46. A shop marks up a $\$14.60$ item by $55\%$. Find the selling price.
47. Reasoning. Explain why the multiplier for a $25\%$ markup is $1.25$, and why a markup multiplier is always greater than 1.
48. Application. A bakery spends $\$8.40$ on ingredients for one cake and marks it up $125\%$. Find the selling price.
49. Error analysis. Asked for the selling price of a $\$50.00$ item with a $30\%$ markup, a student answers $15.00. Explain what the student found and give the selling price.
50. Estimate, then check. An item costs $\$78.40$ and is marked up $25\%$. Estimate the selling price, then compute it exactly.

Exit ticket 3.3

51. An item costs $\$20.00$ and is marked up $30\%$. Find the selling price.
52. An item costs $\$64.00$ and is marked up $25\%$. Find the markup amount.
53. An item costs $\$45.50$ and is marked up $20\%$. Find the selling price.
54. In one or two sentences, explain the difference between a discount and a markup, and what it does to the multiplier.

Lesson 3.4 — Sales Tax and the Total

Tax is added, and the question is always the total

Sales tax is a percent of a purchase price collected by the government. You do not get to choose it, and the store adds it at the register. In Virginia the combined state and local rate is $5.3\%$ in most places, higher in some regions, and there is a reduced rate of $1\%$ on most groceries.

Sales tax works exactly like a markup, arithmetically:

$$\text{tax} = \text{tax rate} \times \text{price}$$ $$\text{total} = \text{price} + \text{tax}$$

A tax problem asks for the total. The tax amount is a step on the way, not the answer. If a problem says "find the total cost," an answer of $2.12 is not a small error — it is an answer to a different question.

The 105.3% shortcut

Since the total is the price plus $5.3\%$ of the price, the total is $105.3\%$ of the price:

$$\text{total} = 1.053 \times \text{price}$$

On a $\$40.00$ purchase, $1.053 \times 40.00 = 42.12$, or $42.12. The two-step route gives $0.053 \times 40.00 = 2.12$ and $40.00 + 2.12 = 42.12$. Same answer.

The shortcut is worth using because it rounds only once. Watch what happens on an $\$18.00$ purchase:

$$1.053 \times 18.00 = 18.954 \rightarrow \$18.95$$

and by the two-step route, $0.053 \times 18.00 = 0.954 \rightarrow \$0.95$, then $18.00 + 0.95 = 18.95$. They agree here, and on nearly every problem you will meet — but the one-step route is the one that cannot drift.

Estimating a total with tax

Virginia's $5.3\%$ is close to the benchmark $5\%$, and $5\%$ is easy: find $10\%$ and halve it. So to estimate the total on a $\$59.80$ purchase, round to $\$60.00$, take $5\%$ to get $3.00, and estimate about $63.00. The exact tax is $0.053 \times 59.80 = 3.1694$, or $3.17, and the exact total is $62.97.

Because $5\%$ is less than $5.3\%$, estimating tax with $5\%$ always understates the tax a little. Whether the whole estimate lands high or low also depends on how you rounded the price.

Tax on more than one item

Tax is charged on the subtotal, so add up the goods first, then tax once. Two tickets at $\$21.50$ each with $6\%$ tax:

$$\text{subtotal} = 2 \times 21.50 = 43.00$$ $$\text{tax} = 0.06 \times 43.00 = 2.58$$ $$\text{total} = 43.00 + 2.58 = 45.58$$

Taxing each ticket separately and adding would give the same answer here, but it is slower and it invites two roundings where one will do.

Worked examples

Example 1 — Tax and total

Find the tax and the total on a $\$40.00$ purchase at a $5.3\%$ rate.

$$0.053 \times 40.00 = 2.12 \qquad 40.00 + 2.12 = 42.12$$

Answer: tax $2.12; total $42.12

Example 2 — Using the one-step multiplier

Find the total on a $\$75.00$ purchase at a $6\%$ rate.

$$1.06 \times 75.00 = 79.50$$

Answer: $79.50

Example 3 — Rounding the tax up

Find the tax and the total on an $\$8.75$ purchase at a $6\%$ rate.

$$0.06 \times 8.75 = 0.525 \rightarrow \$0.53$$

The third decimal place is a 5, so the cents round up.

$$8.75 + 0.53 = 9.28$$

Check: $1.06 \times 8.75 = 9.275 \rightarrow \$9.28$.

Answer: tax $0.53; total $9.28

Example 4 — The reduced grocery rate

Find the total on $\$63.00$ of groceries at the $1\%$ rate.

$1\%$ of $\$63.00$ is $0.63, so the total is $63.00 + 0.63 = 63.63$.

Answer: $63.63

Example 5 — Estimate, then compute

Estimate the total on a $\$79.60$ purchase at $5.3\%$, then compute it exactly.

Round to $\$80.00$ and use $5\%$: $10\%$ is $8.00, so $5\%$ is $4.00, and the total is about $84.00.

$$0.053 \times 79.60 = 4.2188 \rightarrow \$4.22 \qquad 79.60 + 4.22 = 83.82$$

Answer: estimate about $84.00; exact total $83.82

Guided practice

55. Find $5.3\%$ of $\$40.00$.
56. A purchase is $\$40.00$ and the tax rate is $5.3\%$. Find the total.
57. A purchase is $\$25.00$ and the tax rate is $6\%$. Find the tax and the total.
58. A purchase is $\$18.00$ and the tax rate is $5.3\%$. Find the total.
59. A grocery order is $\$63.00$ and the reduced rate is $1\%$. Find the total.
60. Estimate the total on a $\$59.80$ purchase at $5.3\%$ by rounding the price to $\$60.00$ and the rate to $5\%$.

Independent practice

61. Find the tax and the total. a) $\$32.00$ at $5.3\%$ b) $\$75.00$ at $6\%$ c) $\$120.50$ at $5.3\%$ d) $\$14.99$ at $7\%$
62. A purchase is $\$248.00$ and the tax rate is $5.3\%$. Find the total.
63. A purchase is $\$8.75$ and the tax rate is $6\%$. Find the tax and the total.
64. A purchase is $\$52.40$ and the tax rate is $5.3\%$. Find the total.
65. Reasoning. Explain why the total on a purchase taxed at $5.3\%$ is $105.3\%$ of the price.
66. Application. Two concert tickets cost $\$21.50$ each and the tax rate is $6\%$. Find the subtotal, the tax, and the total.
67. Error analysis. Asked for the total cost of a $\$40.00$ purchase at $5.3\%$ tax, a student answers $2.12. Explain what the student found and give the total.
68. Estimate, then check. Estimate the total on a $\$79.60$ purchase at $5.3\%$, then compute it exactly and compare.

Exit ticket 3.4

69. A purchase is $\$30.00$ and the tax rate is $5.3\%$. Find the tax and the total.
70. A purchase is $\$45.00$ and the tax rate is $6\%$. Find the total.
71. A purchase is $\$12.80$ and the tax rate is $5.3\%$. Find the total.
72. Explain why estimating Virginia's $5.3\%$ tax with $5\%$ is close, and whether the estimated tax comes out high or low.

Lesson 3.5 — Tip and the Total

A tip is a percent you choose

A tip, or gratuity, is money added to a bill for service. Unlike tax, the rate is yours to pick — $15\%$, $18\%$, and $20\%$ are the common choices. The arithmetic is the same as tax:

$$\text{tip} = \text{tip rate} \times \text{bill} \qquad \text{total} = \text{bill} + \text{tip}$$

And as with tax, the question asks for the total. Leaving the tip amount as your final answer answers a question nobody asked.

Doing it at the table

Tips get computed in restaurants without paper, so the benchmark route matters more here than anywhere else in the chapter. Everything is built off $10\%$.

On a $\$46.00$ bill: $10\%$ is $4.60. Halve it for $5\%$: $2.30. Add those for $15\%$: $6.90. Double the first for $20\%$: $9.20. A $20\%$ tip makes the total $46.00 + 9.20 = 55.20$, or $55.20.

For $18\%$, the cleanest mental route is $20\%$ minus $2\%$, or just multiply: $0.18 \times 50.00 = 9.00$ on a $\$50.00$ bill.

The multiplier again

The same shortcut works, and for the same reason. A $20\%$ tip makes the total $120\%$ of the bill:

$$\text{total} = 1.20 \times \text{bill}$$

On a $\$72.80$ bill, $1.20 \times 72.80 = 87.36$, or $87.36. By the two-step route, the tip is $0.20 \times 72.80 = 14.56$ and the total is $72.80 + 14.56 = 87.36$.

Tip and tax in the same meal

A restaurant bill can carry both, and this is the one place in the chapter where two percents appear in one problem. That is allowed — the standard names tax and tip together — and it is not the forbidden case, which is chaining two discounts or mixing a discount with a markup.

The convention used in this book: the tip is computed on the pre-tax bill. On a $\$64.00$ meal with $6\%$ tax and a $20\%$ tip,

$$\text{tax} = 0.06 \times 64.00 = 3.84$$ $$\text{tip} = 0.20 \times 64.00 = 12.80$$ $$\text{total} = 64.00 + 3.84 + 12.80 = 80.64$$

Notice that both percents are taken of the same $\$64.00$. Neither one is applied to the other's result.

Worked examples

Example 1 — A benchmark tip

Find a $15\%$ tip on a $\$40.00$ bill and the total.

$10\%$ is $4.00 and $5\%$ is $2.00, so the tip is $6.00.

$$40.00 + 6.00 = 46.00$$

Answer: tip $6.00; total $46.00

Example 2 — A 20% tip with the multiplier

Find the total on a $\$24.60$ bill with a $20\%$ tip.

$$1.20 \times 24.60 = 29.52$$

Answer: $29.52

Example 3 — An 18% tip that needs rounding

Find the tip and the total on a $\$19.99$ bill with an $18\%$ tip.

$$0.18 \times 19.99 = 3.5982 \rightarrow \$3.60$$ $$19.99 + 3.60 = 23.59$$

Answer: tip $3.60; total $23.59

Example 4 — Tax and tip together

A meal costs $\$64.00$. Tax is $6\%$ and the tip is $20\%$ of the pre-tax bill. Find the total.

$$\text{tax} = 0.06 \times 64.00 = 3.84 \qquad \text{tip} = 0.20 \times 64.00 = 12.80$$ $$64.00 + 3.84 + 12.80 = 80.64$$

Answer: $80.64

Example 5 — Estimate, then compute

Estimate the total on a $\$47.30$ bill with a $20\%$ tip, then compute it exactly.

Round to $\$50.00$: $20\%$ is $10.00, so the total is about $60.00.

$$0.20 \times 47.30 = 9.46 \qquad 47.30 + 9.46 = 56.76$$

The price was rounded up by $2.70, so the estimate came out high by more than $3.

Answer: estimate about $60.00; exact total $56.76

Guided practice

73. Find a $15\%$ tip on a $\$40.00$ bill and the total.
74. Find a $20\%$ tip on a $\$35.00$ bill and the total.
75. Find an $18\%$ tip on a $\$50.00$ bill and the total.
76. Find a $15\%$ tip on a $\$62.00$ bill and the total.
77. Find a $20\%$ tip on a $\$24.60$ bill and the total.
78. Estimate a $15\%$ tip on a $\$38.40$ bill by rounding the bill to $\$40.00$.

Independent practice

79. Find the tip and the total. a) $\$56.00$ bill, $15\%$ tip b) $\$83.00$ bill, $20\%$ tip c) $\$27.50$ bill, $18\%$ tip d) $\$46.75$ bill, $15\%$ tip
80. A bill is $\$72.80$ and the tip is $20\%$. Find the total.
81. A bill is $\$19.99$ and the tip is $18\%$. Find the tip and the total.
82. A group's bill is $\$118.00$ and the tip is $18\%$. Find the total.
83. Reasoning. Explain the "find $10\%$, then add half of it" method for a $15\%$ tip, and why it works.
84. Application. A meal costs $\$64.00$. Tax is $6\%$ and the tip is $20\%$ of the pre-tax bill. Find the tax, the tip, and the total.
85. Error analysis. A diner computes a $20\%$ tip on a $\$45.00$ bill as $9.00 and leaves $9.00 on the table as payment for the whole meal. Explain the error and give the amount that should be paid.
86. Estimate, then check. Estimate the total on a $\$47.30$ bill with a $20\%$ tip, then compute it exactly and say how far off the estimate was.

Exit ticket 3.5

87. Find a $15\%$ tip on a $\$20.00$ bill and the total.
88. Find a $20\%$ tip on a $\$41.50$ bill and the total.
89. Find an $18\%$ tip on a $\$36.00$ bill and the total.
90. Explain why a bill-plus-tip total is always more than $100\%$ of the bill.

Lesson 3.6 — Percent Increase and Percent Decrease

A different question

The first five lessons all gave you a percent and asked for an amount. This lesson runs the other way: you are given two amounts and asked for the percent.

If a price moves from an original amount to a new amount, the change is a percent increase when the price went up and a percent decrease when it went down. Either way:

$$\text{percent change} = \frac{\text{amount of change}}{\text{original amount}} \times 100\%$$

where the amount of change is the positive difference between the two amounts.

A price that goes from $\$40.00$ to $\$46.00$ increased by $6.00, and

$$\frac{6}{40} = 0.15 = 15\%.$$

A price that goes from $\$50.00$ to $\$42.00$ decreased by $8.00, and

$$\frac{8}{50} = 0.16 = 16\%.$$

The denominator is always the original

This is the whole lesson in one sentence, and it is where nearly every error lives. A percent change compares the change to where it started, not to where it ended.

Take $\$40.00$ rising to $\$50.00$. The change is $10.00, and dividing by the original gives $\frac{10}{40} = 0.25 = 25\%$. Divide by the new amount instead and you get $\frac{10}{50} = 0.20 = 20\%$ — a different number, and the wrong answer to the question asked.

Both of those percents mean something, which is why the mistake is easy to make. $25\%$ is how much the price went up from $40.00. $20\%$ is how much of the new price the raise accounts for. Only the first is the percent increase.

Here is a consequence worth sitting with. A price going from $40.00 to $50.00 is a $25\%$ increase, but going from $50.00 to $40.00 is a $20\%$ decrease. Same $10.00 of movement, different percents, because the original amount is different each time. Percent increase and percent decrease are not mirror images.

Three steps, in order

1. Find the amount of change: subtract the smaller amount from the larger.
2. Divide by the original amount.
3. Write the quotient as a percent, and name it an increase or a decrease.

On a rent that rises from $\$850$ to $\$901$:

$$901 - 850 = 51 \qquad \frac{51}{850} = 0.06 = 6\%$$

so that is a $6\%$ increase.

Rounding percents. Most problems in this chapter come out to an exact percent or an exact tenth of a percent. When a quotient does not terminate, round the percent to the nearest tenth of a percent and say that you did.

It does not have to be money

Percent change applies to any quantity that has a before and an after. A town's recycling tonnage rising from 1,250 tons to 1,400 tons is an increase of 150 tons, and

$$\frac{150}{1250} = 0.12 = 12\%,$$

a $12\%$ increase. Notice that no dollar sign appeared anywhere, and nothing about the method changed.

Worked examples

Example 1 — A percent increase

A price rises from $\$40.00$ to $\$46.00$. Find the percent increase.

$$46 - 40 = 6 \qquad \frac{6}{40} = 0.15$$

Answer: a $15\%$ increase

Example 2 — A percent decrease

A price falls from $\$80.00$ to $\$60.00$. Find the percent decrease.

$$80 - 60 = 20 \qquad \frac{20}{80} = 0.25$$

Answer: a $25\%$ decrease

Example 3 — A change with cents

A price rises from $\$12.00$ to $\$13.50$. Find the percent increase.

$$13.50 - 12.00 = 1.50 \qquad \frac{1.50}{12.00} = 0.125$$

Answer: a $12.5\%$ increase

Example 4 — Naming the direction

An amount changes from $\$36.00$ to $\$27.00$. Is that an increase or a decrease, and by what percent?

The amount went down, so it is a decrease.

$$36 - 27 = 9 \qquad \frac{9}{36} = 0.25$$

Answer: a $25\%$ decrease

Example 5 — Catching the wrong denominator

A student says that $\$60.00$ rising to $\$75.00$ is a $20\%$ increase, because $\frac{15}{75} = 0.20$. Is that right?

The change is $75 - 60 = 15$, which is correct. But $75$ is the new amount, not the original. Dividing by the original gives

$$\frac{15}{60} = 0.25.$$

Answer: No — it is a $25\%$ increase

Guided practice

91. A price rises from $\$40.00$ to $\$46.00$. Find the amount of increase and the percent increase.
92. A price falls from $\$50.00$ to $\$42.00$. Find the amount of decrease and the percent decrease.
93. An amount rises from $\$25.00$ to $\$30.00$. Find the percent increase.
94. An amount falls from $\$80.00$ to $\$60.00$. Find the percent decrease.
95. A price rises from $\$12.00$ to $\$13.50$. Find the percent increase.
96. An amount changes from $\$36.00$ to $\$27.00$. Say whether it is an increase or a decrease, and by what percent.

Independent practice

97. Find the percent change and name it an increase or a decrease. a) $\$20.00$ to $\$26.00$ b) $\$75.00$ to $\$60.00$ c) $\$48.00$ to $\$54.00$ d) $\$9.60$ to $\$8.40$
98. A monthly rent rises from $\$850.00$ to $\$901.00$. Find the percent increase.
99. Attendance at a school play falls from 240 people to 216 people. Find the percent decrease.
100. A phone plan goes from $\$45.00$ to $\$49.50$ per month. Find the percent increase.
101. A used bike listed at $\$180.00$ sells for $\$153.00$. Find the percent decrease.
102. Reasoning. Explain why the original amount is always the denominator. Use $\$40.00$ rising to $\$50.00$ to show what answer you get if you divide by the new amount instead.
103. Application. A town's recycling rises from 1,250 tons to 1,400 tons in a year. Find the percent increase.
104. Error analysis. A student says $\$60.00$ rising to $\$75.00$ is a $20\%$ increase because $\frac{15}{75} = 0.20$. Explain the error and give the correct percent increase.

Exit ticket 3.6

105. A price rises from $\$50.00$ to $\$58.00$. Find the percent increase.
106. A price falls from $\$120.00$ to $\$96.00$. Find the percent decrease.
107. A price rises from $\$16.00$ to $\$18.40$. Find the percent increase.
108. Explain the difference between "the price decreased by $20\%$" and "the new price is $20\%$ of the original price."

Chapter 3 Review

Vocabulary. percent · benchmark percentage · estimate · discount · discount amount · sale price · complement multiplier · cost · markup · markup amount · selling price · sales tax · subtotal · total · tip · original amount · new amount · amount of change · percent increase · percent decrease

Part A — One discount or one markup and the resulting price (8.CE.1a)

109. A $\$95.00$ item is $20\%$ off. Find the discount amount and the sale price.
110. A $\$42.60$ item is $25\%$ off. Find the sale price.
111. An item costs $\$56.00$ and is marked up $30\%$. Find the selling price.
112. An item costs $\$23.40$ and is marked up $45\%$. Find the selling price.
113. Estimate, then compute. A $\$118.50$ item is $40\%$ off. Estimate the sale price, then find it exactly.
114. Application. A coat priced at $\$68.00$ is $15\%$ off. How much does a shopper save, and what does the coat cost?

Part B — Sales tax, tip, and the resulting total (8.CE.1b)

115. A purchase is $\$64.00$ and the tax rate is $5.3\%$. Find the total.
116. A purchase is $\$31.25$ and the tax rate is $6\%$. Find the tax and the total.
117. A bill is $\$54.00$ and the tip is $18\%$. Find the total.
118. A bill is $\$88.40$ and the tip is $20\%$. Find the total.
119. A grocery order is $\$27.90$ and the reduced rate is $1\%$. Find the total.
120. Application. A meal costs $\$45.60$. Tax is $6\%$ and the tip is $15\%$ of the pre-tax bill. Find the tax, the tip, and the total.

Part C — Percent increase and percent decrease (8.CE.1c)

121. A price rises from $\$80.00$ to $\$92.00$. Find the percent increase.
122. A price falls from $\$150.00$ to $\$120.00$. Find the percent decrease.
123. A price rises from $\$18.00$ to $\$20.70$. Find the percent increase.
124. A club's membership falls from 320 members to 272 members. Find the percent decrease.
125. A price rises from $\$64.00$ to $\$72.00$. Find the percent increase.
126. A jacket's price falls from $\$96.00$ to $\$81.60$. Find the percent decrease.

Part D — Mixed application and reasoning

127. Application. A $\$54.00$ pair of jeans is $25\%$ off, and $5.3\%$ sales tax is charged on the sale price. Find the sale price and the total.
128. Application. A restaurant bill is $\$78.50$ and the diner leaves a $20\%$ tip. Find the total.
129. Reasoning. Explain why the multiplier for a $35\%$ discount is $0.65$ while the multiplier for a $35\%$ markup is $1.35$.
130. Error analysis. A student finds the percent decrease from $\$80.00$ to $\$68.00$ by computing $\frac{12}{68} \approx 0.176$ and answers about $17.6\%$. Explain the error and give the correct percent decrease.
131. Estimate and judge. Without computing exactly, decide whether a $\$29.90$ item at $30\%$ off costs more or less than $\$20.00$, and justify your reasoning. Then compute the sale price exactly.
132. Reasoning. One store raises a price from $\$40.00$ to $\$50.00$. Another store lowers a price from $\$50.00$ to $\$40.00$. Both prices moved by $\$10.00$. Are the two percent changes equal? Explain.

Standards coverage check — Chapter 3

Estimation is required by all three bullets, not just one, so it is taught first in Lesson 3.1 and then practiced inside each application lesson: items 5, 11, 18, 24, 32, 42, 50, 60, 68, 78, 86, 113, and 131 all ask for an estimate before or beside an exact answer.

Two limits from the standard are enforced throughout. Every problem applies exactly one discount or markup — no item chains two discounts, and no item combines a discount with a markup. Every tax and tip problem asks for the resulting total, not just the tax or tip amount.

Grade 7 Chapter 6 covered finding a percent of a number, including benchmark and estimation strategies; this chapter assumes that skill and applies it to money.

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