Math 7 · 7.RP.A.3

Discounts, sales tax, tips and commission

A jacket marked 80 dollars is 25% off, and 8% sales tax is added at the till. What do you actually pay, and why is the answer not simply 17% off?

You need

Two percent changes one after the otherMath 7 Benchmark percents and mental percentMath 6

Where you are 3 in Percent in money situations

Fractions, ratio and percent3

Percent8

Percent in money situations3

Markup and margin

Discounts, sales tax, tips and commission

Simple interest

When you reach for it

Often the amount paid is not the number on the label. A pair of shoes is 30\% off. A bookshop adds sales tax at the till. A café bill has a space for a tip. A salesperson in a bike shop is paid a share of every bike sold. Each time, an amount is worked out as a percent of some other amount, and then either taken away or added on.

That makes all four the same piece of mathematics, the question from the percent equation: a part is a rate times a whole. The work in each situation is deciding two things:

which amount is the whole, the base the percent is taken of, and
whether the part is taken away from it or added to it.

Situation	The whole	The part is
Discount	original price	taken off
Sales tax	price paid	added
Tip	bill	added
Commission	sales	paid to the seller

The lesson takes two days. Part 1 is discounts, worked forwards and backwards. Part 2 is sales tax, tips, commission, and a discount followed by tax.

The procedure

Part 1: discount and sale price

A shop marks a pair of shoes at 60 dollars and puts up a sign: 30\% off. Four words name the amounts involved.

The original price is the price before the sale, here 60 dollars.
The discount rate is the percent taken off, here 30\%.
The amount of discount is how many dollars come off.
The sale price is what is left to pay.

Find the amount of discount. “30\% off” means 30\% of the original price comes off. The original price is the whole because the sign is about the price on the label; there is no other price yet. So the discount is the rate times the original price:

\text{discount} = \text{rate} \times \text{original price}

0.30 \times 60 = 18

Subtract to get the sale price. The sale price is whatever is left once the discount is taken away:

\text{sale price} = \text{original price} - \text{discount}

60 - 18 = 42

The shoes cost 42 dollars in the sale.

Figure 1Thirty percent off a 60 dollar price. Three tenths of the bar come off, and the seven tenths left are the sale price.

Or do it in one multiplication. The figure shows a second way to read the same bar. If 30\% of the price comes off, 100\% - 30\% = 70\% of it is left, so the sale price is 70\% of the original:

0.70 \times 60 = 42

This is allowed because of the distributive property. Taking away 0.30 \times 60 from 60 is

\begin{aligned} 60 - 0.30 \times 60 &= 1 \times 60 - 0.30 \times 60 \\ &= (1 - 0.30) \times 60 \\ &= 0.70 \times 60 \end{aligned}

Write p for the original price, s for the sale price and r for the discount rate as a decimal. The same step works for every rate, so

s = (1 - r)p

The number 1 - r is the multiplier for a decrease, the subject of the multiplier for a percent change. Two steps also give the amount of discount; one step is quicker when only the sale price matters.

Working backwards: the discount rate. Sometimes the original and sale prices are both known and the rate is not. Subtract to get the amount of discount, then ask what percent it is of the original price, as in finding what percent one number is of another:

\text{discount rate} = \frac{\text{discount}}{\text{original price}}

Working backwards: the original price. If the sale price and the rate are known, the sale price is the part that was kept, and the fraction kept is 1 - r. Dividing both sides of s = (1 - r)p by 1 - r gives

p = \frac{s}{1 - r}

This is finding the whole from a part and a percent, with the part being the price paid.

A fixed amount off. Some offers are a number of dollars, such as “10 dollars off”. There is no rate to multiply: the sale price is the original price minus 10, worth the same on every price, while a percent off is worth more on a dearer item.

Part 2: sales tax, tips and commission

Sales tax. In many places a sales tax is added at the till: a percent of the price of what is bought. Every tax rate here is an example. Real rates and rules differ from place to place, and in some countries the tax is already in the marked price.

The tax is a percent of the price being paid, so that price is the whole:

\text{sales tax} = \text{tax rate} \times \text{price}

\text{total} = \text{price} + \text{sales tax}

Since the tax is added, the total is 100\% of the price plus the tax rate. By the same distributive step as before, for a tax rate t the total is (1 + t) times the price. Money is paid in whole cents, so a tax that comes out with more than two decimal places is rounded to the nearest cent.

Tips. A tip is money left for the people who served a meal, worked out as a percent of the bill. The customer chooses the percent; here it is always given, and taken of the bill before any tax. The bill is the whole, and the tip is added on.

Tips are usually worked out in the head, using benchmark percents. 10\% of a bill is the bill with its decimal point moved one place left, because dividing by ten moves every digit one place right. 5\% is half of that, and 20\% is double it.

Commission. Some salespeople are paid a commission: a percent of the value of what they sell. The whole is the value of the sales, and the commission is the part the salesperson receives:

\text{commission} = \text{rate} \times \text{sales}

Nothing is added to the customer’s price; the commission is a share of money already paid. Some salespeople also get a fixed base pay, and their pay is the base pay plus the commission.

A discount and then tax. This lesson takes the tax to be charged on the price actually paid, which after a discount is the sale price; each problem says so. Take the jacket marked 80 dollars, 25\% off, with 8\% tax:

\begin{aligned} \text{sale price} &= 0.75 \times 80 = 60 \\ \text{tax} &= 0.08 \times 60 = 4.80 \\ \text{total} &= 60 + 4.80 = 64.80 \end{aligned}

Figure 2The discount is a quarter of 80. The tax is 8\% of the smaller bar, 60, which is why the two percents cannot simply be subtracted.

Each step was one multiplication, the first by 0.75 and the second by 1.08, so the total is the original price times both multipliers, as in two percent changes one after the other:

\begin{aligned} 0.75 \times 1.08 \times 80 &= 0.81 \times 80 \\ &= 64.80 \end{aligned}

Two facts follow. First, the total is not 25\% - 8\% = 17\% off. The 25\% is a percent of 80, while the 8\% is a percent of the smaller 60; percents of different wholes cannot be combined by subtracting them. The single multiplier 0.81 says the jacket costs 81\% of its marked price, which is 19\% off. Second, the order of the two percents does not change the total. Multiplying 80 by 1.08 first and then by 0.75 gives the same product, by the commutative property of multiplication.

Worked examples

Examples 1 to 4 go with Part 1. Examples 5 to 8 go with Part 2.

Example 1

A bike in the sale

A bike shop takes 15\% off a bike whose original price is 240 dollars. Find the amount of discount and the sale price.

Solution

The discount is 15\% of the original price, which is the whole:

0.15 \times 240 = 36

The amount of discount is 36 dollars. The sale price is what is left:

240 - 36 = 204

The bike costs 204 dollars.

Check in two ways. With benchmark percents, 10\% of 240 is 24 and 5\% is half that, 12, so 15\% is 24 + 12 = 36. With the multiplier, 85\% of the price is left, and 0.85 \times 240 = 204.

Example 2

Finding the discount rate

A pair of sneakers is marked 75 dollars and sells in a sale for 63 dollars. What is the discount rate?

Solution

First the amount of discount, the difference between the two prices:

75 - 63 = 12

The rate is the percent that 12 is of the original price, 75, because a sale sign gives its percent of the label price:

\frac{12}{75} = \frac{4}{25} = \frac{16}{100} = 16\%

The step from \displaystyle \frac{4}{25} to \displaystyle \frac{16}{100} multiplies the top and bottom by 4. The sneakers are 16\% off.

Check: 16\% of 75 is 0.16 \times 75 = 12, the amount of discount.

Example 3

Finding the original price

Rosalind pays 44.20 dollars for a rucksack in a 35\% off sale. What was the original price?

Solution

The 44.20 dollars is not the discount; it is what Rosalind paid, the part of the price that was kept. If 35\% came off, 100\% - 35\% = 65\% was kept. So 44.20 is 65\% of the original price p:

0.65p = 44.20

Dividing both sides by 0.65:

p = \frac{44.20}{0.65} = \frac{4420}{65} = 68

The rucksack was 68 dollars before the sale.

Check: 35\% of 68 is 23.80, and 68 - 23.80 = 44.20, which is what Rosalind paid.

Example 4

Two offers compared

A bike shop gives out a voucher that a customer can use in one of two ways: 30 dollars off, or 25\% off. Which is better on a 96 dollar lock? On a 200 dollar wheel? For what price are the two the same?

Solution

On the lock, 25\% of 96 is 0.25 \times 96 = 24 dollars, less than 30. The fixed 30 dollars off is better: the lock costs 96 - 30 = 66 dollars rather than 96 - 24 = 72.

On the wheel, 25\% of 200 is 50 dollars, more than 30. Now the percent is better: the wheel costs 200 - 50 = 150 dollars rather than 200 - 30 = 170.

The two are worth the same when 25\% of the price p is exactly 30:

0.25p = 30, \qquad p = \frac{30}{0.25} = 120

At 120 dollars both take off 30. Below 120 the 30 dollars off is better, and above 120 the 25\% off is better.

Example 5

Sales tax, and rounding to the cent

Nikolai buys a bike pump priced at 46.80 dollars where the sales tax rate is 7.25\%. Find the tax and the total.

Solution

The tax is 7.25\% of the price. As a decimal, 7.25\% = 0.0725:

0.0725 \times 46.80 = 3.393

A tax is paid in whole cents, so it is rounded to two decimal places. The third decimal place is 3, less than 5, so the tax rounds down to 3.39 dollars. The total is

46.80 + 3.39 = 50.19

dollars.

Check with the multiplier: 1.0725 \times 46.80 = 50.193, which also rounds to 50.19.

Example 6

A tip worked out in the head

Oona and a friend have lunch at a café, and the bill is 42.60 dollars. How much is a 15\% tip, and how much is a 20\% tip? What does each make the total?

Solution

Start from 10\%, which moves the decimal point one place left: 10\% of 42.60 is 4.26.

For 15\%, add half of that again, since 5\% is half of 10\%. Half of 4.26 is 2.13, so the tip is

4.26 + 2.13 = 6.39

dollars, and the total is 42.60 + 6.39 = 48.99 dollars.

For 20\%, double the 10\%: the tip is 2 \times 4.26 = 8.52 dollars, and the total is 42.60 + 8.52 = 51.12 dollars.

Example 7

A salesperson’s commission

Prakash sells bikes and is paid a 6\% commission on his sales. One week he sells bikes worth 3{,}450 dollars. What is his commission? In the same week Mbali, at another shop, earns 168 dollars of commission on sales of 2{,}400 dollars. What is her commission rate?

Solution

Prakash’s commission is 6\% of the value of his sales:

0.06 \times 3{,}450 = 207

He earns 207 dollars of commission.

For Mbali the commission and the sales are known, and the rate is the percent that 168 is of 2{,}400. Dividing top and bottom by 24 makes the bottom 100:

\frac{168}{2{,}400} = \frac{7}{100} = 7\%

Her commission rate is 7\%.

Check: 0.07 \times 2{,}400 = 168.

Example 8

Working back from a receipt

A till receipt from a clothes shop shows a total of 71.28 dollars for a coat bought at 20\% off, with 8\% sales tax on the sale price. What was the coat’s original price?

Solution

Going forwards, the original price p was multiplied by 0.80 for the discount and then by 1.08 for the tax. Together that is one multiplication by

0.80 \times 1.08 = 0.864

so the total is 0.864p:

0.864p = 71.28

Undo the multiplication by dividing both sides by 0.864:

p = \frac{71.28}{0.864} = \frac{71{,}280}{864} = 82.50

The coat’s original price was 82.50 dollars.

Check forwards: 20\% off 82.50 is 0.80 \times 82.50 = 66, the tax on that is 0.08 \times 66 = 5.28, and 66 + 5.28 = 71.28, the total on the receipt.

Boundaries

The rate says what it is a percent of. A discount rate is a percent of the original price. A sales tax is a percent of the price being paid, which after a discount is the sale price. A tip is a percent of the bill, and in this lesson of the bill before tax; if a problem says to tip on the total after tax, the whole is that total instead.

A discount rate is between 0\% and 100\%. At 100\% off the sale price is 0. A rate above 100\% would make the sale price negative, the shop paying the customer, which is not a discount. And at exactly 100\% off, the original price cannot be found from the sale price: the sale price is 0 whatever the original was, and the step of dividing by 1 - r would divide by zero.

Percent changes in a row combine by multiplying, never by adding or subtracting. A discount followed by tax is the product of the two multipliers, and so are two discounts: 25\% off and then a further 20\% off is 0.75 \times 0.80 = 0.60, which is 40\% off, not 45\%. More on this is in two percent changes one after the other.

The order does not matter only when both changes are percents. Two multiplications can be done in either order, but a fixed amount off is a subtraction, and a subtraction and a multiplication do not commute. Take a 10 dollar voucher on a 50 dollar order with 8\% tax. If the tax is worked out on the price paid after the voucher, the total is 1.08 \times (50 - 10) = 43.20 dollars; if the tax is worked out on the full 50 first and the voucher taken off after, it is 1.08 \times 50 - 10 = 44 dollars. Which of these a real till does depends on the shop and on local tax rules, so a problem has to state it; this lesson makes no claim about any real place.

Rounding can move an answer by a cent. The tax is rounded to whole cents and then added. Rounding at a different step, or using one combined multiplier and rounding only at the end, usually gives the same total but can be a cent off it.

Where this goes next

The same shape, a percent of a base added or taken away, runs through the rest of percent in money situations. A shop’s markup is a percent of what the shop paid for an item, added on to make the price it sells for: it works like a tax, with what the shop paid as the whole instead of the price. Simple interest is a percent of an amount borrowed or saved, charged once for each year.

Exercises

Part 1: discount and sale price

Find the amount of discount and the sale price.

A shirt has an original price of 45 dollars and is 20\% off.
A pair of boots has an original price of 130 dollars and is 35\% off.
A bike bell has an original price of 12.80 dollars and is 25\% off.

Part 1: the rate and the original price

A scarf marked 90 dollars is on sale for 72 dollars. What is the discount rate?
Quinlan pays 34 dollars for a hoodie at 15\% off. What was the original price?
A cycling jacket sells for 57.60 dollars after 40\% off. What was the original price?
A bookshop offers either 12 dollars off an order or 30\% off it. Which is better on an order of 32 dollars? On an order of 50 dollars? For what order total are the two the same?

Part 2: sales tax

Find the tax and the total. Round any tax to the nearest cent.

A pair of shoes priced at 64 dollars, with a tax rate of 5\%.
A bike lock priced at 23.50 dollars, with a tax rate of 8\%.
Books priced at 37.40 dollars, with a tax rate of 6.5\%.
A receipt shows a price of 250 dollars and a sales tax of 17.50 dollars. What is the tax rate?

Part 2: tips

A café bill is 18.40 dollars. Find a 15\% tip and the total.
A restaurant bill is 63.50 dollars. Find a 20\% tip and the total.
Rosalind leaves a 9 dollar tip on a 60 dollar bill. What percent of the bill is the tip?

Part 2: commission

A salesperson in a shoe shop is paid a 5\% commission on sales of 4{,}260 dollars. What is the commission?
Prakash is paid a base pay of 400 dollars a week and a 3\% commission on his sales. One week he sells 9{,}500 dollars of bikes. What is his pay that week?
Nikolai earns a commission of 162 dollars on sales of 3{,}600 dollars. What is his commission rate?

Part 2: a discount and then tax

In these, the tax is charged on the price paid after the discount unless the exercise says otherwise.

A jacket has an original price of 120 dollars. It is 30\% off, and then 6\% sales tax is charged on the sale price. Find the sale price, the tax and the total.
A set of bike lights is marked 95 dollars. It is 20\% off, and then 5\% sales tax is charged on the sale price. Find the total.
A receipt shows a total of 48.60 dollars for a shirt bought at 10\% off, with 8\% tax on the sale price. What was the original price?
Mbali has a voucher for 15 dollars off a 70 dollar pair of shoes, and the tax rate is 6\%. Find the total if the tax is worked out after the voucher comes off, and the total if the tax is worked out on the full price and the voucher comes off after. Why are they different?

Reasoning

A price is reduced by 30\%, and then 5\% tax is charged on the sale price. Find the single multiplier from the original price to the total, and say by what percent the total is below the original price. Explain why the answer is not 25\%.
Oona says that taking 15\% off and then adding 6\% tax gives a different total from adding 6\% tax and then taking 15\% off. Explain why she is wrong.
Explain, using the distributive property, why a sale price is always (1 - r) times the original price, where r is the discount rate as a decimal.
Explain why the original price cannot be worked out from the sale price when an item is 100\% off.

Answers

Discount 9 dollars, sale price 36 dollars. 0.20 \times 45 = 9, and 45 - 9 = 36.
Discount 45.50 dollars, sale price 84.50 dollars. 0.35 \times 130 = 45.50, and 130 - 45.50 = 84.50.
Discount 3.20 dollars, sale price 9.60 dollars. 25\% is a quarter, and 12.80 \div 4 = 3.20; 12.80 - 3.20 = 9.60.
20\%. The discount is 90 - 72 = 18, and \displaystyle \frac{18}{90} = \frac{1}{5} = 20\%.
40 dollars. 85\% of the price was kept, so 0.85p = 34 and \displaystyle p = \frac{34}{0.85} = 40.
96 dollars. 60\% was kept, so 0.60p = 57.60 and \displaystyle p = \frac{57.60}{0.60} = 96.
On 32 dollars the 12 dollars off is better, since 30\% of 32 is only 9.60. On 50 dollars the 30\% off is better, since it takes off 15. They are the same at 40 dollars: 0.30p = 12 gives p = 40.
Tax 3.20 dollars, total 67.20 dollars. 0.05 \times 64 = 3.20.
Tax 1.88 dollars, total 25.38 dollars. 0.08 \times 23.50 = 1.88.
Tax 2.43 dollars, total 39.83 dollars. 0.065 \times 37.40 = 2.431, which rounds to 2.43, and 37.40 + 2.43 = 39.83.
7\%. \displaystyle \frac{17.50}{250} = \frac{7}{100} = 7\%.
Tip 2.76 dollars, total 21.16 dollars. 10\% is 1.84 and 5\% is 0.92, and 1.84 + 0.92 = 2.76.
Tip 12.70 dollars, total 76.20 dollars. 10\% is 6.35, doubled is 12.70.
15\%. \displaystyle \frac{9}{60} = \frac{15}{100} = 15\%.
213 dollars. 0.05 \times 4{,}260 = 213.
685 dollars. The commission is 0.03 \times 9{,}500 = 285, and 400 + 285 = 685.
4.5\%. Dividing top and bottom by 36, \displaystyle \frac{162}{3{,}600} = \frac{4.5}{100} = 4.5\%.
Sale price 84 dollars, tax 5.04 dollars, total 89.04 dollars. 0.70 \times 120 = 84, 0.06 \times 84 = 5.04, and 84 + 5.04 = 89.04.
79.80 dollars. The sale price is 0.80 \times 95 = 76, the tax is 0.05 \times 76 = 3.80, and 76 + 3.80 = 79.80.
50 dollars. The combined multiplier is 0.90 \times 1.08 = 0.972, so 0.972p = 48.60 and \displaystyle p = \frac{48.60}{0.972} = 50.
Voucher first: 70 - 15 = 55, the tax is 0.06 \times 55 = 3.30, and the total is 58.30 dollars. Tax on the full price first: 1.06 \times 70 = 74.20, and 74.20 - 15 = 59.20 dollars. They differ by 0.90 dollars, which is 6\% of 15: when the voucher comes off first, no tax is charged on those 15 dollars.
The multiplier is 0.70 \times 1.05 = 0.735, so the total is 73.5\% of the original price, 26.5\% below it. It is not 25\% because the 30\% is a percent of the original price and the 5\% is a percent of the smaller sale price, so the tax is only 5\% of 70\%, that is 3.5\% of the original price, and 30\% - 3.5\% = 26.5\%.
Each change is a multiplication: by 0.85 for the discount and by 1.06 for the tax. Either order multiplies the price by 0.85 and by 1.06, and by the commutative property of multiplication 0.85 \times 1.06 \times p = 1.06 \times 0.85 \times p. Both orders give 0.901p.
The amount of discount is r \times P for an original price P, so the sale price is P - rP. Writing P as 1 \times P and using the distributive property in reverse, 1 \times P - r \times P = (1 - r) \times P.
At 100\% off, r = 1, so the sale price is (1 - 1) \times P = 0 for every original price P. Knowing the sale price is 0 says nothing about P. In the equation (1 - r)P = \text{sale price} the step of dividing by 1 - r would divide by zero, which is not allowed.