Does it matter whether I apply the discount or the tax first?

For the final total, no — multiplication is commutative, so 60 × 0.75 × 1.07 and 60 × 1.07 × 0.75 both equal 48.15. In the real world stores apply the discount first because tax is charged on what you actually pay, and if a problem asks for the intermediate sale price you must follow the stated order. But the final amount comes out the same either way.

Why can't I add two discount percents together?

Because the second discount is taken on the already-reduced price, not the original. Twenty percent off, then 10 percent off, gives 0.80 × 0.90 = 0.72, so you keep 72 percent and save 28 percent — not 30 percent. Multiplying the multipliers always gives the correct combined effect.

How do I know if the multiplier should be more or less than 1?

Ask whether the price is going up or down. Tax, tip, markup, and fees all increase the price, so the multiplier is 1 + r and is greater than 1. Discounts, sales, coupons, and markdowns decrease the price, so the multiplier is 1 - r and is between 0 and 1. If your final answer moved the wrong direction, you used the wrong multiplier.

When should I round in a money problem?

Round at the very end, to the nearest cent, unless the problem says otherwise. If you round in the middle and then keep multiplying, the small rounding error grows and your final answer can be off by a penny or more. Keep the extra decimals in your calculator until the last step.

M7MATH-4.3

Tax, Tip, Discount & Markup

Learn to solve tax, tip, discount, and markup problems in one step using a single multiplier, then chain multipliers for multistep price problems.

What you'll do in this lesson

A voice-first session with the Crimsora tutor on Tax, Tip, Discount & Markup, then targeted practice and FRQs — with the tutor adapting to where you get stuck.

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What this lesson covers

Every price you see at a store has been through a chain of percent changes. A store buys a jacket at wholesale, marks it up, later puts it on clearance at 30 percent off, and then the register adds sales tax. Each of those steps is a percent problem — and each one can be done with a single multiplication instead of two or three separate calculations.

In this lesson you will learn to turn any tax, tip, discount, or markup into one decimal multiplier, and then multiply those multipliers together to handle several steps at once. This is faster, easier to check, and it is exactly how spreadsheets and cash registers do it. You will also see the most common trap in these problems: assuming the order of the steps changes the answer, and mixing up the percent with the multiplier.

From Two Steps to One Multiplier

The slow way to add 8 percent sales tax to a 45 dollar item is to find the tax and then add it:

0.08 \times 45 = 3.60

, then

45 + 3.60 = 48.60

. The fast way uses the distributive property. Since you want the whole price plus 8 percent of it, you want

100\% + 8\% = 108\%

of the price:

45 \times 1.08 = 48.60

Same answer, one step. The number

1.08

is the multiplier. Anything that increases a price (tax, tip, markup) gives a multiplier greater than 1. Anything that decreases a price (discount, sale, coupon) gives a multiplier less than 1, because you keep

100\% - r\%

of the price.

Situation	Percent	Multiplier
8% sales tax	increase 8%	$1.08$
20% tip	increase 20%	$1.20$
45% markup	increase 45%	$1.45$
25% off	decrease 25%	$0.75$
30% discount	decrease 30%	$0.70$
5% off coupon	decrease 5%	$0.95$

Where students go wrong: writing

0.25

as the multiplier for "25 percent off." Multiplying by

0.25

gives the amount of the discount, not the sale price. If the problem asks what you pay, use

0.75

. If it asks how much you saved, use

0.25

. Reading the question carefully decides which one you need.

Also watch the decimal placement: 8 percent is

0.08

, not

0.8

. A quick sanity check catches this — tax should be a small addition, so an answer near 81 dollars for a 45 dollar item is clearly wrong.

Discount, Markup, Tax, and Tip: What Each One Means

These four words describe different real-world situations, but mathematically they are all just percent increase or percent decrease applied to a starting price.

A discount (or sale, or markdown) reduces a price. The starting price is the original or list price, and the result is the sale price.

A markup increases a price. Stores buy at wholesale cost and mark it up to get the retail price they charge customers. A 60 percent markup means the store adds 60 percent of its cost, so retail

= 1.60 \times

cost.

Sales tax is a percent added by the government at the register. It is applied to the price the store actually charges you — which means, at a real store, tax comes after any discount.

A tip (or gratuity) is a percent added voluntarily for service, usually calculated on the food total.

Term	Direction	Applied to	Multiplier form
Discount	down	original price	$1 - r$
Markup	up	store's cost	$1 + r$
Sales tax	up	selling price	$1 + r$
Tip	up	bill before tip	$1 + r$

One more distinction that confuses students: markup percent here is based on the store's cost, not on the selling price. If a store pays 20 dollars and sells for 30 dollars, the markup is 10 dollars, and

10 \div 20 = 0.5

, a 50 percent markup. Always divide the change by the starting amount, which is the cost.

Finally, remember that percents are rates, not fixed amounts. A 15 percent tip on a 40 dollar meal is 6 dollars; on an 80 dollar meal it is 12 dollars. The percent stays the same while the dollar amount doubles.

Chaining Multipliers for Multistep Problems

When a problem has two or more percent steps, you multiply all the multipliers in a row. A 60 dollar sweatshirt is 25 percent off, and then 7 percent sales tax is added:

60 \times 0.75 \times 1.07 = 48.15

You can also do it in stages:

60 \times 0.75 = 45

, then

45 \times 1.07 = 48.15

. Both are correct; the chain just saves rounding trouble in the middle.

A surprising fact: multiplication is commutative, so the order of the multipliers does not change the final answer. Taxing first and then discounting gives the same total. Stores apply the discount first because tax is legally charged on what you actually pay, but arithmetically

60 \times 1.07 \times 0.75

is also

48.15

.

What you may not do is add or subtract the percents. Take 20 percent off, then take another 10 percent off:

100 \times 0.80 \times 0.90 = 72

That is a 28 percent total discount, not 30 percent. The second discount applies to the already-reduced price, so it takes off fewer dollars. Combining the multipliers,

0.80 \times 0.90 = 0.72

, shows this directly: you keep 72 percent, so you saved 28 percent.

Rounding matters in money problems. Round only at the very end, to the nearest cent, unless the problem says otherwise. If you round an intermediate tax amount to the penny and then keep multiplying, your final answer can be off by a cent or two.

One caution about rounding tips: many people round the tip up to a convenient amount. Unless the problem tells you to, compute the exact percent.

Working Backwards to the Original Price

Sometimes you know the final price and need the price before the percent change. Since forward means multiply by the multiplier, backward means divide by it.

A shirt costs 25.80 dollars after 20 percent off. What was the original price? The multiplier for 20 percent off is

0.80

, and

\text{original} \times 0.80 = 25.80 \quad\Rightarrow\quad \text{original} = \frac{25.80}{0.80} = 32.25

A very common wrong move is to add 20 percent back onto 25.80:

25.80 \times 1.20 = 30.96

, which is not 32.25. The reason is that the 20 percent was taken off the larger original price, so adding 20 percent of the smaller sale price gives back too little. Dividing undoes multiplication exactly; adding a percent of a different starting number does not.

The same logic works for tax. If a total of 53.50 dollars includes 7 percent tax, the pre-tax price is

53.50 \div 1.07 = 50

.

A reliable check: take your answer and run it forward. Does

32.25 \times 0.80

equal 25.80? Yes, so the answer is right. This forward check catches almost every error in backward problems, and it takes about ten seconds.

Summary of the two directions:

You know	You want	Operation
starting price	final price	multiply by multiplier
final price	starting price	divide by multiplier

Key terms

Multiplier.: The single decimal you multiply a price by to apply a percent change in one step: $1 + r$ for an increase, $1 - r$ for a decrease, where $r$ is the percent written as a decimal.
Discount.: An amount or percent subtracted from the original price. A 30 percent discount means you pay 70 percent, so the multiplier is $0.70$ .
Markup.: A percent added to a store's cost to set the selling price. A 40 percent markup gives retail price $= 1.40 \times \text{cost}$ .
Sales tax.: A percent of the selling price added at purchase. It is applied after any discounts, using a multiplier such as $1.06$ for 6 percent tax.
Tip (gratuity).: A percent of a bill added for service. A 18 percent tip means the total is $1.18$ times the bill before the tip.
Original price.: The price before a percent change is applied; also called list price. To find it from a final price, divide by the multiplier.
Wholesale cost.: What a store pays for an item before marking it up. Markup percents are calculated using this cost as the starting amount.
Total cost.: The final amount actually paid after all discounts, markups, taxes, and tips have been applied.

Worked example

A bike listed at 240 dollars is on sale for 15 percent off. The store then charges 6.5 percent sales tax on the sale price. Michael also has a coupon for an extra 10 percent off the sale price, applied before tax. What is Michael's total, rounded to the nearest cent? How much did he save compared to the list price plus tax?

Step 1: Turn each percent into a multiplier. 15 percent off means keep 85 percent, so the multiplier is

1 - 0.15 = 0.85

. The extra 10 percent off gives

1 - 0.10 = 0.90

. The 6.5 percent tax gives

1 + 0.065 = 1.065

.

Step 2: Chain them onto the list price.

240 \times 0.85 \times 0.90 \times 1.065

Step 3: Multiply in order.

240 \times 0.85 = 204

. Then

204 \times 0.90 = 183.60

. Then

183.60 \times 1.065 = 195.534

.

Step 4: Round to the nearest cent. The total is 195.53 dollars.

Step 5: Compare to paying list price plus tax.

240 \times 1.065 = 255.60

. The savings are

255.60 - 195.53 = 60.07

dollars.

Check the reasoning: the two discounts combine to

0.85 \times 0.90 = 0.765

, meaning Michael paid 76.5 percent of list price, a 23.5 percent total discount — not 25 percent. That is expected, because the 10 percent coupon applied to the already reduced 204 dollars, not to the full 240 dollars. Notice the tax multiplier does not affect the percent saved, since it multiplies both the discounted and undiscounted prices equally.

Practice questions

A jacket originally priced at 80 dollars is marked 35 percent off. Which single expression gives the sale price?

$80 \times 0.35$
$80 \times 1.35$
$80 \times 0.65$
$80 - 0.35$

Answer: $80 \times 0.65$

A 35 percent discount means you no longer pay 35 percent of the price, so you pay

100\% - 35\% = 65\%

, giving the multiplier

0.65

and a sale price of 52 dollars. The expression

80 \times 0.35

equals 28, which is the amount saved, not the price paid. Multiplying by

1.35

would increase the price, which is what a markup does. Subtracting

0.35

subtracts 35 cents, mixing up a percent with a dollar amount.

A restaurant bill is 46 dollars before tax. Tax is 5 percent, and Dana leaves a 20 percent tip on the pre-tax amount. Find the total amount Dana pays, and explain why you cannot just multiply 46 by 1.25.

Answer: Dana pays 57.50 dollars. Tax: $46 \times 0.05 = 2.30$ . Tip: $46 \times 0.20 = 9.20$ . Total: $46 + 2.30 + 9.20 = 57.50$ . Equivalently, $46 \times 1.25 = 57.50$ works here — because both percents are taken on the same base of 46 dollars, the percents can be added first.

This problem is a special case worth understanding. Normally you cannot add percents, because the second percent applies to a new, changed amount. Here, however, both the 5 percent tax and the 20 percent tip are computed on the same original 46 dollars, so

46 \times 0.05 + 46 \times 0.20 = 46 \times 0.25

by the distributive property, and the total is

46 \times 1.25 = 57.50

. If instead the tip had been calculated on the after-tax total, you would need to chain multipliers:

46 \times 1.05 \times 1.20 = 57.96

, which is different. Always read which amount each percent applies to.

After a 12 percent discount, a pair of headphones costs 61.60 dollars. What was the original price?

Answer: 70 dollars

The multiplier for 12 percent off is

1 - 0.12 = 0.88

, so

\text{original} \times 0.88 = 61.60

. Dividing undoes the multiplication:

61.60 \div 0.88 = 70

. Check it forward:

70 \times 0.88 = 61.60

, so 70 dollars is correct. A common wrong answer is

61.60 \times 1.12 = 68.99

, which fails because the 12 percent was taken from the larger original price, not from the sale price.

FAQ

Does it matter whether I apply the discount or the tax first?: For the final total, no — multiplication is commutative, so $60 \times 0.75 \times 1.07$ and $60 \times 1.07 \times 0.75$ both equal 48.15. In the real world stores apply the discount first because tax is charged on what you actually pay, and if a problem asks for the intermediate sale price you must follow the stated order. But the final amount comes out the same either way.
Why can't I add two discount percents together?: Because the second discount is taken on the already-reduced price, not the original. Twenty percent off, then 10 percent off, gives $0.80 \times 0.90 = 0.72$ , so you keep 72 percent and save 28 percent — not 30 percent. Multiplying the multipliers always gives the correct combined effect.
How do I know if the multiplier should be more or less than 1?: Ask whether the price is going up or down. Tax, tip, markup, and fees all increase the price, so the multiplier is $1 + r$ and is greater than 1. Discounts, sales, coupons, and markdowns decrease the price, so the multiplier is $1 - r$ and is between 0 and 1. If your final answer moved the wrong direction, you used the wrong multiplier.
When should I round in a money problem?: Round at the very end, to the nearest cent, unless the problem says otherwise. If you round in the middle and then keep multiplying, the small rounding error grows and your final answer can be off by a penny or more. Keep the extra decimals in your calculator until the last step.

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