Percent Change & Discounts

Almost every price tag you meet is the result of a percent change. A store buys a jacket, adds a markup, then puts it on the rack. Later the jacket goes on sale and the price comes back down. Both moves are the same piece of math: a percent taken of a starting amount, then added on or taken off.

This topic has two jobs. Given two amounts, find what percent the change was. Given a percent, find the new price. Once you see that a discount and a markup are the same computation with a plus or a minus, the whole topic collapses into one habit: decide what percent of the original is left, then multiply.

Finding the percent of change

Subtract to find the amount of change, divide by the original amount, then multiply by $100$ : percent change $= \dfrac{\text{amount of change}}{\text{original amount}} \cdot 100$ . If the amount went up it is a percent of increase; if it went down it is a percent of decrease. Always name the direction with your answer, because $25\%$ up and $25\%$ down are very different news.

The word original does a lot of work here. It means the amount you started with, not the smaller of the two and not the one that looks nicer. A price going from $20 to $25 is a change of $5 measured against $20, which is a $25\%$ increase. Measure that same $5 against $25 and you get $20\%$ , which answers a different question entirely.

Discounts and markups in one multiplication

A $15\%$ discount takes $15\%$ off, so what is left is $100\% - 15\% = 85\%$ of the original price. That means you can skip the subtraction and go straight to the sale price: multiply the original by $0.85$ . A $15\%$ markup goes the other way. It adds on, so the new price is $115\%$ of the original, or $1.15$ times it.

Both directions follow one rule. Discount of $p\%$ : multiply by $1 - \dfrac{p}{100}$ . Markup of $p\%$ : multiply by $1 + \dfrac{p}{100}$ . Finding the amount taken off and subtracting it gets the same answer, and it is worth doing once so the shortcut makes sense. After that, the single multiplication is faster and leaves fewer places to slip.

Reading the question before you compute

Discount problems ask for two different things and they look alike on the page. How much do you save is the discount amount, $0.30 \cdot 120 = 36$ . How much do you pay is the sale price, $0.70 \cdot 120 = 84$ . Underline which one the question wants before you touch the numbers, because both values are usually sitting in the answer choices.

Watch out for percents stacked on top of each other. Taking $20\%$ off and then another $10\%$ off is not $30\%$ off, because the second discount is taken of the already reduced price: $0.80 \cdot 0.90 = 0.72$ , which is $28\%$ off. Sales tax added after a discount behaves the same way. Handle one change at a time, in the order the problem states.

Worked examples

Example 1: percent of increase

A jacket's price goes from $40 to $50. What is the percent of change?

Find the amount of change

50 - 40 = 10

Divide by the original amount

\dfrac{10}{40} = 0.25

Multiply by

100

to get a percent

0.25 \cdot 100 = 25

Answer: $25\%$ increase.

Example 2: percent of decrease

Attendance at a school play falls from $250$ people to $200$ people. What is the percent of change?

Find the amount of change

250 - 200 = 50

Divide by the original amount

\dfrac{50}{250} = 0.2

Multiply by

100

to get a percent

0.2 \cdot 100 = 20

Answer: $20\%$ decrease.

Example 3: a discount

A $60 pair of shoes is marked $15\%$ off. What is the sale price?

Taking

15\%

off leaves

85\%

100\% - 15\% = 85\%

Write the percent as a decimal

85\% = 0.85

Multiply by the original price

0.85 \cdot 60 = 51

Answer: The sale price is $51.

Example 4: a markup

A store pays $48 for a backpack and marks the price up $25\%$ . What does the backpack sell for?

A markup adds on, so the price is

125\%

of the cost

100\% + 25\% = 125\%

Write the percent as a decimal

125\% = 1.25

Multiply by the store's cost

1.25 \cdot 48 = 60

Answer: The backpack sells for $60.

Try one yourself

Common questions

20\%

off the same as multiplying by

0.8

Yes. Taking $20\%$ off leaves $100\% - 20\% = 80\%$ of the price, and $80\%$ written as a decimal is $0.8$ . A $45 item at $20\%$ off costs $0.8 \cdot 45 = 36$ dollars. Multiplying by $0.2$ instead gives $9, which is the amount saved, not the price paid.

Does

20\%

off followed by

10\%

off equal

30\%

off?

No. The second discount is taken of the reduced price, not the original. Multiply the two factors: $0.80 \cdot 0.90 = 0.72$ , so you pay $72\%$ of the original and the total discount is $28\%$ , not $30\%$ .

How do I find the original price when I only know the sale price?

Work backward from the multiplier. If a $15\%$ discount was applied, the sale price is $0.85$ times the original, so divide instead of multiply: an item on sale for $51 started at $\dfrac{51}{0.85} = 60$ dollars.

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Percent Change & Discounts

Finding the percent of change

Discounts and markups in one multiplication

Reading the question before you compute

Worked examples

Example 1: percent of increase

Example 2: percent of decrease

Example 3: a discount

Example 4: a markup

Try one yourself

Common questions

Keep learning

Percent Change & Discounts

Percent of a Number

Simple Interest