Percent Increase and Decrease, Step by Step

Percent change equals the change divided by the original value, times 100. The most common wrong answer comes from dividing by the new value instead of the old one. The same idea covers discounts, tax, raises, and "percent increase" questions in grade 7 math, which the Common Core standards list as percent increase and decrease, markups and markdowns, tax, and gratuities (7.RP.A.3).

1. The formula

Percent change = (new value - original value) / original value x 100

A positive result is an increase. A negative result is a decrease. The original value is always the one you started from, which is usually the "before" number in the story.

2. Example 1: a price goes up

A video game costs $40 and later costs $46.

1. Change: 46 - 40 = 6
2. Divide by the original: 6 / 40 = 0.15
3. Convert to a percent: 0.15 x 100 = 15%

The price rose 15%. Dividing 6 by the new price instead gives 6 / 46 = 13.04%, which is the wrong answer to this question.

3. Example 2: a quantity goes down

A class has 80 students signed up for a club in September and 60 in March.

1. Change: 60 - 80 = -20
2. Divide by the original: -20 / 80 = -0.25
3. Convert: -25%

Enrollment fell by 25%.

4. The multiplier shortcut

The grade 7 expressions standard gives a useful pattern: a + 0.05a = 1.05a, so "increase by 5%" is the same as "multiply by 1.05" (7.EE.A.2). The same logic works for decreases.

For a decrease, subtract the percent from 100% and turn it into a decimal. A 30% discount leaves 70% of the price, so you multiply by 0.70.

5. Example 3: a sale price, then tax

A jacket costs $60 and is marked 30% off. A sales tax of 8% is added after the discount.

1. Sale price: 60 x 0.70 = $42.00
2. Price with tax: 42 x 1.08 = $45.36

Compare this with a common slip: adding the percents. A 30% discount and an 8% tax do not combine into "22% off." That would give 60 x 0.78 = $46.80, which is $1.44 too high. The discount applies to $60, but the tax applies to $42, so they act on different amounts.

6. Example 4: find the original price

A pair of shoes costs $48 after a 20% discount. What was the original price?

The shoes cost 80% of the original, because 100% - 20% = 80%. So 0.80 x original = 48.

Original = 48 / 0.80 = $60.

The grade 6 standards describe this as solving problems that give a part and the percent, and ask for the whole (6.RP.A.3.c). Check by going forward: 60 x 0.80 = 48.

A frequent mistake is to take 20% of $48 and add it back, which gives $57.60. That fails because 20% was taken from $60, not from $48.

7. Why +25% and -25% do not cancel

Start with $100 and lose 25%: 100 x 0.75 = $75. Now gain 25% on the new amount: 75 x 1.25 = $93.75. You end $6.25 short.

To get back from $75 to $100, you need a gain of 25 / 75 = 33.3%. A decrease and an increase of the same percent do not cancel because each one uses a different starting value.

A second case: a drop from 60 to 45 is -15 / 60 = -25%. A rise from 45 back to 60 is 15 / 45 = 33.3%.

8. Percent versus percentage points

If a rate moves from 10% to 12%, it rose by 2 percentage points. The percent increase is 2 / 10 = 20%. Homework sometimes asks for one and sometimes the other, so read the wording: "percentage points" means subtract; "percent increase" means divide by the original.

9. Common mistakes

• Dividing by the new value. The denominator is the original value.
• Adding percents from different steps. Apply each change one after the other.
• Reversing a percent by using the same percent. Work out the multiplier and divide by it.
• Forgetting the sign. A decrease should come out negative, or be described with the word "decrease."
• Rounding too early. Keep the full decimal until the final step.

10. Key takeaways

• Percent change = change / original x 100.
• Increasing by p% means multiplying by 1 + p/100. Decreasing means multiplying by 1 - p/100.
• To find an original price, divide the final price by the multiplier.
• Discounts and taxes act on different amounts, so they cannot be added together.
• Equal up and down percents do not cancel, because the base changes.