Math Core

Prices rise, populations shrink, and stores mark things up and down. A percent change measures a change relative to the starting amount. In this lesson you'll find percent changes, write them as a single multiplication, and use equations to work backward, which is exactly how percent problems appear in algebra.

Finding a percent change

Percent change

\text{percent change} = \frac{\text{amount of change}}{\text{original amount}}

Write the result as a percent. It's a percent increase if the amount went up and a percent decrease if it went down.

Worked example: Same change, different percents

A quantity goes from 64 to 80. What is the percent change?
A quantity goes from 80 to 64. What is the percent change?

Solutions. Both changes are $80 - 64 = 16$ .

The original is 64: $\dfrac{16}{64} = 0.25$ , a $25\%$ increase.
The original is 80: $\dfrac{16}{80} = 0.2$ , a $20\%$ decrease.

The change is the same, but the starting amounts differ, so the percents differ.

Common mistake

Always divide by the original amount, the value before the change. Dividing by the new amount is the most common mistake with percent change.

Percent change as a multiplier

If a price $x$ increases by $12\%$ , the new price is the original plus $12\%$ of it:

x + 0.12x = 1x + 0.12x = 1.12x.

Combining $1x$ and $0.12x$ turns two steps into one multiplication. A decrease works the same way: $x - 0.12x = 0.88x$ .

Multipliers

An increase of $r$ (as a decimal) multiplies the original by $1 + r$ .
A decrease of $r$ (as a decimal) multiplies the original by $1 - r$ .

For example, a $30\%$ discount multiplies by $0.70$ , and a $6\%$ tax multiplies by $1.06$ .

Worked example: A discount, then tax

A jacket is priced at $85. It is on sale for $30\%$ off, and then $6\%$ sales tax is added. What is the final cost?

Apply each multiplier in order:

\begin{aligned} \text{sale price} &= 0.70 \times 85 = 59.50 \\ \text{with tax} &= 1.06 \times 59.50 = 63.07 \end{aligned}

The final cost is $63.07.

Working backward with an equation

If you know the amount after a change, let $x$ be the original, write the multiplier equation, and solve.

Worked example: Finding the original

A town's population fell by $15\%$ to 3,400 people. What was the population before the drop?

After a $15\%$ decrease, $85\%$ remains, so

\begin{aligned} 0.85x &= 3400 \\ x &= \frac{3400}{0.85} = 4000 \end{aligned}

The original population was 4,000. Check: $15\%$ of 4,000 is 600, and $4000 - 600 = 3400$ . ✓

Common mistake

Don't "undo" a $15\%$ decrease by adding $15\%$ of 3,400. That gives $3400 + 510 = 3910$ , which is wrong, because the $15\%$ was taken from the original 4,000, not from 3,400. Divide by the multiplier instead.

Changes in a row

When one percent change follows another, multiply by each multiplier in turn. The overall multiplier is their product.

Worked example: Up 10%, then down 10%

A price rises $10\%$ and then falls $10\%$ . Is it back where it started?

The overall multiplier is

1.10 \times 0.90 = 0.99.

The final price is $99\%$ of the original, a $1\%$ decrease overall. The drop is taken from a larger amount than the rise was, so it removes more.

Tip

To find an overall percent change, pick an easy starting value like 100. In the example above, 100 becomes 110, then $110 - 11 = 99$ . That's a $1\%$ decrease.