Math Core

"Ticket prices went up 20%." "The team's attendance dropped 15%." A percent change tells you how big a change is compared to where you started, which makes changes of different sizes easy to compare.

Measuring a change with a percent

Suppose a phone plan goes from $40 to $50 a month, and a concert ticket goes from $200 to $210. Both went up $10. But $10 is a big jump for a $40 plan and a small one for a $200 ticket. A percent change captures that difference by comparing the change to the original amount.

Phone plan: $\dfrac{10}{40} = 0.25 = 25\%$ increase.
Concert ticket: $\dfrac{10}{200} = 0.05 = 5\%$ increase.

Percent change

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

Then write the decimal as a percent. If the amount went up, it's a percent increase. If it went down, it's a percent decrease.

The amount of change is always a positive difference: new minus original for an increase, original minus new for a decrease.

Worked example: Percent increase

A school club had 25 members last year and 34 members this year. What is the percent increase?

The change is $34 - 25 = 9$ members. The original amount is 25.

\frac{9}{25} = 0.36 = 36\%

Membership increased by 36%.

Worked example: Percent decrease

A bike was priced at $250. Now it costs $190. What is the percent decrease?

The change is $250 - 190 = 60$ dollars. The original price is 250.

\frac{60}{250} = 0.24 = 24\%

The price decreased by 24%.

Common mistake

Always divide by the original amount, not the new one. In the bike example, $\dfrac{60}{190} \approx 32\%$ is wrong, because the change is measured from where the price started, $250.

Finding the new amount

Sometimes you know the percent change and need the new amount. You could find the change and then add or subtract it. There's also a one-step shortcut.

If something increases by $20\%$ , the new amount is the whole original ( $100\%$ ) plus $20\%$ more, which is $120\%$ of the original. If something decreases by $20\%$ , what's left is $100\% - 20\% = 80\%$ of the original.

One-step multipliers

Increase by $p\%$ : multiply the original by $(100\% + p\%)$ , written as a decimal.
Decrease by $p\%$ : multiply the original by $(100\% - p\%)$ , written as a decimal.

An increase of $15\%$ means multiplying by $1.15$ . A decrease of $15\%$ means multiplying by $0.85$ .

Worked example: New amount after a change

A town of 4,800 people grows by $5\%$ . What is the new population?
A lake's water level was 30 feet. After a dry summer, it fell by $12\%$ . What is the new level?

Solutions.

$1.05 \times 4{,}800 = 5{,}040$ people. (Check the long way: $5\%$ of 4,800 is 240, and $4{,}800 + 240 = 5{,}040$ .)
$0.88 \times 30 = 26.4$ feet. (Check: $12\%$ of 30 is 3.6, and $30 - 3.6 = 26.4$ .)

Working backward to the original

If you know the new amount and the percent change, the original is the missing whole. Write an equation with the multiplier and solve.

Worked example: Find the original amount

After a $25\%$ increase, a gym charges $45 a month. What did it charge before?

An increase of $25\%$ means the new price is $1.25$ times the original, $x$ .

\begin{aligned} 1.25x &= 45 \\ x &= \frac{45}{1.25} = 36 \end{aligned}

The original price was $36. Check: $25\%$ of 36 is 9, and $36 + 9 = 45$ .

Tip

A decrease can never be more than $100\%$ (you can't lose more than everything), but an increase can. If something doubles, it increased by $100\%$ . If it triples, it increased by $200\%$ .