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Master: Percent

Ratio and rates · step 4 of 5

A question to start

A shop raises the price of a 100 dollar jacket by 20%. A month later, it has a sale: 20% off. Is the jacket back to 100 dollars? Most people say yes. It is not: it costs 96 dollars. And if the shop had put on the sale first and the rise second, it would still cost 96 dollars. Percent is everywhere (prices, taxes, test marks, news headlines) and it is full of little traps like this one. This page gives you one tool, the multiplier, that handles all of them.

The idea

Percent means "out of a hundred"

Percent comes from the Latin per centum, "for each hundred". The sign \(\%\) means "divided by 100":

\[35\% = \frac{35}{100} = 0.35.\]

A percent is a ratio with the second part fixed at 100, which makes different things easy to compare. Converting:

  • percent to decimal: divide by 100 (move the point two places left): \(7\% = 0.07\), \(150\% = 1.5\);
  • decimal to percent: multiply by 100: \(0.625 = 62.5\%\);
  • fraction to percent: divide, then multiply by 100: \(\tfrac38 = 0.375 = 37.5\%\).

Some to know by heart: \(\tfrac12 = 50\%\), \(\tfrac14 = 25\%\), \(\tfrac34 = 75\%\), \(\tfrac15 = 20\%\), \(\tfrac1{10} = 10\%\), \(\tfrac13 \approx 33.3\%\).

Percent of an amount

"Of" means multiply. \(35\%\) of 240 is \(0.35 \times 240 = 84\). A mental route: 10% of 240 is 24, so 30% is 72, 5% is 12, and 35% is \(72 + 12 = 84\).

One amount as a percent of another

To write \(A\) as a percent of \(B\), divide and multiply by 100:

\[\frac{A}{B} \times 100\%.\]

18 out of 72 is \(\tfrac{18}{72} \times 100\% = 25\%\).

Multipliers: the one tool

To increase an amount by \(p\%\), you keep the whole 100% and add \(p\%\), so you multiply by \(1 + \tfrac{p}{100}\). To decrease by \(p\%\), multiply by \(1 - \tfrac{p}{100}\).

Change Multiplier
up 20% 1.20
up 5% 1.05
up 100% (doubled) 2
down 20% 0.80
down 15% 0.85
down 100% (all gone) 0

Multipliers make three hard problems easy.

Repeated changes multiply. Up 20% and then down 20% is \(1.20 \times 0.80 = 0.96\): a 4% loss overall. The order does not matter, because \(0.80 \times 1.20\) is also 0.96. The two 20%s are not percents of the same thing: the rise is 20% of 100, the fall is 20% of 120.

Reverse percent. If the price after a 15% discount is 68 dollars, then \(0.85 \times \text{original} = 68\), so the original was \(68 \div 0.85 = 80\) dollars. Divide by the multiplier to go back. (Taking 15% of 68 and adding it back on is wrong, because the 15% was of the original price, not of 68.)

Percent change. To find the percent by which something changed, compare the change with the original:

\[\text{percent change} = \frac{\text{new} - \text{original}}{\text{original}} \times 100\%.\]

A positive answer is an increase; a negative one is a decrease.

Percent or percentage points?

If a bank's interest rate goes from 4% to 5%, it rose by 1 percentage point, the plain difference. But it rose by 25%, because 1 is a quarter of 4. News reports sometimes use whichever sounds bigger. A careful reader asks which one is meant.

Percents bigger than 100

Nothing stops a percent from passing 100. Mars's surface gravity is about 38% of Earth's, so Earth's is about \(1 \div 0.38 \approx 2.63\) times Mars's, or 263% of it. Both statements describe the same two numbers; only the "of what?" changed.

Worked examples

1. The jacket. \(100 \times 1.20 = 120\), then \(120 \times 0.80 = 96\) dollars. Overall multiplier \(0.96\), a 4% decrease.

2. Percent change. A town grew from 250 thousand to 290 thousand people. By what percent?

  • Change: \(290 - 250 = 40\) thousand.
  • \(\dfrac{40}{250} \times 100\% = 16\%\) increase.
  • Check with a multiplier: \(250 \times 1.16 = 290\).

3. Reverse percent with tax. A bill including 13% sales tax comes to 50.85 dollars. What was the price before tax?

  • The multiplier for adding 13% is 1.13.
  • \(50.85 \div 1.13 = 45\) dollars before tax. The tax was \(50.85 - 45 = 5.85\) dollars.

4. Three changes in a row. A share price rises 10%, then falls 10%, then rises 10%. Overall change?

  • \(1.10 \times 0.90 \times 1.10 = 1.089\).
  • That is an 8.9% increase overall, not 10%.

Try it

  1. Find 35% of 240 without a calculator.
  2. A price of 45 dollars goes up by 12%. What is the new price?
  3. After a 25% increase, a ticket costs 75 dollars. What did it cost before?
  4. A population falls from 8000 to 6800. What is the percent change?
  5. An interest rate goes from 4% to 5%. Describe the change in percentage points and in percent.
  6. A price is cut by 30% and then the new price is cut by another 20%. Is that the same as a 50% cut?
Answers
  1. 10% is 24, so 30% is 72; 5% is 12. \(72 + 12 = 84\).
  2. \(45 \times 1.12 = 50.40\) dollars.
  3. \(75 \div 1.25 = 60\) dollars. (Check: \(60 \times 1.25 = 75\).)
  4. \(\dfrac{6800 - 8000}{8000} \times 100\% = \dfrac{-1200}{8000} \times 100\% = -15\%\), a 15% decrease.
  5. Up 1 percentage point. As a percent of the old rate, \(\dfrac{1}{4} \times 100\% = 25\%\) higher.
  6. No. \(0.70 \times 0.80 = 0.56\), so the price is 56% of the original: a 44% cut, not 50%.

Common mistake

Using the new amount as the base. Percent change is always measured against the original. Going from 80 to 100 is a 25% increase (\(\tfrac{20}{80}\)), but going back from 100 to 80 is a 20% decrease (\(\tfrac{20}{100}\)). The same gap of 20 is a different percent of a different starting point. This is the whole secret of the jacket.

Where it leads