Skip to content

Percent and percentage points

Ratio, rates and money · level 6, lesson 3 of 7

A question to start

A bank raises its savings interest rate from 4% to 5%. Two newspapers report it.

  • Newspaper A: "Bank raises rate by 1%"
  • Newspaper B: "Bank raises rate by 25%"

One of them sounds like a tiny change and the other like a big one. Which is right?

In a way, both are trying to say something true, but newspaper A has used the wrong word. The rate went up by 1 percentage point, and that is a rise of 25 percent. This lesson is about the difference, which matters every time you read about interest rates, taxes, test scores, polls or health risks.

The idea

When the things being compared are already percents

Usually a percent compares an amount with a base: 18 out of 24 is 75%. But some quantities are already percents: an interest rate, a tax rate, a test score, the share of votes a party gets. When a percent changes, there are two ways to measure the change, exactly as in level 5:

  • The absolute change is the plain difference, new minus old. When the things being compared are percents, this difference is measured in percentage points (sometimes "points" for short). From 4% to 5% is \(5 - 4 = 1\) percentage point.
  • The relative change compares the difference with the old value, as a percent. From 4% to 5% is \(\dfrac{5 - 4}{4} = \dfrac14 = 25\%\).

So:

\[4\% \to 5\%: \quad \text{up } 1 \text{ percentage point}, \quad \text{up } 25\%.\]

Newspaper A said "1%" when it meant "1 percentage point". Newspaper B is correct. A careful writer would say "the rate rose by 1 percentage point, from 4% to 5%", which gives the reader everything.

Why both are useful

Percentage points tell you the change in the actual thing. If you have 10 000 dollars in the bank, a rise from 4% to 5% earns you 1 percentage point more, which is 100 dollars more a year. That is the number that reaches your pocket.

Percent tells you how big the change is compared with where it started. Your interest goes up from 400 dollars to 500 dollars a year, a 25% rise in your interest. Both numbers are right; they answer different questions.

4% 5% +1 pt 1 block of 4: 25% more each block is one percentage point

Small bases make big percents

When the starting percent is small, a tiny change in points can be a huge change in percent. Suppose a rare side effect of a medicine happens to 1 person in 10 000 (that is 0.01%), and a new medicine raises it to 2 in 10 000 (0.02%).

  • A headline could say: "New medicine doubles the risk!" That is true: up 100%.
  • But the change is only \(0.02 - 0.01 = 0.01\) percentage points: 1 extra person in every 10 000.

In health, the percentage-point change is called the absolute risk change, and the percent change is the relative risk change. Both are true. Only reporting the relative one can scare people; only reporting the absolute one can hide a real danger in a big population. Good reporting gives both. (More on reading data honestly in Statistics.)

Basis points

Banks talk about very small changes in rates, so they use an even smaller unit: a basis point is one hundredth of a percentage point. A rise from 5.00% to 5.25% is 0.25 percentage points, or 25 basis points.

Percents bigger than 100

A percent can be more than 100. The trick is to keep straight "percent of" and "percent more than".

  • Of: if this year's sales are 3 times last year's, they are 300% of last year's.
  • More than: the increase is 2 times last year's, so they are 200% more than last year's.

Mars is a good example. Gravity at Mars's surface is about 3.71 m/s² and at Earth's about 9.81 m/s². So

  • Mars's gravity is \(\dfrac{3.71}{9.81} \approx 0.378\), about 38% of Earth's;
  • Earth's gravity is \(\dfrac{9.81}{3.71} \approx 2.64\), about 264% of Mars's, or 164% more than Mars's.

Same two numbers, different base, very different-looking percents.

A percent of a percent

When you take a percent of a group that is itself a percent of a bigger group, multiply. If 40% of a class take French and 25% of those also take Spanish, then \(0.40 \times 0.25 = 0.10\): 10% of the whole class take both. Not \(40\% - 25\% = 15\%\) and not \(25\%\).

Worked examples

1. The bank. A rate goes from 4% to 5%.

  • Points: \(5 - 4 = 1\) percentage point.
  • Percent: \(\dfrac{1}{4} = 25\%\) higher.

2. A test score. Sam's score goes from 60% to 75%.

  • Points: \(75 - 60 = 15\) percentage points.
  • Percent: \(\dfrac{15}{60} = 25\%\) better.

3. Going down. The unemployment rate falls from 6.0% to 5.4%.

  • Points: \(5.4 - 6.0 = -0.6\), down 0.6 percentage points.
  • Percent: \(\dfrac{-0.6}{6.0} = -0.10\), down 10%. (One in ten of the people who were unemployed now has a job, roughly speaking.)

4. Risk. A risk goes from 1 in 10 000 to 2 in 10 000.

  • Relative: doubled, up 100%.
  • Absolute: up 0.01 percentage points.

5. Gravity. Mars has about 38% of Earth's gravity. How do you describe Earth compared with Mars?

  • \(9.81 \div 3.71 \approx 2.64\): Earth's gravity is about 264% of Mars's, which is about 164% more.

6. A percent of a percent. 40% of a class take French; 25% of those take Spanish too.

  • \(0.40 \times 0.25 = 0.10\): 10% of the class take both.

Try it

  1. An interest rate goes from 3% to 4.5%. Describe the change in percentage points and in percent.
  2. A politician's approval falls from 52% to 39%. Describe the change both ways.
  3. A test score goes from 80% to 88%. How many percentage points up? How many percent up?
  4. A company's sales this year are 3 times last year's. Complete: this year's sales are ___% of last year's and ___% more than last year's.
  5. A new safety rule lowers a risk from 4% to 3%. Give the absolute change and the relative change. Which sounds more impressive?
  6. 60% of students in a school bike to school, and 30% of the bikers ride an e-bike. What percent of all the students ride an e-bike?
  7. A loan rate drops from 5.25% to 5.00%. Give the change in percentage points, in basis points, and in percent.
  8. A year on Mars is about 687 Earth days; an Earth year is about 365.25 days. A Mars year is about ___% of an Earth year, or ___% longer.
Answers
  1. Up \(4.5 - 3 = 1.5\) percentage points. In percent: \(\dfrac{1.5}{3} = 50\%\) higher.
  2. Down \(52 - 39 = 13\) percentage points. In percent: \(\dfrac{13}{52} = 25\%\) lower.
  3. Up 8 percentage points. In percent: \(\dfrac{8}{80} = 10\%\).
  4. 300% of last year's, and 200% more than last year's.
  5. Absolute: down 1 percentage point. Relative: \(\dfrac{1}{4} = 25\%\) lower. "Risk cut by 25%" sounds more impressive than "risk cut by 1 point", though they describe the same change.
  6. \(0.60 \times 0.30 = 0.18\), so 18% of all students.
  7. Down 0.25 percentage points, which is 25 basis points. In percent: \(\dfrac{0.25}{5.25} \approx 0.048\), about 4.8% lower.
  8. \(\dfrac{687}{365.25} \approx 1.88\): about 188% of an Earth year, or about 88% longer.

Common mistake

Saying "percent" when you mean "percentage points". From 4% to 5% is up 1 percentage point, not up 1%. If you are not sure, give both numbers and the before and after.

Subtracting percents of different groups. "40% take French and 25% of those take Spanish" means \(0.40 \times 0.25 = 10\%\) of the class, because the 25% is a percent of the French group, not of the class.

Where it leads