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What percent means: a rate per hundred

One carousel fills 18 of its 25 seats and a bigger one fills 34 of its 50. How can you tell which is fuller when the two wholes are not the same size?

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Why this exists

Thiago runs carousels at a fair. On the noon ride his small carousel filled 18 of its 25 seats and his big one filled 34 of its 50. The big one carried more riders, but that does not say which was fuller: 18 out of 25 and 34 out of 50 are parts of wholes of different sizes.

The way out is to measure both against the same whole. If each carousel had exactly 100 seats and kept the same rate, how many would be filled? Multiplying 18 out of 25 by 4 gives 72 out of 100, and multiplying 34 out of 50 by 2 gives 68 out of 100. Now the two sit side by side, and the small carousel was fuller. Counting “for each hundred” is so useful that it has its own name and its own sign: a percent.

Definition

Day 1: for each hundred

A percent is a rate per hundred: it says how many parts there are for every 100 of a whole. The word comes from the Latin per centum, “for each hundred”, the same cent as in 100 years to a century or 100 cents to a dollar. The sign \% is read “percent”.

So 72\% means 72 for every 100. As a part-to-whole ratio it is 72 : 100, and as a fraction it is \displaystyle \frac{72}{100}, seventy-two hundredths. The whole, whatever its size, counts as 100\%.

The picture of a percent is a hundred grid: a square cut into 10 rows of 10 small squares, 100 in all, standing for the whole. A percent is the number of small squares shaded. Figure 1 shows 37\%: three full rows of 10 and 7 squares more.

Figure 1A hundred grid with 37 of its 100 squares shaded: 37\%.

Day 2: the percent bar

When the whole is not 100 things, a percent bar matches percents to amounts. It is a long rectangle standing for the whole, with two scales: percents along the top, from 0\% to 100\%, and the amount along the bottom, from 0 to the whole. It is a The double number line drawn as a bar, with 0\% lined up with 0 and 100\% lined up with the whole.

The marks come from folding. Fold a paper strip in half and the crease is 50\%, lined up with half the whole. Fold it in half again and the new creases are 25\% and 75\%. Cutting the bar into ten equal parts gives a mark every 10\%, each step lined up with a tenth of the whole. Figure 2 is the bar for a carousel with 40 seats, folded twice.

0\%
0
25\%
10
50\%
20
75\%
30
100\%
40
\text{percent}
\text{seats}
Figure 2The percent bar for a carousel of 40 seats. The first fold makes the dark line at 50\%; the second fold makes the lighter lines at 25\% and 75\%.

Reading straight down from a percent gives that percent of the whole: in Figure 2, 75\% of 40 seats is 30 seats. Reading up from an amount gives the percent that amount is of the whole.

Intuition

A percent rescales the whole to 100. A wormery of 20 worms, one of 100 and one of 1000 are very different sizes, but “40\% are red wigglers” means the same thing in each: if the whole were shrunk or stretched to 100 worms, 40 of them would be red wigglers. In the small wormery that is 8 worms, in the middle one 40, in the big one 400. The percent describes the mix, not the count.

That is why percents compare so well. Any part-to-whole rate can be scaled to a rate per hundred, and once two rates both say “out of 100”, the bigger number is the bigger share.

The grid shows a percent directly when the whole has 100 things, or when the whole can be shared evenly among the 100 squares: with a whole of 20, each thing gets 5 squares (Example 2). The bar works for any whole, because its bottom scale stretches to fit. The same 50\% mark sits over 20 seats on a carousel of 40 and over 30 call-outs at a lifeboat station that had 60.

Results

A percent is a number of hundredths.

37\% = \frac{37}{100} = 0.37

Hundredths are the second place after the decimal point, so a whole-number percent below 100 fills the first two places after the point: 37\% = 0.37 and 9\% = 0.09. Fractions that are not already hundredths are turned into decimals by Converting between fractions and decimals.

The whole is 100\%. Since \displaystyle \frac{100}{100} = 1, one hundred percent is one whole. Half the whole is 50\%, a quarter is 25\%, and a tenth is 10\%.

Parts that make up a whole have percents adding to 100\%. If 35\% of the call-outs came at night, the other 65\% came by day.

Equal rates give equal percents. Equivalent part-to-whole ratios have the same percent, whatever the size of their wholes: 7 out of 20, 35 out of 100 and 70 out of 200 are all 35\%.

Why it is true

A percent is a ratio whose second amount is 100, so finding a percent is finding the equivalent ratio with 100 as the whole. Equivalent ratios come from multiplying or dividing both amounts by the same number (Scaling a ratio up or down). For 7 out of 20, multiply both by 5, since 20 \times 5 = 100: the rate is 35 out of 100, which is 35\%. For 70 out of 200, divide both by 2: again 35 out of 100. Every ratio equivalent to these scales to the same rate per hundred, which is the last result.

The bar is a double number line, so what holds there holds here. 0\% lines up with 0 because none of the whole is 0 of it, and 100\% lines up with the whole because all of the whole is the whole. A fold cuts both scales at the same place, so half of 100\% sits over half the whole, and a tenth of 100\% sits over a tenth of the whole. Matching points are equivalent ratios, so each amount on the bottom is the number of hundredths of the whole named above it.

The parts add to 100\% because shading every part of the whole shades all 100 squares of its grid.

Worked examples

Examples 1 to 3 go with the first day, 4 to 7 with the second.

Example 1

Desta’s hundred worms

Desta counts a sample from her wormery: exactly 100 worms, of which 37 are red wigglers, 48 are tiger worms and 15 are nightcrawlers. Write each kind as a percent, as a fraction and as a decimal, and check that the parts make the whole.

Solution

The whole is already 100, so each count is already a rate per hundred. 37 red wigglers out of 100 worms is 37\%, the shaded squares of Figure 1. As a fraction it is \displaystyle \frac{37}{100}, thirty-seven hundredths, which as a decimal is 0.37. In the same way the tiger worms are \displaystyle 48\% = \frac{48}{100} = 0.48 and the nightcrawlers are \displaystyle 15\% = \frac{15}{100} = 0.15.

Every worm is one of the three kinds, so the percents must make the whole:

37\% + 48\% + 15\% = 100\%

They do, and the counts agree: 37 + 48 + 15 = 100 worms.

Example 2

Night call-outs at the lifeboat station

Nuala’s lifeboat station had 20 call-outs last month, and 7 of them came at night. What percent of the call-outs came at night, and what percent by day?

Solution

The whole is 20, not 100, so scale the rate to one per hundred. Since 20 \times 5 = 100, multiply both amounts by 5:

7 : 20 = 35 : 100

On the hundred grid, give each call-out a block of 5 squares, as in Figure 3. The 7 night call-outs fill 7 blocks, which is 35 squares, so 35\% of the call-outs came at night.

Figure 320 call-outs shared among 100 squares: each call-out gets a block of 5. Seven blocks are 35 squares.

The day call-outs are the other 20 - 7 = 13 blocks, 13 \times 5 = 65 squares, so 65\%. Check. 35\% + 65\% = 100\%.

Example 3

Which carousel is fuller?

Thiago’s small carousel has 25 seats and filled 18 on the noon ride. His big carousel has 50 seats and filled 34. Which was fuller?

Solution

“Fuller” asks about the share of seats filled, a part-to-whole rate, so compare per hundred. For the small carousel 25 \times 4 = 100, so multiply both amounts by 4: 18 : 25 = 72 : 100, which is 72\%. For the big carousel 50 \times 2 = 100, so multiply both by 2: 34 : 50 = 68 : 100, which is 68\%.

The small carousel was fuller, 72\% against 68\%, even though the big one carried more riders, 34 against 18. The counts answer “how many riders”; the percents answer “how full”.

Example 4

Folding the carousel bar

Thiago’s middle carousel has 40 seats. On one ride 75\% of the seats were filled. How many riders were on it?

Solution

Draw a percent bar with 0\% to 100\% along the top and 0 to 40 seats along the bottom, as in Figure 2. Folding the bar in half puts 50\% over half of 40, which is 20 seats. Folding again cuts each half in two, so 25\% sits over 10 seats and 75\% sits over 20 + 10 = 30 seats. Read straight down from 75\%:

75\% \text{ of } 40 \text{ seats is } 30 \text{ seats}

There were 30 riders. Check. 75\% is three quarters of the bar, and three quarters of 40 is 3 \times 10 = 30.

Example 5

A year of call-outs in tenths

Nuala’s station had 60 call-outs in a year: 30\% were to swimmers, 65\% to boats, and the other 5\% to walkers cut off by the tide. How many call-outs were of each kind?

Solution

Cut the bar into tenths, as in Figure 4. Each 10\% step lines up with 60 \div 10 = 6 call-outs. 30\% is 3 steps, so there were 3 \times 6 = 18 call-outs to swimmers.

0\%
0
10\%
6
20\%
12
30\%
18
40\%
24
50\%
30
60\%
36
70\%
42
80\%
48
90\%
54
100\%
60
\text{percent}
\text{call-outs}
Figure 4The percent bar for 60 call-outs, cut into tenths. Each 10\% step lines up with 6 call-outs.

5\% is half of a 10\% step, so it lines up with half of 6, which is 3 call-outs to walkers. 65\% is 6 steps and a half, so the boats had 6 \times 6 + 3 = 39 call-outs.

Check. 18 + 39 + 3 = 60, the whole, just as 30\% + 65\% + 5\% = 100\%.

Example 6

Reading the percent from the bar

Lyra’s wormery holds 80 worms, and 28 of them are nightcrawlers. What percent are nightcrawlers?

Solution

This time the amount is known and the percent is wanted, so read the bar upwards. On a bar from 0 to 80 worms, each 10\% step lines up with 80 \div 10 = 8 worms. 28 is not a whole number of 8s: it sits between 24, under 30\%, and 32, under 40\%, exactly halfway. Halfway lines up with halfway, so 28 worms sits under 35\%.

The same answer comes from a smaller step. Halving the 10\% step gives 5\% for every 4 worms, and 28 = 7 \times 4, so 28 worms is 7 \times 5\% = 35\%.

So 35\% of Lyra’s worms are nightcrawlers. Check. Dividing both amounts of 28 : 80 by 4 gives 7 : 20, and multiplying both by 5 gives 35 : 100.

Example 7

Working back to the whole

On a busy ride, 45 riders filled 75\% of the seats on Thiago’s newest carousel. How many seats does that carousel have?

Solution

Now the whole is the unknown, but the bar still works. 45 sits under 75\%, and the number of seats is whatever sits under 100\%. 75\% is three quarter-steps of 25\%, so each quarter of the bar lines up with 45 \div 3 = 15 seats. The whole bar is four quarters:

4 \times 15 = 60 \text{ seats}

Check. A quarter of 60 is 15, and three quarters is 45. Finding the whole from a part and its percent, for percents that do not fold so neatly, is the work of Finding the whole from a part and a percent.

Boundaries

A percent needs a whole. “35\%” alone says nothing about how many: 35\% of 20 call-outs is 7, and 35\% of 60 call-outs is 21. Two percents compare shares, not counts, unless they are percents of the same whole.

A part of a whole is at most 100\%. A carousel cannot fill more than all its seats, and no kind of worm can be more than the whole wormery.

Above 100\%. A percent can pass 100 when it compares one amount with another, rather than a part with its whole. If Nuala’s station had 40 call-outs one year and 60 the next, the second year is 150\% of the first: 150 call-outs for every 100 the year before. Percents above 100 and below 1 works with these.

Below 1\%. A percent can also be less than one. 0.5\% means half of one for each hundred, which is 1 for every 200: if 0.5\% of the 2000 worms in a big wormery are nightcrawlers, there are 10 of them.

Counts come whole. The bar is exact, but 35\% of a 30-seat carousel sits over 10.5 seats, and no ride fills half a seat. The rate is fine; the count only makes sense when the numbers allow it.

Friendly numbers, for now. Reading the bar is quick when the whole splits evenly into halves, quarters or tenths. A percent such as 36\% of 75 worms, where 75 splits into neither quarters nor tenths of whole worms, needs the arithmetic of Finding a percent of a quantity.

Where this goes next

Turning any fraction into a percent, \displaystyle \frac{3}{8} for example, is Converting among fractions, decimals and percents, which reaches hundredths by division when no whole-number scaling lands on 100. Benchmark percents and mental percent turns the folds of this lesson, 50\%, 25\% and 10\%, into tools for working percents in your head. Finding a percent of a quantity finds a percent of any amount, and Finding the whole from a part and a percent runs the bar backwards, as Example 7 did. Later, Finding what percent one number is of another and Solving percent problems with an equation or a proportion write the bar’s matching as a single equation, percents give the relative frequencies inside Two-way tables, and a percentile names a data value by the percent of the data at or below it.

Exercises

Day 1: rates per hundred

  1. In Ansel’s sample of 100 worms, 62 are tiger worms. Write the share of tiger worms as a percent, as a fraction and as a decimal.
  2. Write 9\% as a fraction and as a decimal.
  3. Write 0.4 as a percent.
  4. On a hundred grid, 4 full rows and 3 more squares are shaded. What percent is shaded, and what percent is not?
  5. A lifeboat station had 20 call-outs, and 11 of them were to boats with engine trouble. What percent is that?
  6. A carousel with 50 seats has 27 filled. What percent of the seats are filled, and what percent are empty?
  7. Keanu’s wormery holds 25 worms, and 9 of them are red wigglers. What percent are red wigglers?
  8. Over two years a lifeboat station had 200 call-outs, and 46 came at night. What percent came at night?
  9. A wormery holds 300 worms, and 102 are nightcrawlers. What percent are nightcrawlers?
  10. One carousel has 20 seats with 13 filled; another has 50 seats with 31 filled. Which is fuller?

Day 2: the percent bar

Exercises 11 and 12 use the carousel of Figure 2 (40 seats), and Exercises 13 and 14 the station of Figure 4 (60 call-outs). Mark the bar more finely where a percent is not drawn.

  1. How many seats are 50\% of the carousel, and how many are 25\%?
  2. How many seats are 10\% of the carousel, and how many are 90\%?
  3. How many call-outs are 40\% of the year’s, and how many are 90\%?
  4. How many call-outs are 5\% of the year’s, and how many are 15\%?
  5. A wormery holds 200 worms. Fold its percent bar into quarters. How many worms are 25\% of them, and how many are 75\%?
  6. Lyra’s wormery holds 80 worms. How many worms are 45\% of them?
  7. At the station of Figure 4, 21 of the 60 call-outs were to swimmers. What percent is that?
  8. On the carousel of Figure 2, 34 seats are filled. What percent is that?
  9. A wormery holds 120 worms, and 30\% of them are tiger worms. How many tiger worms are there?
  10. A station’s 12 call-outs to stranded walkers were 20\% of all its call-outs. How many call-outs did it have?
  11. A carousel has 24 seats. How many seats are 25\% of them, and how many are 12.5\%?

Reasoning

  1. Explain why 100\% of the worms in a wormery means all of them, whatever the number of worms.
  2. Wormery A has 40 red wigglers out of 200 worms. Wormery B has 30 red wigglers out of 100 worms. Which has the larger share of red wigglers? Explain why comparing 40 with 30 does not answer the question.
  3. Explain why the 50\% mark of a percent bar always lines up with half the whole, whatever the whole is.
  4. A grid has 50 squares instead of 100. How many squares should be shaded to show 30\%? Explain.
Answers
  1. 62\%, \displaystyle \frac{62}{100} and 0.62. The whole is 100, so 62 out of 100 is already a rate per hundred.
  2. \displaystyle \frac{9}{100} and 0.09. Nine hundredths: the 9 sits in the hundredths place.
  3. 40\%. 0.4 = 0.40, forty hundredths, which is 40 for every 100.
  4. 43\% shaded and 57\% not. 4 \times 10 + 3 = 43 of the 100 squares are shaded, and 100 - 43 = 57 are not.
  5. 55\%. Multiply both amounts by 5: 11 : 20 = 55 : 100.
  6. 54\% filled and 46\% empty. Multiply both by 2: 27 : 50 = 54 : 100; then 100\% - 54\% = 46\%.
  7. 36\%. Multiply both by 4: 9 : 25 = 36 : 100.
  8. 23\%. Divide both by 2: 46 : 200 = 23 : 100.
  9. 34\%. Divide both by 3: 102 : 300 = 34 : 100.
  10. The 20-seat carousel. 13 : 20 = 65 : 100, which is 65\%, while 31 : 50 = 62 : 100, which is 62\%.
  11. 20 seats and 10 seats. One fold halves 40, and a second fold halves 20.
  12. 4 seats and 36 seats. Each tenth of 40 is 4, and 90\% is 9 tenths: 9 \times 4 = 36.
  13. 24 and 54. Each 10\% step is 6 call-outs: 4 \times 6 = 24 and 9 \times 6 = 54.
  14. 3 and 9. 5\% is half of a 6-call-out step, and 15\% is one step and a half: 6 + 3 = 9.
  15. 50 worms and 150 worms. Two folds cut 200 into quarters of 50, and 75\% is three quarters: 3 \times 50 = 150.
  16. 36 worms. 10\% of 80 is 8 worms, so 5\% is 4 worms, and 45\% is nine 5\% steps: 9 \times 4 = 36.
  17. 35\%. Each 10\% step is 6 call-outs, and 21 sits halfway between 18, under 30\%, and 24, under 40\%.
  18. 85\%. Each 10\% step is 4 seats, and 34 sits halfway between 32, under 80\%, and 36, under 90\%.
  19. 36 tiger worms. 10\% of 120 is 12, and 30\% is 3 \times 12 = 36.
  20. 60 call-outs. 20\% lines up with 12, and 100\% is five 20\% steps: 5 \times 12 = 60.
  21. 6 seats and 3 seats. Two folds cut 24 into quarters of 6; a third fold halves the 25\% part, so 12.5\% lines up with 3.
  22. 100\% means 100 for every 100 of the whole: each hundred of the whole is counted in full, so nothing is left out. As a fraction, \displaystyle \frac{100}{100} = 1, one whole, whether the wormery has 20 worms or 2000.
  23. Wormery B. For A, dividing both amounts by 2 gives 40 : 200 = 20 : 100, which is 20\%; B is 30\%. The counts 40 and 30 are out of different wholes: A has more red wigglers only because it has twice as many worms. Per hundred worms, B has 30 red wigglers and A only 20.
  24. The bar is a double number line with 0\% over 0 and 100\% over the whole. Folding it in half puts the crease in the middle of both scales at once, so half of 100\%, which is 50\%, sits over half the whole. Half the whole compared with the whole is the ratio 1 : 2, and 1 : 2 = 50 : 100 for any whole.
  25. 15 squares. With 50 squares standing for the whole, each square stands for two squares of a hundred grid, 2\%, so 30\% needs 30 \div 2 = 15 squares. Halving both amounts gives the same thing: 30 : 100 = 15 : 50.
kg1128 · updated 2026-10-12 · CC-BY-SA-4.0