Ten blocks on a bar: percents as fractions
Draw the whole as a bar of 10 equal blocks: each block is 10%. Many percents are simple fractions of that bar, like 25% = 1/4 and 12.5% = 1/8, and the fraction is often the fastest way to the answer.
Guess first
Guess first: a bag holds 2,000 g of flour. A recipe needs 12.5% of the bag. How many grams is that?
Show answer
Correct answer: C
12.5% is one eighth, so 2,000 ÷ 8 = 250 g. Check: 10% of 2,000 = 200 and 2.5% of 2,000 = 50, and 200 + 50 = 250.
Option A: 25 g is ten times too small: the decimal point slipped one place.
Option B: 160 g divides 2,000 by 12.5 (2,000 ÷ 12.5 = 160). Dividing by the percent does not give a share: 12.5% means 12.5 out of every 100.
Option D: 1,750 g is what is left in the bag (2,000 − 250 = 1,750), not what the recipe needs.
Hint: 12.5% is half of 25%, and 25% is a quarter.
If you said 250 g by halving a quarter, you used a fraction instead of a formula. This lesson turns that into a habit, with a picture.
An everyday picture
A chocolate bar of 10 equal pieces makes percents visible: the whole bar is 100%.
1 piece of 10 10%
one block of the bar
5 pieces 50%
half the bar, 1/2
2½ pieces 25%
a quarter of the bar, 1/4
1¼ pieces 12.5%
an eighth of the bar, 1/8
A percent tells you how much of the bar; the fraction tells you how to cut it.
See it: the ten-block bar
Here is the bar of 10 blocks, drawn as two rows of 5 like a chocolate bar, with 4 blocks shaded:
4 shaded blocks are 4 × 10% = 40%, which is 4/10 = 2/5 of the bar. Now cut the same bar into 8 equal blocks instead (two rows of 4) and shade 3:
Each block is now 100% ÷ 8 = 12.5%, so 3 blocks are 3 × 12.5% = 37.5%, which is 3/8 of the bar.
| Percent | Fraction | Blocks of a 10-block bar |
|---|---|---|
| 10% | 1/10 | 1 |
| 20% | 1/5 | 2 |
| 25% | 1/4 | 2½ |
| 50% | 1/2 | 5 |
| 12.5% | 1/8 | 1¼ |
| 33⅓% | 1/3 | 3⅓ |
| 16⅔% | 1/6 | 1⅔ |
| 37.5% | 3/8 | 3¾ |
| 62.5% | 5/8 | 6¼ |
| 75% | 3/4 | 7½ |
Use the bar to find 62.5% of 480:
- 62.5% = 5/8: cut the bar into 8 equal blocks and take 5.
- One block is 480 ÷ 8 = 60.
- Five blocks are 5 × 60 = 300.
Try it with paper. Fold a strip of paper in half three times and open it: you have 8 equal parts, each 12.5%. Colour 3 parts and you have 37.5%.
Find the rule
What is 87.5% of 560? Use 87.5% = 7/8.
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Correct answer: A
One eighth of 560 = 560 ÷ 8 = 70, so seven eighths = 7 × 70 = 490.
Option B: 70 is the rest of 560 (560 − 490 = 70), one eighth, not seven.
Option C: 49 is ten times too small: the decimal point slipped one place.
Option D: 6.4 divides 560 by 87.5 (560 ÷ 87.5 = 6.4). A percent of a number is a share of it, not a division by the percent.
Hint: Divide by 8, then multiply by 7.
On a ten-block bar, 3 blocks (30%) are worth 87. What is the whole bar worth?
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Correct answer: B
3 blocks = 87, so 1 block (10%) = 87 ÷ 3 = 29, and the whole bar of 10 blocks = 10 × 29 = 290.
Option A: 26.1 is 30% of 87, but 87 is already the 30% part of the bar.
Option C: 29 is one block, 10% of the bar; the whole bar is 10 blocks.
Option D: 2,610 multiplies 87 by 30. Share 87 among the 3 blocks instead.
Hint: Find one block first.
45 out of 120 is what percent? Simplify the fraction first.
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Correct answer: A
45/120 = 3/8 (divide both by 15), and 3/8 = 37.5%. Check: 37.5% of 120 = 45.
Option B: 62.5% is the part that is not 45, the other five eighths.
Option C: 266⅔% divides 120 by 45: the whole is 120, so divide by 120.
Option D: 3.75% is ten times too small: the decimal point slipped one place.
Hint: Divide 45 and 120 by 15.
Name the rule
- Percent to fraction: x% = x ÷ 100, then simplify. 37.5% = 37.5 ÷ 100 = 3/8.
- A fraction of a number: divide by the bottom, multiply by the top. 3/8 of 720 = 720 ÷ 8 × 3.
- The ten-block bar: each block is 10%. Cut it into 8 and each block is 12.5%.
- The whole from a part: find one block's value, then multiply by the number of blocks in the bar.
A percent is a fraction with 100 below: 12.5% = 12.5/100 = 1/8. Dividing by 8 is quicker than multiplying by 0.125, and it is the same multiplication.
Worked, half-done, solo
Question
What is 62.5% of 480?
Solution
- Turn the percent into a fraction
62.5% = 62.5/100 = 5/8.
- Find one eighth
480 ÷ 8 = 60.
- Take five eighths
5 × 60 = 300.
- Check another way
50% of 480 = 240 and 12.5% of 480 = 60, so 240 + 60 = 300.
Answer: 300
Watch out: 480 ÷ 62.5 = 7.68 is not a share of 480. A percent of a number is the number times the fraction.
Now finish one that is half done.
A tank is 20% full. After 150 litres are added it is 45% full. So the 150 litres fill 45% − 20% = 25% of the tank, a quarter of it. What is the tank's capacity?
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Correct answer: A
25% is a quarter, so the whole tank is 4 × 150 = 600 litres. Check: 20% of 600 = 120 and 45% of 600 = 270, and 270 − 120 = 150.
Option B: 333⅓ litres treats the 150 litres as 45% of the tank (150 ÷ 45 × 100). The 150 litres fill only the gap from 20% to 45%.
Option C: 750 litres treats the 150 litres as 20% of the tank (150 ÷ 20 × 100).
Option D: 37.5 litres is 25% of 150, but 150 litres is already the 25% part.
Hint: The 150 litres are a quarter of the tank.
And one on your own.
A cricket team won 14 of its 21 matches. What percent of its matches did it win?
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Correct answer: A
14/21 = 2/3 (divide both by 7), and 2/3 = 66⅔%.
Option B: 33⅓% is the share of matches it did not win.
Option C: 150% divides 21 by 14: the whole is 21 matches, so divide by 21.
Option D: 6⅔% is ten times too small: the decimal point slipped one place.
Hint: Divide 14 and 21 by 7.
The traps
- The percent read as a denominator: 12.5% is not 1/12 and 33⅓% is not 1/33. Go through ÷ 100: 12.5/100 = 1/8.
- Divided by the percent: 2,000 ÷ 12.5 is not a share of 2,000.
- One block taken as the bar: find one block, then count all the blocks.
- The rest instead of the part: 5/8 is the other part when 3/8 is asked.
- The wrong whole: the fraction is of the number after "of".
Sunil worked out 16⅔% of 270 as 270 ÷ 16 = 16.875. What went wrong?
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Correct answer: A
16⅔% = 1/6, because 16⅔/100 = 1/6. So 16⅔% of 270 = 270 ÷ 6 = 45. Sunil divided by 16, as if the percent were a denominator.
Option B: 1/16 is 6.25%, not 16⅔%.
Option C: 270 × 16⅔ = 4,500 is far more than 270 itself, so it cannot be a part of it.
Option D: 1/3 is 33⅓%, twice 16⅔%.
Hint: 16⅔% is half of 33⅓%.
Mixed check
What is 33⅓% of 1,350?
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Correct answer: A
33⅓% = 1/3, and 1,350 ÷ 3 = 450.
Option B: 900 is the other two thirds (1,350 − 450 = 900).
Option C: 40.5 divides 1,350 by 33⅓, which is not a share.
Option D: 45 is ten times too small: the decimal point slipped one place.
Hint: 33⅓% is one third.
From lesson 6:
A price falls 20% and then rises 10% twice. What is the overall change?
Show answer
Correct answer: A
0.8 × 1.1 × 1.1 = 0.968, so the price ends at 96.8% of the start: a 3.2% decrease.
Option B: No change adds the percents (−20 + 10 + 10 = 0). Each change is of a different price.
Option C: A 3.2% increase has the right size but the wrong direction: 96.8% of the start is a fall.
Option D: A 10% increase uses only the last change.
Hint: Multiply 0.8, 1.1 and 1.1.
Quick reference
Percents as fractions
- Percent to fraction
- x ÷ 100, then simplify 37.5% = 3/8
- Common fractions
- 50% = 1/2, 25% = 1/4, 12.5% = 1/8 33⅓% = 1/3, 16⅔% = 1/6
- A fraction of a number
- divide by the bottom, multiply by the top 3/8 of 720: 720 ÷ 8 × 3 = 270
- Ten-block bar
- 1 block = 10% 3 blocks = 87, so the bar = 290
- Whole from a part
- one block × the number of blocks 25% = 150, so 4 × 150 = 600
Why it works: a percent is a fraction with 100 below; simplified, it divides quickly and gives the same answer.
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