Percents — Print
Per cent means per hundred, so every percent is a fraction with a hundred underneath.
What the practice questions cover
These questions use 2 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Percent — Parts per hundred. 37% = 37100 = 0.37.
- Percent increase — How much something grew, as a percentage of what it was.
Shade sixty-three squares of a hundred-square grid.
How much is shaded? Sixty-three out of a hundred — which is what "sixty-three per cent" says out loud.
The same picture is also 63100, and also 0.63. One grid, three names.
Nothing has been converted. They were always the same amount, written differently.
A percent is not a new kind of number. It is a fraction whose bottom is always a hundred, which is exactly what makes percents easy to compare.
Three ways to write one amount
63%, 63100 and 0.63 are the same amount in three notations.
Percent to decimal: divide by 100, which moves the point two places left.
Decimal to percent: multiply by 100, moving it two places right.
Two places, either way
The only question is which direction. Percents are the bigger-looking numbers, so going to a decimal always makes the number smaller.
Finding a percent of something
Turn the percent into a decimal and multiply.
25% of 80 is 0.25 × 80 = 20.
For friendly percents, mental shortcuts are faster: 10% is a tenth, 50% is a half, 25% is a quarter.
Build from 10%
Find 10% by moving the point one place, then scale. 30% is three of those; 5% is half of one.
Discounts and increases in one step
A 20% discount leaves 80%, so multiply by 0.8 — one step, not two.
A 15% increase leaves 115%, so multiply by 1.15.
Working out the change and then subtracting gives the same answer with more chances to slip.
The multiplier carries the whole story
The 1 in 1.15 keeps what you already had; the .15 adds the growth. Multiplying by 0.15 alone throws the original away.
Percent change
Percent change compares the change with the original, not with the new value.
A price rising from $40 to $52 has changed by $12, and 1240 = 0.3, so 30%.
Dividing by the new price instead gives a different and wrong answer.
Always over the original
Up 30% then down 30% does not return you to the start, because the second percent is taken of a larger number.
Working backwards
If a sale price is $36 after 25% off, then $36 is 75% of the original.
So divide: 36 ÷ 0.75 = 48.
Adding 25% back onto $36 would give $45, which is not the price it started at.
Undo by dividing
Multiplying got you there, so dividing gets you back. Adding the percent back on is the commonest error in this unit.
Worked examples
A percent of a number
Find 35% of 60.
- 10% of 60 is 6. Move the point one place.
- So 30% is 18, and 5% is 3. Scale from the friendly one.
- 18 + 3 = 21. Or directly: 0.35 × 60 = 21.
21
A discount
A $60 coat is reduced by 20%. What is the sale price?
- A 20% discount leaves 80%. What you keep, not what comes off.
- Multiply by 0.8: 60 × 0.8. One step.
- = $48. Check: the saving is $12, which is a fifth of $60.
$48
Working backwards
After 25% off, a game costs $36. What was the original price?
- 25% off leaves 75%, so $36 is 0.75 of the original. Identify what the $36 represents.
- Divide: 36 ÷ 0.75. Undo the multiplication.
- = $48. Check: 25% of 48 is 12, and 48 - 12 = 36.
$48
Words to know
- Percent
- A number out of a hundred, written with %.
- Percent of a number
- A part found by multiplying, as in 25% of 80.
- Percent increase
- How much something grew, as a percentage of what it was.
- Discount
- An amount taken off a price, usually given as a percent.
- Multiplier
- The single number you multiply by, such as 0.8 for 20% off.
Mistakes to watch for
Multiplying by the discount instead of what is left. A 20% discount on $60 means multiplying by 0.8 to get $48 — multiplying by 0.2 gives $12, which is the saving, not the price.
Adding the percent back to reverse a discount. $36 after 25% off came from $48, found by dividing by 0.75. Adding 25% of $36 gives $45, which is simply a different number.
Dividing by the new value for percent change. The change is always compared with the original: $40 to $52 is 1240 = 30%, not 1252.
Moving the decimal point the wrong way. 63% is 0.63, not 6.3. Percents look bigger, so the decimal must come out smaller.
Assuming up then down by the same percent returns to the start. A $100 item up 20% is $120; down 20% from there is $96, because the second percent is taken of a larger amount.
"Per cent" is Latin for "by the hundred". Roman emperors levied a tax of 1100 on goods sold, and the idea stuck.
The % symbol evolved from a scribbled abbreviation of "per 100" — you can still see the two zeros in it.
Percentages above 100 are perfectly sensible: a price that triples has risen by 200%, because the increase is twice the original.
Problem 1
Write 63% as a decimal.
Worked answer
0.63
Move the point two places left
Problem 2
Write 0.4 as a percent. Give just the number.
Worked answer
40
0.4 × 100 = 40
Problem 3
Find 25% of 80.
Worked answer
20
80 ÷ 4 = 20
Problem 4
Find 10% of 340.
Worked answer
34
34.0
Problem 5
Find 50% of 96.
Worked answer
48
96 ÷ 2 = 48
Problem 6
Find 35% of 60.
Worked answer
21
10% = 6, so 30% = 18
5% = 3
18+3 = 21
Problem 7
Find 15% of 240.
Worked answer
36
24 + 12 = 36
Problem 8
A $60 coat is reduced by 20%. Find the sale price.
Worked answer
48
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Problem 9
A $45 game is reduced by 10%. Find the sale price.
Worked answer
40.5
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Problem 10
A $80 price rises by 15%. Find the new price.
Worked answer
92
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Problem 11
Write 35 as a percent. Give just the number.
Worked answer
60
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Problem 12
A price rises from $40 to $52. Find the percent increase.
Worked answer
30
Change = 12
1240 = 0.3
Problem 13
A price falls from $50 to $35. Find the percent decrease.
Worked answer
30
Change = 15
1550 = 0.3
Problem 14
After 25% off, a game costs $36. What was the original price?
Worked answer
48
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Problem 15
30% of a number is 21. Find the number.
Worked answer
70
21 ÷ 0.3 = 70
Problem 16
A $100 item rises 20% then falls 20%. What is the final price?
- $96
- $100
- $104
- $80
Worked answer
$96
Up to $120
20% of 120 is 24
120-24 = 96