Percentages as multipliers, and ratios
What this lesson covers
- Percentages are a way of expressing a fraction of 100.
- Percentages can be used to proportionally increase or decrease a quantity by multiplication and can be presented as decimal multipliers. A percentage increase can be described by the additional percentage or the percentage of the final amount compared to the original amount (e.g. a 20% increase represents 120% of the original amount). A percentage decrease can be described by the percentage lost or the percentage of the final amount compared to the original amount (e.g. a 20% decrease represents 80% of the original amount).
- Ratios show part-to-part or part-to-whole comparisons of two or more quantities.
A percentage is an instruction
- Per cent means per hundred. Nothing more mysterious than that.
- So a percentage is a fraction with the bottom fixed at 100.
- And a fraction of something is an instruction to multiply.
Three ways to write the same thing
- 45 percent is 45 out of 100.
- As a fraction that is 45 over 100, which cancels to 9 over 20.
- As a decimal, 0.45. Divide by a hundred and you are there.
| Form | Written | Note |
|---|---|---|
| 45% | 45 out of 100 | |
| As a fraction | 45/100 | cancels to 9/20 |
| As a decimal | 0.45 | divide by 100 |
| Per cent | means per hundred | that is all |
Finding a part of something
- 15 percent of 240. The word "of" means multiply.
- So it is 0.15 times 240, which is 36.
- A fraction of an amount works exactly the same way.
| Question | Working | Answer |
|---|---|---|
| 15% of 240 | 0.15 × 240 | 36 |
| 3/8 of 240 | 240 ÷ 8 × 3 | 90 |
| 35% of 260 | 0.35 × 260 | 91 |
| The word "of" | means multiply | every time |
This out of that, as a percentage
- 18 out of 40. Divide first: 18 over 40 is 0.45.
- Times by a hundred and it is 45 percent.
- It does not have to come out tidy. 7 out of 30 is 23.3 percent.
| Question | Dividing | As a percent |
|---|---|---|
| 18 out of 40 | 18 ÷ 40 = 0.45 | 45% |
| As a fraction | 18/40 | 9/20 |
| 7 out of 30 | 7 ÷ 30 = 0.2333… | 23.3% |
| Divide, then | times by 100 |
A rise or a fall is one multiplication
- Going up 20 percent means you end with 120 percent of it.
- So multiply by 1.2. One multiply, not an add-on afterwards.
- Going down 20 percent leaves 80 percent, so multiply by 0.8.
| Change | Multiplier | Same as saying |
|---|---|---|
| Up 20% | × 1.2 | 120% of it |
| Down 20% | × 0.8 | 80% of it |
| Up 5% | × 1.05 | 105% of it |
| Down 35% | × 0.65 | 65% of it |
Two sentences, one multiplication
- "A 20 percent increase" and "120 percent of the original".
- Those are the same instruction in different words.
- Being able to swap between them is most of this topic.
Up 20 percent, then down 20 percent
- A hundred dollars goes up 20 percent, to 120.
- Then down 20 percent. You do not get a hundred back.
- You get 96, because that 20 percent was 20 percent of 120.
| Step | Working | Amount |
|---|---|---|
| Start | $100 | |
| Up 20% | × 1.2 | $120 |
| Down 20% | × 0.8 | $96 |
| Why | the 20% was of $120 | a bigger amount |
Finding the whole amount back
- 20 percent of some amount is 30. What was the amount?
- It got to 30 by multiplying by 0.2, so divide by 0.2.
- 150. And check it forwards: 0.2 times 150 is 30.
| Question | Working | Answer |
|---|---|---|
| 20% of it is 30 | 30 ÷ 0.2 | 150 |
| Check | 0.2 × 150 = 30 | correct |
| 15% of it is 45 | 45 ÷ 0.15 | 300 |
| Going back | divide, never multiply |
The one that catches everybody
- After a 20 percent rise a jacket costs 120 dollars.
- Taking 20 percent off 120 gives 96, and it is wrong.
- It arrived by multiplying by 1.2, so divide by 1.2. A hundred.
| Step | What happens | Result |
|---|---|---|
| After a 20% rise | the price is $120 | |
| Tempting | take 20% off $120 | $96, wrong |
| It got here by × 1.2 | so divide by 1.2 | $100 |
| Check | 100 × 1.2 = 120 | correct |
Which direction are you going?
- Going with the change, multiply.
- Coming back from it, divide by the same multiplier.
- Every percentage question is one of those two.
A ratio, and the number nobody sees
- 3 to 2 compares two parts against each other.
- But there are five parts altogether, and five is the useful number.
- So 3 to 2 is three fifths and two fifths of the whole.
| Written | Means | Value |
|---|---|---|
| 3 : 2 | part to part | two groups |
| Parts altogether | 3 + 2 | 5 |
| As part to whole | 3/5 and 2/5 | |
| Not 3/2 and 2/3 | those are something else |
Sharing in a ratio
- Share 40 in the ratio 3 to 2. Add them: five parts.
- Forty divided by five is eight, so one part is eight.
- Three parts is 24, two parts is 16, and they add back to 40.
| Step | Working | Result |
|---|---|---|
| Share 40 in 3 : 2 | 5 parts | |
| One part | 40 ÷ 5 | 8 |
| Three parts, two parts | 3 × 8, 2 × 8 | 24 and 16 |
| Check | 24 + 16 | 40 |
The check that costs five seconds
- Add your shares back up. They must give the original.
- Run a percentage answer forwards again and see if it lands.
- Both catch the wrong-direction error, which is the common one.
What to take away
- Seven lines, and the fourth and fifth are the marks.
| A percentage | is an instruction to multiply |
| "of" means multiply | 15% of 240 is 0.15 × 240 |
| This out of that | divide, then times 100 |
| Up 20% is × 1.2 | which is 120% of the original |
| Going back | divide by the multiplier |
| Up then down by 20% | does not return |
| A ratio 3 : 2 | has five parts in it |
Your turn: four questions
- Pause here and work all four.
- Question 3 is the backwards one. Divide.
- Question 4 starts by adding the two numbers in the ratio.
| Question | |
|---|---|
| 1 | Find 35% of 260. |
| 2 | 24 is what percentage of 80? |
| 3 | 15% of an amount is 45. Find it. |
| 4 | Share $72 in the ratio 5 : 4. |
Show answer
1. 91. Multiply by 0.35, because "of" means multiply. 2. 30%. Divide 24 by 80 to get 0.3, then times by a hundred. 3. 300. This is the backwards direction, so divide rather than multiply: 45 ÷ 0.15 = 300. Check it by going forwards again, 0.15 × 300 = 45. 4. $40 and $32. Five and four make nine parts, and $72 ÷ 9 is $8 a part. Five parts is $40 and four parts is $32, and they add back to $72, which is the check worth doing every time.
The answers
- 0.35 times 260 is 91.
- 24 over 80 is 0.3, so 30 percent.
- 45 divided by 0.15 is 300.
- Nine parts of eight dollars: 40 dollars and 32 dollars.
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