Percentages as multipliers, and working backwards
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 change is one multiplication
- Most people do a rise of 20% in two steps: find it, then add it.
- It works, and it is why they cannot do the reverse.
- One multiplication does the whole thing.
Percent means per hundred
- That is not a memory aid. It is the definition.
- 20% is twenty per hundred, which is 20/100, which is 0.2.
- So every percentage is already a decimal you can multiply by.
| Percentage | Reading | As a decimal |
|---|---|---|
| 20% | 20 per hundred | 20/100 = 0.2 |
| 75% | 75 per hundred | 75/100 = 0.75 |
| 5% | 5 per hundred | 5/100 = 0.05 |
| 150% | 150 per hundred | 150/100 = 1.5 |
Finding a part: just multiply
- 15% of 60 is 0.15 times 60, which is 9.
- No formula, no method. One multiplication.
- A fraction works the same way: three eighths of 64 is 24.
| Question | Working | Answer |
|---|---|---|
| 15% of 60 | 0.15 × 60 | 9 |
| 40% of 250 | 0.4 × 250 | 100 |
| 3/8 of 64 | 0.375 × 64 | 24 |
Raising by 20%, both ways
- The long way: 20% of 80 is 16, and 80 plus 16 is 96.
- The short way: 80 times 1.2 is 96. Same answer.
- 1.2 because you keep all of it and add a fifth more. 120%.
| Question | Method | Result |
|---|---|---|
| 80 raised by 20% | the long way | 80 + 16 = 96 |
| 80 raised by 20% | the multiplier | 80 × 1.2 = 96 |
| Why 1.2? | the original plus a fifth of it | 100% + 20% = 120% |
And cutting by 12%
- If you lose 12%, what is left is 88% of what you had.
- So multiply by 0.88. 250 becomes 220.
- Not subtract 0.12. Multiply by 0.88.
| Question | Method | Result |
|---|---|---|
| 250 cut by 12% | the long way | 250 − 30 = 220 |
| 250 cut by 12% | the multiplier | 250 × 0.88 = 220 |
| Why 0.88? | what is left after losing 12% | 100% − 12% = 88% |
The ones worth recognising instantly
- Up 10% is times 1.1. Up 5% is times 1.05.
- Down 25% is times 0.75. Down 8% is times 0.92.
- Rises go above 1. Falls go below it. Always.
| Change | Multiplier | Which is |
|---|---|---|
| Up 10% | × 1.1 | 110% of it |
| Up 5% | × 1.05 | 105% of it |
| Down 25% | × 0.75 | 75% of it |
| Down 8% | × 0.92 | 92% of it |
Now the question that catches everyone
- 20% of an amount is 30. What was the amount?
- Most people answer 36. Thirty, plus 20% of thirty.
- That is wrong, and here is the way to see why.
| Step | What is happening | Value |
|---|---|---|
| 20% of an amount is 30 | the common wrong answer | 30 + 6 = 36 |
| What actually happened | × 0.2 took us to 30 | |
| So undo it | 30 ÷ 0.2 | 150 |
| Check forwards | 0.2 × 150 | 30, correct |
The same thing with a price tag
- A jacket costs $69 after a 15% rise. What was it before?
- Multiplying by 1.15 got us to 69, so divide by 1.15 to get back.
- Sixty dollars. And check it: 60 times 1.15 is 69.
| Step | What is happening | |
|---|---|---|
| A price is $69 after a 15% rise | so × 1.15 gave 69 | |
| Undo it | 69 ÷ 1.15 | $60 |
| Check forwards | 60 × 1.15 | $69, correct |
Always run it forwards afterwards
- A reversed calculation is easy to get backwards.
- So take your answer and do the original change to it.
- If you do not land on the number you started with, try again.
Turning one number into a percentage of another
- 18 out of 24. Divide the part by the whole: 0.75.
- Then multiply by a hundred. Seventy-five percent.
- The part goes on top. That is the only thing to get right.
| Question | Working | Answer |
|---|---|---|
| 18 out of 24 | 18 ÷ 24 = 0.75 | 75% |
| 7 out of 20 | 7 ÷ 20 = 0.35 | 35% |
| The rule | divide the part by the whole | then × 100 |
Why the part goes on top
- A percentage answers: how much of the whole is this?
- So the whole is what you are dividing by.
- Get it upside down and you will get an answer over 100%.
One last idea: what a ratio compares
- A class of 30: eighteen boys, twelve girls.
- Boys to girls is 18 to 12, which is 3 to 2. Part to part.
- Boys to the class is 18 to 30, which is 3 to 5. Part to whole.
| Comparison | Kind | Ratio |
|---|---|---|
| 18 boys, 12 girls | part to part | 18:12 = 3:2 |
| 18 boys in a class of 30 | part to whole | 18:30 = 3:5 |
| Same class | different question | check which is asked |
Which one a question wants is not always said
- Both of those describe the same class truthfully.
- They are different numbers because they answer different questions.
- Next lesson is entirely about working with these.
What to take away
- Six things, and the fourth one is where the marks are.
| Percent | means per hundred, so it is a fraction of 100 |
| A rise of 20% | is one multiplication by 1.2 |
| A fall of 20% | is one multiplication by 0.8 |
| To go backwards | divide by the multiplier, do not subtract |
| Always check | by running your answer forwards |
| A ratio | compares part to part, or part to whole |
Your turn
- Pause here and try all four.
- Answers are in the description.
| Question | |
|---|---|
| 1 | Find 35% of 80. |
| 2 | Increase 45 by 20%. |
| 3 | 30% of an amount is 21. What is the amount? |
| 4 | What percentage of 48 is 12? |
Show answer
1. 28, from 0.35 × 80. 2. 54, from 45 × 1.2. 3. 70, from 21 ÷ 0.3. Check it forwards: 0.3 × 70 is 21. If you got 27.3 you added 30% of 21 instead of undoing the multiplication. 4. 25%, from 12 ÷ 48 = 0.25.
The answers
- 0.35 × 80, then 45 × 1.2.
- Question 3 is 21 ÷ 0.3, not 21 plus 30%.
- Check forwards: 0.3 × 70 is 21.
| Answer | |
|---|---|
| 1 | 28 |
| 2 | 54 |
| 3 | 70 |
| 4 | 25% |
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