Using a multiplier
🎯 In this topic you will
- Use a multiplier to calculate a percentage increase or decrease
🧠 Key Words
- multiplier
Show Definitions
- multiplier: A number by which another number is multiplied.
In this section, you will learn a more efficient way to calculate percentage increases and decreases.
Suppose you want to increase$275$ by 65%.
You start with $275 = 100\%$.
Then 65% of $275 = 178.75$ and the total is $453.75$.
$100\% + 65\% = 165\%$
You can find 165% of $275$ in a single calculation.
$165\% = 1.65$ and so $165\% \text{ of } 275 = 1.65 \times 275 = 453.75$
This is the value after the increase of 65%.
To increase the value by 65% you used a multiplier of 1.65.
Now suppose you want to decrease$275$ by 54%.
Again $275 = 100\%$.
So $275 - 54\% = 100\% - 54\% = 46\%$
$46\% = 0.46$ and so $46\% \text{ of } 275 = 0.46 \times 275 = 126.50$
This is the value after a decrease of 54%.
To decrease the value by 54% you used a multiplier of 0.46.
In general,
$\text{original value} \times \text{multiplier} = \text{new value}$
You can also write this as $\text{multiplier} = \tfrac{\text{new value}}{\text{original value}}$.
❓ EXERCISE 10.2
1. What multiplier would you use to:
a) increase a value by 63%
b) decrease a value by 63%
c) increase a value by 103%
d) decrease a value by 88%
👀 Show answer
b) × 0.37
c) × 2.03
d) × 0.12
2. Match each percentage change to the correct multiplier.
(The first one is done for you: A and ii)
A. 50% increase
B. 80% increase
C. 80% decrease
D. 120% increase
E. 20% decrease
F. 20% increase
i. ×0.2 ii. ×1.5 iii. ×1.8
iv. ×1.2 v. ×0.8 vi. ×2.2
👀 Show answers
A → ii (×1.5)
B → iii (×1.8)
C → i (×0.2)
D → vi (×2.2)
E → v (×0.8)
F → iv (×1.2)
3. Write the multiplier for:
a) an increase of 45%
b) an increase of 245%
c) a decrease of 45%
👀 Show answer
b) × 3.45
c) × 0.55
4. Here are some multipliers. Write the percentage change in each case:
a) ×0.75
b) ×1.22
c) ×3.33
d) ×0.33
e) ×0.03
👀 Show answer
b) 22% increase
c) 233% increase
d) 67% decrease
e) 97% decrease
5. Increase each of these numbers by 85%:
a) 40
b) 180
c) 12
👀 Show answer
b) 180 × 1.85 = 333
c) 12 × 1.85 = 22.2
6. Find the value of 45 kg after the following changes:
a) an increase of 20%
b) an increase of 170%
c) a decrease of 60%
👀 Show answer
b) 45 × 2.7 = 121.5
c) 45 × 0.4 = 18
7.
a. The mass of a girl is 26.5 kg. Several years later her mass has increased by 62%. Calculate her new mass. Round your answer to 1 d.p.
b. A man has a mass of 172.4 kg. He reduces his mass by 38%. Calculate his new mass.
👀 Show answers
8.
a. Increase 964 by 65%
b. Increase 357 by 195%
c. Decrease 560 by 84%
👀 Show answers
8a. Increase = 65% of 964 = 0.65 × 964 = 626.6 New value = 964 + 626.6 = 1590.6
8b. Increase = 195% of 357 = 1.95 × 357 = 696.15 New value = 357 + 696.15 = 1053.2
8c. Decrease = 84% of 560 = 0.84 × 560 = 470.4 New value = 560 − 470.4 = 89.6
9. Change each length by the percentage shown.
| Length (mm) | Change | New length (mm) |
|---|---|---|
| 90 | 180% increase | |
| 240 | 12% increase | |
| 660 | 70% decrease | |
| 320 | 7% decrease |
👀 Show answers
9a. 90 mm, 180% increase → New length = 90 × (1 + 1.80) = 90 × 2.8 = 252 mm
9b. 240 mm, 12% increase → New length = 240 × 1.12 = 268.8 mm
9c. 660 mm, 70% decrease → New length = 660 × 0.30 = 198 mm
9d. 320 mm, 7% decrease → New length = 320 × 0.93 = 297.6 mm
10. An athlete has a resting pulse rate of 60 beats per minute. During a race, this increases to 160 beats per minute.
a) Calculate the percentage increase.
b) Calculate the percentage decrease after the race, when his pulse rate falls from 160 to 60 beats per minute.
👀 Show answers
11.
a. Increase 96 by 25%
b. Decrease 200 by 40%
c. Increase 60 by 100%
d. Decrease 240 by 50%
👀 Show answers
11a. 96 × 1.25 = 120
11b. 200 × 0.60 = 120
11c. 60 × 2.00 = 120
11d. 240 × 0.50 = 120
12
a. The population of a town increases from 63,200 by 17%. Calculate the new population.
b. The population of a city increases from 7.35 million to 12.82 million. Calculate the percentage change.
c. The population of an island is 4,120. The population decreases by 16.5%. Calculate the new population.
👀 Show answers
12a. Increase = 17% of 63,200 = 0.17 × 63,200 = 10,744 New population = 63,200 + 10,744 = 73,944
12b. Change = 12.82 − 7.35 = 5.47 million Percentage change = (5.47 ÷ 7.35) × 100 = 74.4%
12c. Decrease = 16.5% of 4,120 = 0.165 × 4,120 = 679.8 ≈ 680 New population = 4,120 − 680 = 3,440
🧠 Think like a Mathematician
Task: Investigate changes in population over time and use percentage change to identify patterns and make predictions.
| Year | Population (millions) |
| 1950 | 554 |
| 1960 | 660 |
| 1970 | 828 |
| 1980 | 1000 |
| 1990 | 1177 |
| 2000 | 1291 |
| 2010 | 1369 |
Questions:
👀 show answer
- a) i. $\dfrac{828-554}{554} \times 100 \approx 49.5\%$ ii. $\dfrac{1291-828}{828} \times 100 \approx 56.0\%$ iii. $\dfrac{1177-660}{660} \times 100 \approx 78.3\%$ iv. $\dfrac{1369-554}{554} \times 100 \approx 147.1\%$
- b) The greatest percentage increase was between 1960 and 1990 (78.3%).
- c) The increase from 2000 to 2010 was $1369 - 1291 = 78$ million (~6%). If growth slowed similarly, the 2020 population could be around $1369 + 78 \approx 1447$ million (≈1.45 billion). This assumes the slowing trend continues.
❓ EXERCISE 14
14. Prices in a shop are reduced. Copy and complete this table.
| Original price | Percentage reduction | Reduced price |
|---|---|---|
| $280 | 20% | |
| $420 | 45% | |
| $620 | $217 | |
| $750 | $705 |
👀 Show answers
Row 1: $280 with 20% reduction → 280 × 0.80 = $224
Row 2: $420 with 45% reduction → 420 × 0.55 = $231
Row 3: $620 reduced to $217 → Reduction = (620−217) ÷ 620 × 100 = 65%
Row 4: $750 reduced to $705 → Reduction = (750−705) ÷ 750 × 100 = 6%
15. The height of a tree is 3.65 m. Find the new height if the height increases by:
a) 15% b) 132% c) 260%
👀 Show answers
16. The depth of water in a well is decreasing. Calculate the percentage reduction from:
a) Monday to Tuesday b) Tuesday to Thursday c) Monday to Friday
| Day | Depth (m) |
|---|---|
| Monday | 5.75 |
| Tuesday | 5.10 |
| Wednesday | 4.31 |
| Thursday | 3.58 |
| Friday | 2.46 |
👀 Show answers
17. Here are two sentences:
- The population of India is 407% of the population of the USA.
- The population of India is 307% more than the population of the USA.
a) Explain why both these sentences can be correct.
b) Compare your explanation with a partner’s. Can you improve your explanation?
👀 Show answers
18. Marcus says: “When 650 is increased by 184%, the answer is 1846.”
a) Describe two different ways to check that Marcus is correct.
b) Which way do you think is better? Give a reason.
👀 Show answers
Method 1: Increase = 184% of 650 = 1.84 × 650 = 1196 New value = 650 + 1196 = 1846
Method 2: Use multiplier = 1 + 1.84 = 2.84 650 × 2.84 = 1846
b) The multiplier method is usually better because it is quicker and avoids two-step calculations.
⚠️ Be careful! Multipliers for % Change
- Build the multiplier from the original. Increase by P% → 1 + P/100; decrease by P% → 1 − P/100.
Example: −54% ⇒ 1 − 0.54 = 0.46 (not 0.54!). - Don’t split into two steps when one will do. “Increase 275 by 65%” = 275 × 1.65, not “find 65% then add” (though both should match).
- Use the correct base when finding % change. Multiplier = new ÷ original. Then % change = (multiplier − 1) × 100%.
Example: 2300 → 2850: 2850/2300 = 1.239 ⇒ +23.9% (not 2850−2300 over 2850). - Successive changes multiply, not add. +20% then +20% ⇒ 1.2 × 1.2 = 1.44 (overall +44%), not +40%.
- +P% then −P% never returns you to the start (for P>0). Net factor = (1+P/100)(1−P/100) = 1 − (P/100)² < 1.
- Distinguish “increase by 195%” vs “is 195% of”.
“Increase by 195%” ⇒ factor 2.95 (original + 1.95×original).
“Is 195% of” ⇒ factor 1.95. - Delay rounding the multiplier. Keep extra decimals in intermediate steps; round only at the end to avoid drift in answers.
- Quick sense-checks. Multiplier > 1 ⇒ result bigger; < 1 ⇒ smaller; = 1 ⇒ unchanged. Compare with friendly fractions (10%, 25%, 50%).