Percentage increases and decreases
🎯 In this topic you will
- Calculate percentage increases and decreases
- Write a change in value as a percentage
🧠 Key Words
- absolute change
- percentage decrease
- percentage increase
Show Definitions
- absolute change: The actual numerical difference between two values, without considering direction.
- percentage decrease: How much a value goes down, expressed as a percentage of the original amount.
- percentage increase: How much a value goes up, expressed as a percentage of the original amount.
The price of a train journey increases from $75$ to $105$.
The price increase is $105 - 75 = 30$.
To find the percentage increase, you must write the increase as a percentage of the original price.
That is $\tfrac{30}{75} \times 100\% = 0.4 \times 100\% = 40\%$
🔎 Reasoning Tip
$30 \div 75 = 0.4$
Suppose the price decreases from $75$ to $60$. The decrease is $15$.
You can write this as a percentage of the original price in a similar way:
$\tfrac{15}{75} \times 100\% = 0.2 \times 100\% = 20\%$
The percentage decrease is 20%.
For an increase or a decrease, $75$ is the denominator of the fraction.
❓ EXERCISES
1.
a. Find 15% of $70
b. Increase $70 by 15%
c. Decrease $70 by 15%
👀 Show answer
b. $70 + 10.5 = 80.5$
c. $70 - 10.5 = 59.5$
2.
a. Find 80% of 3200 people
b. Increase 3200 by 80%
c. Decrease 3200 by 80%
👀 Show answer
b. $3200 + 2560 = 5760$
c. $3200 - 2560 = 640$
3.
a. Find 2% of 19.00 kg
b. Increase 19.00 kg by 2%
c. Decrease 19.00 kg by 2%
👀 Show answer
b. $19 + 0.38 = 19.38$ kg
c. $19 - 0.38 = 18.62$ kg
4. Sarah has saved $240. How much will she have if she increases her savings by:
a. 10% b. 50% c. 70% d. 100% e. 120%
👀 Show answer
b. $240 + 50\% = 240 + 120 = 360$
c. $240 + 70\% = 240 + 168 = 408$
d. $240 + 100\% = 240 + 240 = 480$
e. $240 + 120\% = 240 + 288 = 528$
5. The population of a town is 45,000. It is expected to rise by 85% in the next ten years. Estimate the population in ten years’ time.
👀 Show answer
New population = $45000 + 38250 = 83250$
6. Show that:
a. 81 is 135% of 60 b. 60.8 is 190% of 32 c. 308 is 220% of 140
👀 Show answer
b. $190\% \times 32 = 1.9 \times 32 = 60.8$ ✅
c. $220\% \times 140 = 2.2 \times 140 = 308$ ✅
7.
a. What percentage of 950 is 380?
b. What percentage of 380 is 950?
👀 Show answer
b. $ \tfrac{950}{380} \times 100 = 250\% $
8.
a. What percentage of 40 years is 8 years?
b. What percentage of 8 years is 40 years?
👀 Show answer
b. $\tfrac{40}{8} \times 100 = 500\%$
9. A metal bar is 1.80 m long. It is heated and the length increases by 0.5%.
a. What is the absolute increase in length?
b. How long is the bar now?
👀 Show answer
b. $1.80 + 0.009 = 1.809$ m
10. Work out:
a. 20% of 60 km b. 90% of 60 km c. 170% of 60 km d. 260% of 60 km
👀 Show answer
b. $90\% \times 60 = 54$ km
c. $170\% \times 60 = 102$ km
d. $260\% \times 60 = 156$ km
11. Copy and complete this table.
| Amount | 40% | 140% | 280% | 420% |
|---|---|---|---|---|
| $20 | $8 | $84 | ||
| 50 kg | ||||
| 90 m | 126 m |
👀 Show answer
| Amount | 40% | 140% | 280% | 420% |
|---|---|---|---|---|
| $20 | $8 | $28 | $56 | $84 |
| 50 kg | 20 kg | 70 kg | 140 kg | 210 kg |
| 90 m | 36 m | 126 m | 252 m | 378 m |
12. The mass of a child is 22 kg. In the next 10 years, this mass increases by 150%.
a. Find 150% of 22 kg.
b. Find the mass after 10 years.
👀 Show answer
b. $22 + 33 = 55$ kg
13. A shop lists its prices in a table:
| Item | Price |
|---|---|
| Table | $280 |
| Armchair | $520 |
| Bed | $1040 |
a. In a sale, all the prices are reduced by 30%. Calculate the sale prices.
b. How much would you save if you bought all three items in the sale?
👀 Show answer
Table: $280 - 30\% = 280 \times 0.7 = 196$
Armchair: $520 \times 0.7 = 364$
Bed: $1040 \times 0.7 = 728$
b. Total savings = $280 + 520 + 1040 - (196 + 364 + 728) = 1840 - 1288 = 552$
14. Electricity costs are rising by 8%.
The table shows the costs for one year for four customers.
Copy the table and fill in the last column to show the costs for one year after the price rise.
| Customer | Cost before the rise | Absolute change ($) | Cost after the rise |
|---|---|---|---|
| A | $415 | ||
| B | $629 | ||
| C | $1390 |
👀 Show answer
| Customer | Cost before the rise | Absolute change ($) | Cost after the rise |
|---|---|---|---|
| A | $415 | $33.20 | $448.20 |
| B | $629 | $50.32 | $679.32 |
| C | $1390 | $111.20 | $1501.20 |
15. A garage is reducing the prices of cars. Calculate the new prices.
| Model | Old price ($) | Decrease (%) | Absolute change ($) | New price ($) |
|---|---|---|---|---|
| Ace | 15,800 | 2.0 | ||
| Beta | 21,300 | 12.0 | ||
| Carro | 24,200 | 0.5 |
👀 Show answer
| Model | Old price ($) | Decrease (%) | Absolute change ($) | New price ($) |
|---|---|---|---|---|
| Ace | 15,800 | 2.0 | 316 | 15,484 |
| Beta | 21,300 | 12.0 | 2,556 | 18,744 |
| Carro | 24,200 | 0.5 | 121 | 24,079 |
16. Mia sees this sign in a shop window:
She says: "The original price of a coat was $120 so the price is now $84."
a. Explain the calculation that Mia has done and why her statement is incorrect.
b. What is the price of the coat now?
👀 Show answer
a) Mia worked out 70% of $120 (which is $84) and assumed this was the new price. Her mistake is that the coat is reduced by 70%, so $84 is the amount of the discount, not the final price.
b) New price = $120 − $84 = $36.
🧠 Think like a Mathematician
Task: Explore what happens when a number is increased by $P\%$ and then decreased by $P\%$. Does the final result depend on $P$?
Questions:
👀 show answer
- a) Start with 2000. Increase by 50%: $2000 \times 1.5 = 3000$. Then decrease by 50%: $3000 \times 0.5 = 1500$. Final number = 1500.
- b) Increase by 25%: $2000 \times 1.25 = 2500$. Decrease by 25%: $2500 \times 0.75 = 1875$. Final number = 1875.
- c) For any $P > 0$, the final number is always less than the original 2000. The loss is proportional to $P^2$. General formula: $2000 \times (1 + \tfrac{P}{100})(1 - \tfrac{P}{100}) = 2000 \times (1 - \tfrac{P^2}{10000})$.
- d) Increasing and then decreasing by the same percentage never returns you to the starting number (unless $P = 0$). The result is always lower than the original.
❓ EXERCISE 18
18. A shop is selling a phone for $80. The shop increases the price by 10%.
a) Find the new price.
After two weeks, the shop decreases the new price by 10%. Read what Arun and Sofia say:
- Arun: "The price will go back down to $80."
- Sofia: "The price now will be less than $80."
b) Explain why Arun is wrong and Sofia is correct.
c) Find the price of the phone after the decrease.
👀 Show answer
a) 10% of $80 = $8. New price = $80 + $8 = $88.
b) Arun is wrong because a 10% decrease is taken from the new price ($88), not the original $80. Sofia is correct because the decrease brings the price below $80.
c) 10% of $88 = $8.80. New price = $88 − $8.80 = $79.20.
19. The same shop is selling a television for $400.
a) The shop increases the price by 20%. Find the new price.
b) The shop increases the price by a further 20%. Here are three statements:
- The new price is $560
- The new price is more than $560
- The new price is less than $560
Which statement is correct? Give a reason for your answer.
c) Show your answer to a partner. Is he or she convinced by your explanation?
👀 Show answer
20. Sofia has savings of $500. She spends some money and says: “My savings have decreased by 150%.”
a. Is it possible for her savings to decrease by more than 100%?
Arun has 500 g of rice. He says: “I cooked some rice and the amount I have has decreased by 150%.”
b. What can you say about this statement?
👀 Show answer
a) No, a decrease cannot be more than 100% because that would mean she owes money. The maximum possible decrease is 100%, leaving her with nothing.
b) Arun’s statement is also incorrect. He cannot lose more than 100% of his rice. A decrease of 150% would mean he has negative rice, which is not possible.
⚠️ Be careful! Percentage Increase/Decrease & Change
- Always use the original as the base. Percentage change = $\dfrac{\text{new}-\text{original}}{\text{original}}\times100\%$ — not over the new value.
Example: $75\to105$: $\dfrac{30}{\color{black}{75}}=40\%$ (not $\dfrac{30}{105}$). - Successive changes multiply, they don’t add. +20% then +20% is factor $1.2\times1.2=1.44$ → +44%, not +40%.
- +P% then −P% ≠ 0% overall. The final factor is $ (1+\tfrac{P}{100})(1-\tfrac{P}{100})=1-\big(\tfrac{P}{100}\big)^2 < 1$.
Example: $80 \xrightarrow{+10\%} 88 \xrightarrow{-10\%} 79.2$ (below $80$). - Increase/decrease by P% with a single multiplier. New value = original × $(1+\tfrac{P}{100})$ (increase) or $(1-\tfrac{P}{100})$ (decrease).
Example: decrease 37%: factor $0.63$; increase 84%: factor $1.84$. - Distinguish % change from percentage points. Going from 40% to 50% is a +10 percentage points change, which is a +25% relative increase.
- Large decreases have limits. You cannot decrease by more than 100% of a positive amount (that would make it negative).
- Sense-check with easy fractions/decimals. 25% = $\tfrac14$, 10% = $\tfrac{1}{10}$, 5% = $\tfrac{1}{20}$, 1% = $\tfrac{1}{100}$.
- Reverse problems: If $y$ is $P\%$ of $x$, then $x=\dfrac{y}{P/100}$. Don’t subtract the percent from $y$.
Example: “$81$ is $135\%$ of what?” $x=\dfrac{81}{1.35}=60$.