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Daily Maths

Daily Maths

Percentage increase and decrease

Percentages- Formula Core + Extended 10 min read

On this page
  1. In this note
  2. Formula bank
  3. The multiplier
  4. Increasing or decreasing by a percentage
  5. Finding the percentage increase or decrease
  6. One change after another Extended
  7. Ways they test it
  8. Mistakes to avoid
  9. Summary
  10. Practice
  11. Can I do this?
  12. Quiz

In this note

Syllabus 1.13
  • 1.13Increase or decrease an amount by a percentage, using a multiplier.
  • 1.13Find the percentage increase or decrease: change ÷ original × 100.
  • 1.13Handle one percentage change after another (repeated change). Extended

Before you start: percentages, fractions and decimals.

Formula bank

Increase by r%Learn it\(\text{new}=\text{original}\times\left(1+\dfrac r{100}\right)\)+15% → × 1.15; +3% → × 1.03.Use it when tax, pay rise, growth, "increase by".
Decrease by r%Learn it\(\text{new}=\text{original}\times\left(1-\dfrac r{100}\right)\)−12% → × 0.88.Use it when discount, sale, loss, "decrease by".
Percentage changeLearn it\(\dfrac{\text{change}}{\text{original}}\times100\)change = new − old; original = the value BEFORE.Use it when "find the percentage increase/decrease".
Repeated changeLearn it\(\text{original}\times m_1\times m_2\); same change n times: \(\times m^{n}\)multiply the multipliers, never add the percentages.Use it when "then", "a further", "each year".

The multiplier

The original amount is 100%. After a 20% increase you have \(100\%+20\%=120\%\) of it; after a 20% decrease you have \(100\%-20\%=80\%\). As decimals those are the multipliers 1.2 and 0.8.

Three bars on the same scale: 100%, 120% and 80% of the originalOriginal: 100%100%After a 20% increase: 120%100%+20%After a 20% decrease: 80%80%−20%
All three bars use the same scale.

General methodHow to write the multiplier for any change

  1. Increase: 100% + the percentage. Decrease: 100% − the percentage.
  2. Change it to a decimal (÷ 100).

+3% → 103% → 1.03 (not 1.3). −12% → 88% → 0.88. +300% → 400% → 4.

Try it yourself

1
Multiplier for an increase of 4%.
Show answer

Answer 1.04

2
Multiplier for a decrease of 25%.
Show answer

Answer 0.75

3
A price is multiplied by 0.93. By what percentage did it go down?
Hint
0.93 = 93%.
Show answer

Answer 7%

Increasing or decreasing by a percentage

General methodHow to increase or decrease any amount by a percentage

  1. Calculator: new amount = original × multiplier.
  2. No calculator: find the percentage with 10%, 5%, 1% pieces, then ADD it (increase) or TAKE IT AWAY (decrease).
  3. Money: 2 decimal places.

Example Increase $64 by 15%.

Multiply by 1.15 (or find 15% and add it).

  1. \(64\times1.15=73.60\)
  2. check: \(10\%=6.40\), \(5\%=3.20\), \(64+9.60=73.60\)

Answer $73.60

Try it yourself

1
Decrease $850 by 12%.
Hint
× 0.88.
Show answer

Answer $748

2
Increase 40 by 300%.
Hint
Increase BY 300% means 400% of it.
Show answer

Answer 160

3
Increase 18 by 40%.
Show answer

Answer 25.2

Finding the percentage increase or decrease

General methodHow to find any percentage change

  1. Change = bigger − smaller.
  2. Original = the value BEFORE the change (the old price, the earlier year, what was paid).
  3. \(\dfrac{\text{change}}{\text{original}}\times100\).
  4. Say whether it is an increase or a decrease.

Calculator shortcut: new ÷ original gives the multiplier. 1.2 means a 20% increase; 0.86 means a 14% decrease.

Example A village grows from 160 people to 200. Find the percentage increase.

Change ÷ original × 100; the original is 160.

  1. change \(=200-160=40\)
  2. \(\tfrac{40}{160}\times100=25\)

Answer 25% increase

Try it yourself

1
A phone's value falls from $250 to $215. Percentage decrease?
Show answer

Answer 14%

2
A price rises from $72 to $90. Percentage increase?
Hint
Divide by 72, the original.
Show answer

Answer 25%

One change after another Extended

The second change is a percentage of the NEW amount, so you multiply the multipliers. Adding the percentages is always wrong: up 10% then down 10% is \(1.1\times0.9=0.99\), which is 1% LESS than you started with.

General methodHow to do any repeated percentage change

  1. Write a multiplier for each change.
  2. Multiply the original by all of them: \(\text{original}\times m_1\times m_2\). The same change n times: \(\times m^{n}\).
  3. Overall percentage change: multiply the multipliers and read the result (0.72 = 28% decrease; 1.026 = 2.6% increase).

Example A car costs $12 000 and loses 15% of its value every year. Find its value after 4 years.

Multiply by 0.85 four times: \(0.85^{4}\).

  1. \(12\,000\times0.85^{4}=6264.075\)

Answer $6264.08

Try it yourself

1
A coat is reduced by 20%, then by a further 10%. What is the total percentage reduction?
Hint
\(0.8\times0.9\).
Show answer

Answer 28%

2
Population grows 5% a year for 3 years. Total percentage growth (3 s.f.)?
Hint
\(1.05^{3}\).
Show answer

Answer 15.8%

Ways they test it

Type 1 Increase or decrease by r%

"Decrease 480 by 15%." (2 marks)

Spot it "increase by" / "decrease by", or tax, discount, pay rise.

  1. Multiplier (or find r% and add/subtract).
  2. Answer with units.

Type 2 Percentage change

"Ravi bought a car for $18 500 and sold it for $15 355. Find the percentage decrease." (2 marks)

Spot it "percentage increase/decrease", "percentage profit/loss".

  1. Change ÷ original × 100.
  2. Original = the first value.

For example $18 500 → $15 355

  1. change \(=3145\)
  2. \(\tfrac{3145}{18\,500}\times100=17\)

Answer 17% decrease

Type 3 Two changes, one after the other

"Population up 8% in 2024, down 5% in 2025. Overall percentage change?" (2 marks)

Spot it "then", "in the next year", "overall".

  1. Multiply the multipliers.
  2. Read the result as a change.

For example +8% then −5%

  1. \(1.08\times0.95=1.026\)

Answer 2.6% increase

Type 4 Same change every year

"A machine loses 18% of its value each year..."

Spot it "each year", "every month", "depreciation".

  1. \(\times m^{n}\).
  2. Round only at the end.

Type 5 Show someone is wrong

"Amina says up 25% then down 20% is up 5%. Show she is wrong."

Spot it "show that" with a claim.

  1. Work out the real value with multipliers.
  2. Compare with the claim.

For example $64, +25% then −20%

  1. \(64\times1.25=80\)
  2. \(80\times0.8=64\)

Answer back to $64, not $67.20

Mistakes to avoid

  • ✗ increase by 3%: × 1.3✓ × 1.033% = 0.03.
  • ✗ decrease by 12%: × 0.12✓ × 0.880.12 finds the decrease; what is LEFT is 88%.
  • ✗ % change = change ÷ new✓ change ÷ originalCompare with what you started with.
  • ✗ +10% then −10% = no change✓ × 1.1 × 0.9 = 0.99: 1% downThe second change acts on the new amount.
  • ✗ 20% off then 10% off = 30% off✓ 0.8 × 0.9 = 0.72: 28% offMultiply the multipliers.

Summary

  • Increase by r%: × (1 + r/100). Decrease by r%: × (1 − r/100).
  • Percentage change = change ÷ ORIGINAL × 100.
  • Repeated change: multiply the multipliers; same change n times: \(m^{n}\).
  • Increase BY 300% = × 4; 300% OF = × 3.

If you see …

  • "increase / decrease by r%"× the multiplier.
  • "percentage increase / decrease"change ÷ original × 100.
  • "then", "each year"multiply the multipliers.

Practice

Practice: easy, medium, hard

Easy

1
Increase $250 by 4%.
Show working
  1. \(250\times1.04=260\)

Answer $260

2
Decrease 200 by 25%.
Show working
  1. \(200\times0.75=150\)

Answer 150

3
A ticket rises from $40 to $46. Percentage increase?
Show working
  1. \(\tfrac{6}{40}\times100=15\)

Answer 15%

Medium

1
Increase 60 by 35% without a calculator.
Hint
10% = 6, 5% = 3.
Guide me

What is it asking? the new value

What do we know? 60, +35%

Plan 35% = 30% + 5%, then add

Show working
  1. \(35\%=18+3=21\)
  2. \(60+21=81\)

Answer 81

2
Shoes cost $1500. In a sale they cost $1260. Percentage reduction?
Guide me

What is it asking? % decrease

What do we know? original $1500

Plan change ÷ original × 100

Show working
  1. \(\tfrac{240}{1500}\times100=16\)

Answer 16%

3
A gym had 5600 members. Membership rose by 3.5%. How many members now?
Guide me

What is it asking? the new number

What do we know? +3.5%

Plan × 1.035

Show working
  1. \(5600\times1.035=5796\)

Answer 5796

Hard

1
A machine costs $16 000. Its value falls 18% in year 1 and 10% in year 2. Find (a) its value after 2 years (b) the overall percentage decrease.
Hint
(b) \(0.82\times0.9=0.738\).
Guide me

What is it asking? a value and an overall %

What do we know? −18% then −10%

Plan multiply the multipliers; compare with 16 000

Show working
  1. (a) \(16\,000\times0.82\times0.9=11\,808\)
  2. (b) \(0.738\): \(100-73.8=26.2\)

Answer (a) $11 808 (b) 26.2%

PartAnswer / workingMarksGuidance
(a)\(16\,000\times0.82\times0.9\)M1
11 808A1
(b)\(\tfrac{11\,808}{16\,000}\)M1
26.2A1not 28
Total: 4 marks
2
Fatima earns $45 000. After a 12% pay rise her pay falls by 12%. What does she earn now?
Guide me

What is it asking? her final pay

What do we know? +12% then −12%

Plan × 1.12 × 0.88

Show working
  1. \(45\,000\times1.12=50\,400\)
  2. \(50\,400\times0.88=44\,352\)

Answer $44 352

PartAnswer / workingMarksGuidance
\(45\,000\times1.12\times0.88\)M1
44 352A1
Total: 2 marks
3
A rectangle's length goes up 20% and its width goes down 15%. Percentage change in its area?
Hint
Area = length × width, so multiply the multipliers.
Guide me

What is it asking? % change in area

What do we know? ×1.2 and ×0.85

Plan area multiplier = 1.2 × 0.85

Show working
  1. \(1.2\times0.85=1.02\)
  2. 2% increase

Answer 2% increase

PartAnswer / workingMarksGuidance
\(1.2\times0.85\)M1
2% increaseA1
Total: 2 marks

Can I do this?

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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