Cambridge Lower Secondary · Mathematics 0862 · Stage 8 · Number
Percentage Increase, Decrease and Absolute Change
Increase and decrease amounts by a percentage with and without a calculator, use multipliers, and find and compare absolute and percentage changes.
British & IGCSE › Checkpoint 8 › Checkpoint 8 Maths — Complete Course
- Understand percentage increase and percentage decrease, and absolute change — increase or decrease an amount by a percentage, and find the change as an amount and as a percentage of the original (8Nf.05)
Prices go up, sales take money off, populations grow, a phone battery runs down. There are two ways to describe any change: the absolute change (the actual amount it went up or down by, in dollars, kilograms, people…) and the percentage change (how big the change is compared with the original amount). This page teaches both, and shows why they can tell different stories.
Key words
| Word / phrase | Meaning / use |
|---|---|
| percentage (%) | نسبة مئوية — a number out of 100: 15% = 15/100 = 0.15 |
| original amount | القيمة الأصلية — the amount before the change; percentages of change are always of this amount |
| increase | زيادة — go up |
| decrease | نقص / تخفيض — go down |
| absolute change | التغير المطلق — the actual amount of the change: new amount − original amount |
| percentage change | التغير بالنسبة المئوية — the change as a percentage of the original: change ÷ original × 100 |
| multiplier | المضروب فيه (المعامل) — the single number you multiply by: increase by 15% → × 1.15 |
| discount | خصم — a decrease in price, often given as a percentage |
1. Absolute change and percentage change
Absolute change = new amount − original amount. It has the same units as the amounts. A positive answer is an increase; a negative answer is a decrease.
Percentage change = (change ÷ original amount) × 100. It compares the change with where you started.
| Shop A | Shop B | |
|---|---|---|
| Original price | $20 | $200 |
| New price | $25 | $210 |
| Absolute change | $25 − $20 = $5 | $210 − $200 = $10 |
| Percentage change | 5 ÷ 20 × 100 = 25% | 10 ÷ 200 × 100 = 5% |
So "Which price rose more?" has two possible answers. Shop B had the bigger absolute increase; Shop A had the bigger percentage increase. In a Checkpoint question, always say which one you mean, and back it up with numbers.
2. Increasing an amount by a percentage
Method 1 — find the percentage, then add it on. This is best without a calculator. Build the percentage from easy parts: 10% (÷ 10), 5% (half of 10%), 1% (÷ 100), 50% (÷ 2), 25% (÷ 4).
- 10% of $60 = $6, 5% of $60 = $3, so 15% = $6 + $3 = $9.
- Add it on: $60 + $9 = $69.
Method 2 — use a multiplier. The new amount is 100% + 15% = 115% of the original, and 115% = 1.15. So multiply by 1.15: $60 × 1.15 = $69. This is one step on a calculator.
3. Decreasing an amount by a percentage
Method 1: find the percentage and subtract it. Method 2: the new amount is 100% − 20% = 80% of the original, so multiply by 0.8.
| Change | New amount is | Multiplier |
|---|---|---|
| increase by 15% | 115% | × 1.15 |
| increase by 7% | 107% | × 1.07 |
| increase by 40% | 140% | × 1.4 |
| increase by 100% (double) | 200% | × 2 |
| decrease by 20% | 80% | × 0.8 |
| decrease by 35% | 65% | × 0.65 |
| decrease by 3% | 97% | × 0.97 |
| decrease by 12.5% | 87.5% | × 0.875 |
Quick check: an increase multiplier is more than 1; a decrease multiplier is less than 1. An increase of 7% is × 1.07, not × 1.7 (that would be a 70% increase).
- التغير المطلق = الجديد − الأصلي (بالجنيه أو الكيلو…).
- النسبة المئوية للتغير = (التغير ÷ الأصلي) × ١٠٠ — دايماً اقسم على القيمة الأصلية، مش الجديدة.
- زيادة ١٥٪ يعني اضرب في ١٫١٥، ونقص ٢٠٪ يعني اضرب في ٠٫٨.
4. Finding the percentage change
- Find the absolute change: new − original.
- Divide by the original amount.
- Multiply by 100 and write %. Say whether it is an increase or a decrease.
| Original → new | Absolute change | ÷ original × 100 | Percentage change |
|---|---|---|---|
| 80 → 92 | +12 | 12 ÷ 80 × 100 | 15% increase |
| 250 → 215 | −35 | 35 ÷ 250 × 100 | 14% decrease |
| 1.6 kg → 1.2 kg | −0.4 kg | 0.4 ÷ 1.6 × 100 | 25% decrease |
| 30 → 60 | +30 | 30 ÷ 30 × 100 | 100% increase (it doubled) |
Without a calculator write the change as a fraction of the original and make the denominator 100: 12/80 = 3/20 = 15/100 = 15%. And 35/250 = 7/50 = 14/100 = 14%.
5. One change after another
A price of $500 goes up by 10%, then the new price goes down by 10%. Is it back to $500?
- Up 10%: $500 × 1.1 = $550.
- Down 10%: $550 × 0.9 = $495.
- It is not back to $500, because the second 10% is a percentage of $550, not of $500. Overall: absolute change −$5, percentage change 5 ÷ 500 × 100 = 1% decrease.
This is a favourite "Is he correct? Explain" question. Always ask: a percentage of which amount?
Worked examples
Worked example 1 (no calculator) — A gym membership costs $240 a year. The price increases by 15%. Work out the new price. [2 marks]
- 10% of 240 = 24; 5% = 12; so 15% = 36. (1 mark for 36 seen.)
- New price = 240 + 36 = $276.
Common mistake: giving $36 as the final answer. The question asks for the new price, so add the increase on.
Worked example 2 (calculator) — A car is worth $8400. Its value decreases by 35%. Work out its new value. [2 marks]
100% − 35% = 65%, so the multiplier is 0.65. $8400 × 0.65 = $5460. Check: 35% of 8400 = 2940, and 8400 − 2940 = 5460 ✓.
Worked example 3 — The number of students in a school went from 2400 to 2760. (a) Work out the absolute change. (b) Work out the percentage change. [3 marks]
- (a) 2760 − 2400 = 360 students (an increase).
- (b) 360 ÷ 2400 × 100 = 0.15 × 100 = 15% increase. Without a calculator: 360/2400 = 36/240 = 3/20 = 15/100.
Common mistake: dividing by the new number (360 ÷ 2760 = 13.0…%). The percentage change always uses the original.
Worked example 4 — Hoda's bag of flour had a mass of 1.6 kg. After baking it had a mass of 1.2 kg. Work out the percentage decrease in the mass. [2 marks]
Change = 1.6 − 1.2 = 0.4 kg. Percentage = 0.4 ÷ 1.6 × 100 = 0.25 × 100 = 25% decrease. (0.4/1.6 = 4/16 = 1/4.)
Worked example 5 — Two shops sell the same phone. Shop P: was $300, now 20% off. Shop Q: was $280, now $50 off. Which shop is cheaper now? Show your working. [2 marks]
- Shop P: $300 × 0.8 = $240.
- Shop Q: $280 − $50 = $230.
- Shop Q is cheaper (by $10). Notice: Shop P's discount is $60 (bigger absolute discount), but the starting prices were different, so you must compare the final prices.
Worked example 6 — Youssef says, "If a price goes up by 10% and then down by 10%, it ends up the same." Use a price of $500 to show that Youssef is wrong. [2 marks]
$500 × 1.1 = $550 (1 mark). $550 × 0.9 = $495, which is not $500, so Youssef is wrong. The 10% decrease is taken from the bigger amount, $550, so it removes more ($55) than the increase added ($50).
- Giving only the increase (e.g. $36) when the question asks for the new amount.
- Using the wrong multiplier: increase by 7% is × 1.07, not × 1.7 or × 0.07; decrease by 35% is × 0.65, not × 0.35.
- Dividing by the new amount instead of the original when finding a percentage change.
- Mixing up absolute and percentage change — the absolute change has units ($, kg, people); the percentage change has %.
- Thinking that +10% then −10% gets you back to the start.
- Forgetting to say whether the change is an increase or a decrease.
- Non-calculator paper: "Increase 80 by 15%" — build the percentage from 10%, 5% and 1%, and show those steps for method marks.
- Calculator paper: real-life contexts (prices, populations, masses) — use a multiplier and write the calculation, e.g. 8400 × 0.65.
- Two-part questions: "(a) Work out the absolute change (b) Work out the percentage change." Give units in (a) and % in (b).
- "Which is the better deal?" or "Which had the greater increase?" — calculate both, then write a clear sentence comparing them.
- "Is he correct? Explain your answer." — use a numerical example as your evidence (Thinking and Working Mathematically: convincing).
- "Draw a ring around" the correct multiplier or the correct new amount.
Practice
- Absolute change = new − original (with units). Percentage change = change ÷ original × 100 (with %).
- Always compare with the ORIGINAL amount.
- Increase by r%: multiply by (100 + r)/100, e.g. 15% → × 1.15. Decrease by r%: multiply by (100 − r)/100, e.g. 20% → × 0.8.
- No calculator: build percentages from 10%, 5% and 1%.
- A bigger absolute change is not always a bigger percentage change — and +10% then −10% does not return to the start.
All pages in this course
Start here
Number
- Order of Operations and Calculating with Integers
- Factors, Multiples, Primes, HCF and LCM
- Types of Number and Index Laws
- Squares, Cubes and Roots (Positive and Negative)
- Multiplying and Dividing by 0.1 and 0.01, and Significant Figures
- Fractions: Recurring Decimals, Mixed Numbers and Order of Operations
- Percentage Increase, Decrease and Absolute Change
- Ordering and Calculating with Decimals
- Ratio and Proportion
Algebra
- Algebraic Expressions: Simplifying, Expanding and Factorising
- Writing Expressions and Formulae, and Changing the Subject
- Equations and Inequalities
- Sequences and the nth Term
- Functions and Straight-line Graphs (y = mx + c)
- Reading and Interpreting Real-life Graphs
Geometry and measure
- Quadrilaterals and Regular Polygons
- Circumference of a Circle, and Miles and Kilometres
- Area of Parallelograms and Trapezia
- 3D Shapes: Euler's Formula and Plans and Elevations
- Volume of Prisms and Surface Area
- Angles in Triangles and on Parallel Lines
- Constructions with Ruler and Compasses
- Bearings and Midpoints
- Translation, Reflection, Rotation and Enlargement
Statistics and probability
- Data Collection and Sampling
- Representing Data
- Averages, Range and Comparing Distributions
- Probability: Complementary and Combined Events
Practice tests