% Percentages — T Level / Level 3 Engineering Maths
Percentage means "out of 100". Engineers use percentages every day — efficiency ratings, material composition, tolerance errors, scale factors and percentage changes in dimensions all rely on this skill.
T Level Core 4.1Per cent = out of 100Multiplier methodReverse percentagesEngineering context
Key Idea% means "out of 100" — here's what common percentages look like on a scale
💡 Per cent literally means "per hundred" — from Latin per centum. 37% means 37 out of every 100. It's just a fraction with 100 on the bottom!
Why percentages?3 reasons engineers need them
Efficiency — "this motor is 85% efficient" means 85 out of every 100 joules of input becomes useful output
Composition — "an alloy is 72% steel" compares to a standard base of 100
Changes — "load increased by 15%" expresses growth or reduction on a universal scale
📌 In T Level exams, percentages appear in: efficiency calculations, material composition, tolerance as % of dimension, and percentage change questions.
The big pictureThree things you need to master
① Converting: % ↔ decimal ↔ fraction
② Finding a % of an amount
③ % increase / decrease
④ Expressing one thing as a % of another
⑤ Reverse % (find the original value)
✅ Master the multiplier method — one technique handles increase, decrease AND reverse percentage!
🔄Converting%, decimal, fraction
🔢Find a %of an amount
📈% Changeincrease & decrease
✖️Multiplierfastest method
📐Express as %A as % of B
🔁Reverse %find original
⚙️Engineeringreal uses
🔄
Converting Between % , Decimals & Fractions
These three are all different ways to write the same number. You need to switch between them instantly.
The Conversion TriangleHow the three forms connect
% → DecimalDivide by 100 (move decimal 2 places left)
75% ÷ 100 = 0.75
8% ÷ 100 = 0.08
12.5% ÷ 100 = 0.125
130% ÷ 100 = 1.3
💡 Just shift the decimal point TWO places to the left. 45% → 0.45
⚠️ 8% → 0.08 NOT 0.8 — you need two moves, not one!
Decimal → %Multiply by 100 (move decimal 2 places right)
0.35 × 100 = 35%
0.07 × 100 = 7%
0.125 × 100 = 12.5%
1.2 × 100 = 120%
💡 Shift the decimal point TWO places to the right. 0.6 → 60%
% → FractionWrite over 100, then simplify
75% = 75/100 = 3/4
40% = 40/100 = 2/5
12% = 12/100 = 3/25
8% = 8/100 = 2/25
💡 Write the percentage over 100, then simplify using HCF.
Quick ReferenceCommon conversions — memorise these for the exam
Percentage
Decimal
Fraction
Memory aid
50%
0.5
1/2
half
25%
0.25
1/4
quarter
75%
0.75
3/4
three quarters
10%
0.1
1/10
one tenth
20%
0.2
1/5
one fifth
33.3%
0.333...
1/3
one third
66.7%
0.667...
2/3
two thirds
1%
0.01
1/100
one hundredth
100%
1.0
1/1
the whole thing
150%
1.5
3/2
one and a half times
🔄 Try it — Converter%, Decimal & Fraction
👆Try: 75% · 8% · 12.5% · switch to Decimal and try 0.35 · 0.07 · 1.2
🔢
Finding a Percentage of an Amount
The most common percentage calculation — "what is X% of Y?" Three methods, same answer.
FormulaPercentage of amount = (percentage ÷ 100) × amount
Three MethodsFinding 30% of £240 — all three give the same answer
💡 For T Level exams, Method ① (decimal multiplier) is fastest and least error-prone. Learn it as your default!
Example — Find 35% of 420 kg
Convert: 35% = 35 ÷ 100 = 0.35
Multiply: 0.35 × 420
Calculate: 0.35 × 400 = 140 0.35 × 20 = 7
Answer:147 kg ✅
⚙️ Engineering — 8% of a 650N load (safety factor)
Context: A safety margin requires 8% of the design load to be added as a buffer. Design load = 650N.
Convert: 8% = 0.08
Multiply: 0.08 × 650 = 52
Answer:Buffer = 52N Total load = 702N ✅
🔢 Try it — Find a Percentage of an AmountAny % of any value
% of
👆Try: 35% of 420 kg · 8% of 650 N · 17.5% of 280 · 130% of 200
📈
Percentage Increase & Decrease
When a value goes up or down by a percentage — the step-by-step method and the faster multiplier method.
📌 Why does this work? If you keep 100% and add 20%, you have 120% = 1.2 of the original. If you remove 15%, you keep 85% = 0.85 of the original.
Multiplier TableCommon multipliers — this is exam gold
Operation
Multiplier
Because…
Increase by 5%
× 1.05
100% + 5% = 105% = 1.05
Increase by 10%
× 1.10
100% + 10% = 110% = 1.10
Increase by 20%
× 1.20
100% + 20% = 120% = 1.20
Increase by 25%
× 1.25
100% + 25% = 125% = 1.25
Increase by 50%
× 1.50
100% + 50% = 150% = 1.50
Decrease by 5%
× 0.95
100% − 5% = 95% = 0.95
Decrease by 10%
× 0.90
100% − 10% = 90% = 0.90
Decrease by 15%
× 0.85
100% − 15% = 85% = 0.85
Decrease by 20%
× 0.80
100% − 20% = 80% = 0.80
Decrease by 30%
× 0.70
100% − 30% = 70% = 0.70
Increase 480 by 35%
Multiplier: 1 + 35/100 = 1.35
Calculate: 480 × 1.35
Answer:648 ✅
Decrease 750 by 22%
Multiplier: 1 − 22/100 = 0.78
Calculate: 750 × 0.78
Answer:585 ✅
⚙️ Engineering — Steel rod expands by 0.8% when heated
Original: rod = 2400 mm at room temperature
Multiplier: 1 + 0.8/100 = 1.008
New length: 2400 × 1.008 = 2419.2 mm ✅
Expansion: 2419.2 − 2400 = 19.2 mm thermal expansion
✖️ Try it — Multiplier MethodOne multiplication
👆Try: Increase 480 by 35% · Decrease 750 by 22% · Increase 2400 by 0.8% — notice how the multiplier is built automatically.
📐
Expressing One Value as a Percentage of Another
"A is what percentage of B?" — used for efficiency ratings, error rates, composition analysis.
FormulaA as a % of B = (A ÷ B) × 100
Visual Method28 as a percentage of 80 — three steps
Example — 45 as a % of 120
Fraction: 45/120
Divide: 45 ÷ 120 = 0.375
× 100: 0.375 × 100 = 37.5% ✅
⚙️ Motor efficiency: output 340W from 400W input
Fraction: 340/400
Divide: 340 ÷ 400 = 0.85
× 100: 0.85 × 100 = 85% efficiency ✅
⚠️ Common mistake: Putting the numbers the wrong way round. Always put the "part" (A) on top and the "whole" (B) on the bottom. 45 as a % of 120 = 45/120, NOT 120/45!
📐 Try it — Express as a PercentageA as % of B
as % of
👆Try: 45 as % of 120 · 340 as % of 400 (motor efficiency) · 12 as % of 400 (defect rate)
🔁
Reverse Percentage
You're given the value AFTER a percentage change. You need to work back to find the ORIGINAL value before the change.
The Key IdeaYou always divide by the multiplier — never subtract the percentage!
FormulaOriginal = New value ÷ multiplier
⚠️ Never subtract the percentage directly! After a 15% increase to give £552: the original is NOT 552 − 15% of 552. The correct method is 552 ÷ 1.15 = £480.
Example — After 20% increase, price = £360. Find original.
Multiplier: increase 20% → ×1.20
Reverse: original = 360 ÷ 1.20
Answer:£300 ✅
Example — After 30% decrease, value = 245. Find original.
Multiplier: decrease 30% → ×0.70
Reverse: original = 245 ÷ 0.70
Answer:350 ✅
⚙️ Engineering — A component has been machined down by 8%. Final length = 460mm. Find original length.
Multiplier: decrease 8% → ×0.92
Reverse: original = 460 ÷ 0.92
Answer:500 mm ✅
🔁 Try it — Reverse PercentageFind the original value
👆Try: After +20%, value=£360 · After −30%, value=245 · After −8%, length=460 mm (machined component)
⚙️
Applications
Where this topic is used in engineering, manufacturing, maintenance and daily life.
🏭Manufacturing:Defect rate: 12/400 = 3%. Quality control target <1%.
🔧Maintenance:Efficiency η=(P_out/P_in)×100%. Pump at 85% efficiency.
A 3m aluminium bar expands by 0.6% when heated. Find the new length.
New length = 3000 × 1.006 = 3018 mm Expansion = 18mm
🔧 T Level Exam Style — Compressor Performance
Problem: A compressor is rated at 92% efficiency. It consumes 5.5 kW of power. Find the useful power output, then find what input power would be needed to produce 5.0 kW of useful output.
A component weighs 2.3 kg after having 15% of its mass removed by machining. What was its original mass? Give your answer to 3 significant figures.
[4]
Decrease of 15% → multiplier = 0.85 Original = 2.3 ÷ 0.85 = 2.7058... = 2.71 kg (3 s.f.) ✅
Q15
A CNC machine produces 480 components in a shift. 3.5% are rejected as defective. How many pass quality control? Then: if the pass rate needs to reach 97.5%, how many more components per shift must pass?