QR: Percentages, Ratios & Proportions
Percentage Calculations
Percentages are the single most commonly tested topic in QR. Master these core operations:
Finding a Percentage of a Value
X% of Y = (X/100) × Y
Example: 23% of 450 = 0.23 × 450 = 103.5
Percentage Change
Percentage change = ((New − Old) / Old) × 100
Example: Price rose from $80 to $92. Change = ((92−80)/80) × 100 = (12/80) × 100 = 15% increase
Reverse Percentages
If a value AFTER a percentage increase/decrease is given and you need the original:
After a 20% increase, the price is $120. Original = 120 / 1.20 = $100
After a 15% decrease, the price is $170. Original = 170 / 0.85 = $200
Successive Percentages
A 10% increase followed by a 10% decrease does NOT return to the original value.
100 → +10% → 110 → −10% → 99. Net effect: 1% decrease.
For successive changes, multiply the multipliers: 1.10 × 0.90 = 0.99 (i.e., 1% decrease).
Ratios
A ratio A:B means “for every A units of one thing, there are B units of another.”
Key operations:
- Simplifying: 12:8 = 3:2 (divide both by HCF)
- Sharing in a ratio: Share 600 in the ratio 2:3. Total parts = 5. Each part = 120. Answer: 240 and 360.
- Finding one quantity: If A:B = 3:5 and A = 45, then one part = 45/3 = 15, so B = 5 × 15 = 75.
Proportional Reasoning
Direct proportion: if one quantity doubles, the other doubles too. If 5 items cost $30, then 8 items cost $30 × (8/5) = $48.
Inverse proportion: if one quantity doubles, the other halves. If 4 workers take 12 days, then 6 workers take 12 × (4/6) = 8 days.
Unit Rate Method
The most reliable approach for proportion questions:
- Find the rate for ONE unit (divide)
- Multiply by the desired quantity
Example: 350ml of juice costs $2.80. What does 500ml cost?
Cost per ml = $2.80 / 350 = $0.008. Cost of 500ml = 500 × $0.008 = $4.00.