Percentages: Formulas, Shortcuts & Tricks for SSC and Bank Exams

Percentage means per hundred

x% = x/100. Most percentage questions become quick once you convert the percentage into a simple fraction. Learn this table by heart.

Example: 37.5% of 640 = 3/8 × 640 = 240.

Three basic question types

• Find x% of y: y × x/100. Example: 15% of 260 = 26 + 13 = 39 (10% plus 5%).
• Find the whole from a part: whole = part × 100/x. Example: 35% of a number is 84, so the number = 84 × 100/35 = 240.
• Find what percent A is of B: A/B × 100. Example: 42 out of 60 = 7/10 = 70%.

Percentage increase and decrease

% change = change ÷ original value × 100. Always divide by the original (old) value. Example: a price goes from Rs 250 to Rs 290. Change = 40, so increase = 40/250 × 100 = 16%.

Use multiplying factors: +20% means × 1.2, −15% means × 0.85. To reverse a change, divide by the factor. Example: after a 20% rise the price is Rs 600; the original was 600 ÷ 1.2 = Rs 500 (not 600 × 0.8 = 480).

Successive changes: a + b + ab/100

For two changes one after another, net % change = a + b + ab/100. Increases are positive, decreases negative. A positive answer is a net increase, a negative one a net decrease.

• Up 30% then down 10%: 30 − 10 + (30 × −10)/100 = 20 − 3 = +17%. Check: 100 → 130 → 117.
• Two successive discounts of 25% and 20%: −25 − 20 + (25 × 20)/100 = −40, a single discount of 40%. Check: 100 → 75 → 60.
• Same percentage up and then down (or down then up): net loss = x²/100 %. Up 30% then down 30%: 9% loss. Check: 100 → 130 → 91.

More than / less than (the base changes)

Reason: the percentage is always taken on the value you compare against. Put the base value at 100 and work it out if you forget the formula.

Expenditure = price × consumption

• Price rises r%: to keep spending the same, cut consumption by r/(100 + r) × 100 %. Example: price up 25%, cut 25/125 = 20%.
• Price falls r%: consumption can rise by r/(100 − r) × 100 % for the same spending. Example: price down 20%, consumption can rise 20/80 = 25%.
• Price and consumption both change: use successive change. Price +10%, consumption −10%: 10 − 10 − 1 = −1%, so spending falls 1%.

Population growth and depreciation

After n years: P × (1 + r/100)^n for growth, P × (1 − r/100)^n for depreciation. Each year's change is on the new figure, like compound interest.

• Growth: 8,000 people growing 5% a year for 2 years = 8,000 × 1.05 × 1.05 = 8,820.
• Depreciation: a Rs 50,000 machine losing 10% a year is worth 50,000 × 0.9 × 0.9 = Rs 40,500 after 2 years.
• Going back in time: population now 13,310 after 3 years of 10% growth. 3 years ago = 13,310 ÷ 1.331 = 10,000.

Marks and pass percentage

Pass marks = pass % × maximum marks. Example: pass mark is 40%. A student scores 180 and fails by 20 marks. So pass marks = 200 = 40% of maximum, and maximum = 200 × 100/40 = 500.

Elections

• Two candidates: the loser's % = 100 − winner's %. Margin % = difference of the two percentages. Example: winner gets 58%, wins by 2,240 votes. Margin = 58 − 42 = 16% of total = 2,240, so total = 2,240 × 100/16 = 14,000.
• With invalid votes: 10% of votes are invalid; the winner gets 60% of valid votes and wins by 1,440. Margin = 20% of valid = 1,440, so valid = 7,200. Valid is 90% of total, so total = 7,200 ÷ 0.9 = 8,000.

Marked price and discount

Discount is always on the marked price; profit is always on the cost price. If goods are marked x% above cost and sold at d% discount, profit % = x − d − xd/100. Example: marked 25% above cost, 12% discount: 25 − 12 − 3 = 10% profit. Check: cost 100, marked 125, sold at 125 × 0.88 = 110.

Speed tricks

• 10%: move the decimal one place left. 1%: two places. 5% = half of 10%. 15% = 10% + 5%.
• Take a convenient base of 100 for any 'by what percent' question; the answer comes straight out.
• Convert to fractions: 12.5% → 1/8, 162/3% → 1/6, 6.25% → 1/16. Dividing is faster than multiplying by decimals.
• Check options by size: a 10% rise for 2 years must be a little over 20%, never exactly 20%.

Common traps