14.1 Percentages & Ratio

Percentages & Ratio

Quick Answer: A percentage is a fraction out of 100. Percentage change is the difference divided by the original. Successive percentage changes multiply - they do not add. A 20% rise followed by a 20% fall leaves a net 4% decrease.

What Is a Percentage?

A percentage is a fraction with denominator 100. The symbol % means /100, so 35% = 35/100 = 0.35. To convert any fraction to a percentage, multiply by 100: 3/5 = 0.6 x 100 = 60%. To convert a percentage to a decimal, divide by 100: 12.5% = 0.125.

Percentage of a Quantity

To find x% of a quantity Q, compute (x/100) x Q. Example: 15% of 240 = 0.15 x 240 = 36. This is the single most common percentage operation on the test.

Percentage Change

Percentage change = (new value - original value) / original value x 100. A positive result is an increase; a negative result is a decrease. The denominator is always the original value for that step.

Worked example: A salary rises from Rs 40,000 to Rs 46,000. Change = (46,000 - 40,000) / 40,000 x 100 = 6,000 / 40,000 x 100 = 15% increase.

Ratio and Proportion

A ratio compares two quantities in the same units and is written a:b (read 'a to b'). A proportion equates two ratios: a/b = c/d.

Worked example: If a:b = 3:5 and b = 20, then 3/5 = a/20, so a = (3 x 20)/5 = 12.

Direct and Inverse Proportion

• Direct proportion: y = kx. Both quantities move in the same direction. If 2 kg of rice costs Rs 300, 5 kg costs Rs 750 (300 x 5/2).
• Inverse proportion: xy = k. One quantity rises while the other falls. If 3 workers build a wall in 10 days, 5 workers take (3 x 10)/5 = 6 days.

Test tip: 'more workers, fewer days' is the classic fingerprint of inverse proportion. 'More items, more cost' is direct.

Successive Percentages

When two percentage changes apply one after the other, multiply the factors:

Final = Original x (1 + p1/100) x (1 + p2/100)

Worked example: A price is Rs 1,000. It rises 20%, then falls 20%.

1. After the rise: 1,000 x 1.20 = Rs 1,200
2. After the fall: 1,200 x 0.80 = Rs 960
3. Net change: (960 - 1,000) / 1,000 x 100 = -4%

A 20% up then 20% down is NOT a 0% net change - it is a 4% decrease. The fall is computed on the larger base (1,200), so it removes more than the rise added.

The Reverse Percentage Trap

If a value after a 25% increase is Rs 1,250, the original is NOT 1,250 x 0.75 = 937.5. That is the most common mistake on the test. The correct setup recognises that the new value equals the original times 1.25:

Original x 1.25 = 1,250 -> Original = 1,250 / 1.25 = Rs 1,000.

Rule: to reverse a percentage increase of r%, divide by (1 + r/100). To reverse a decrease of r%, divide by (1 - r/100). Never multiply by the complement.

Percentage-Fraction Equivalents

Memorising these speeds up mental arithmetic - most PAF GD Pilot math items are timed, and converting 1/8 to 12.5% on sight saves seconds.

Arithmetic vs Geometric Chain

Adding percentages (the arithmetic approach) only works when the base is the same. When the base changes - successive discounts, successive markups - you must use the geometric (multiplicative) chain.

Key Traps to Avoid

1. Reverse percentage - divide by (1 + rate), never multiply by (1 - rate).
2. Successive percentages - multiply factors; never add them when bases differ.
3. Percentage points vs percentage - a rise from 10% to 15% is 5 percentage points, but a 50% relative increase (5/10 x 100).
4. Sign of change - a fall after a rise is computed on the larger base, which is why up-then-down never returns to the start.