3.5 Percentage Increase and Decrease

Figure: A jump from 40% to 50% is +10 percentage points, but a +25% relative increase.

%Increase = (New Value - Original Value)/(Original Value) × 100

%Decrease = (Original Value - New Value)/(Original Value) × 100

The original (initial) value is always the base (the denominator) — this is the single most common source of error in this entire topic. When a value increases by r%, the new value equals Original × (1 + r/100). When it decreases by r%, the new value equals Original × (1 − r/100).

Worked Example 1: The price of an item rises from ₹250 to ₹300. Find the percentage increase. Increase = 300 − 250 = 50. % increase = (50/250) × 100 = 20%.

Worked Example 2: A quantity falls from 80 to 68. Find the percentage decrease. Decrease = 80 − 68 = 12. % decrease = (12/80) × 100 = 15%.

Worked Example 3: A’s income is 20% more than B’s income. By what percentage is B’s income less than A’s income? Let B = 100, so A = 120 (since A is 20% more than B). Now B is less than A by (120 − 100) = 20, and this 20 must be compared with A (the new base), not B: % less = (20/120) × 100 = 50/3 % = 16.67%. This is the classic trap: “20% more” and “16.67% less” describe the same pair of numbers, just viewed from different bases. Never assume the reverse percentage is the same number.

Worked Example 4: The runs scored by a batsman in a match increased from 45 to 63 in his next innings. By what percentage did his score increase, and what would his score have been if it had instead increased by the same percentage from 80? % increase = [(63−45)/45] × 100 = (18/45) × 100 = 40%. If a score of 80 increases by the same 40%, new score = 80 × 1.40 = 112. This two-part style — first find the % rate from one pair of numbers, then apply that same rate to a different base — appears frequently in DI-linked percentage questions.

Worked Example 5: A’s income is 25% more than B’s income, and C’s income is 20% less than B’s income. By what percentage is A’s income more than C’s income? Let B = 100. Then A = 125 (25% more than B) and C = 80 (20% less than B). A is more than C by (125 − 80)/80 × 100 = 45/80 × 100 = 56.25%. Notice the base here is C (80), the smaller of the two values being compared — always the value you are comparing against, not A itself.

Worked Example 6: The height of a plant increased from 150 cm to 180 cm in one year. If it grows by the same percentage the following year, find its height at the end of the second year. % increase in year 1 = [(180 − 150)/150] × 100 = (30/150) × 100 = 20%. Height at end of year 2 = 180 × 1.20 = 216 cm.