How to Solve Percentage Problems Step by Step

Solving Percentage Problems

Percentage problems fall into a few standard patterns. Recognizing which pattern you're dealing with makes the calculation straightforward.

Percentage of a Number

"What is 15% of 240?" Multiply the number by the percentage divided by 100: 240 × 0.15 = 36. This pattern appears in tip calculations, discount amounts, tax computation, and commission calculations.

Percentage Change

"A price went from $80 to $92. What's the percentage increase?" Formula: ((new - old) / old) × 100 = ((92 - 80) / 80) × 100 = 15%. For decreases, the result is negative. Common mistake: using the new value as the base instead of the old value. A drop from 100 to 80 is a 20% decrease, but a rise from 80 to 100 is a 25% increase — the percentages are not symmetric.

Reverse Percentage

"A shirt costs $68 after a 15% discount. What was the original price?" The discounted price is 85% of the original: $68 / 0.85 = $80. This pattern appears in tax-inclusive pricing: if the price including 20% tax is $120, the pre-tax price is $120 / 1.20 = $100.

Percentage Points vs Percentages

An interest rate rising from 5% to 6% is a 1 percentage point increase but a 20% increase. These are different things, and confusing them is a common source of miscommunication in business and finance. "Percentage points" refers to the arithmetic difference between two percentages.

Compound Percentages

A 20% increase followed by a 20% decrease does NOT return to the original value. $100 + 20% = $120. $120 - 20% = $96. The net effect is a 4% decrease. This is because the decrease is calculated on the larger post-increase value. For repeated compounding, use the formula: final = initial × (1 + rate)^periods.