Percentages: name the base before calculating

Percent means per hundred

A percentage compares an amount with a reference base. Twenty-four percent means 24/100, or 0.24. The relation is part = rate × base, with the rate written as a decimal. The most useful first question is 'Percent of what?' If 18 out of 60 participants finish an activity, the fraction is 18/60, so the percentage is 30%. The denominator is everyone in the stated group, not only those who finish. Naming the base prevents many otherwise correct-looking calculations from answering the wrong question.

Worked example: a discount and its reverse

A fictional shop marks a bag at PHP 960 and offers a 15% discount. The discount is 0.15 × 960 = PHP 144, so the sale price is 960 - 144 = PHP 816. You can calculate the sale price directly as 0.85 × 960 because 85% remains. Now reverse the problem: if a bag costs PHP 816 after a 15% discount, its original price is 816 ÷ 0.85 = PHP 960. Adding 15% of 816 would give the wrong original price because it uses the sale price as the base.

Percent change uses the starting value

A club increases its membership from 48 to 60. The increase is 12 members, and the percent increase is 12/48 × 100 = 25%. If membership later falls from 60 back to 48, the decrease is still 12, but the percent decrease is 12/60 × 100 = 20%. Equal numerical changes need not have equal percentage changes. A rise from 40% to 55% is 15 percentage points; relative to 40%, it is a 37.5% increase. State which comparison is intended.

Successive changes multiply

An amount of PHP 1,000 increases by 10% and then decreases by 10%. The first change gives 1,000 × 1.10 = 1,100. The second gives 1,100 × 0.90 = 990. The final amount is PHP 10 below the start. In one line, the multiplier is 1.10 × 0.90 = 0.99. Two changes are applied to different bases, so the percentages do not simply cancel. The same principle applies to successive discounts, provided the wording says the later discount applies to the already-reduced price.

Check what the problem leaves out

A textbook price calculation may tell you to ignore taxes, delivery charges or rounding. Real bills can include those items, so do not treat a practice scenario as a quotation or current commercial offer. In a test problem, follow the stated assumptions. Estimate first: a 15% discount on almost PHP 1,000 should be around PHP 150, which makes PHP 144 plausible.

• Use change divided by original value for percentage increase or decrease.
• Divide by the remaining multiplier when reconstructing an original amount.
• Keep percentage points separate from percent change when both values are already percentages.

Sources & further reading