Percentages Made Clear: Methods, Checks, and Practice

Start With the Meaning

A percentage is a quantity expressed out of one hundred. If 18 out of 60 questions are correct, the percentage is 18/60 × 100 = 30%. Always identify the base before calculating: 20% of 50 is different from 20% of 80.

To convert a percentage to a fraction, divide by 100 and simplify. To convert a fraction to a percentage, multiply by 100. These simple conversions make mental estimates faster during timed practice.

Increase and Decrease

If a quantity increases by r percent, multiply it by 1 + r/100. A 15% increase changes 200 to 200 × 1.15 = 230. If it decreases by r percent, multiply by 1 − r/100. A 15% decrease changes 200 to 170.

A common error is to assume that a 20% increase followed by a 20% decrease returns to the original value. Starting at 100, the increase gives 120 and the decrease gives 96. The combined change is a 4% decrease.

For successive changes a and b, the net percentage change is a + b + ab/100 when both are expressed with their signs. Use a negative value for a decrease, then estimate the result with an actual number to catch sign errors.

Compare Parts With the Correct Base

If A is 25% more than B, then A = 1.25B and B is not 25% less than A. The decrease from A back to B is 25/125 × 100 = 20%. This difference matters in ratio and data-interpretation questions.

When a question says a score rises from 40 to 50, the increase is 10 points but the percentage increase is 10/40 × 100 = 25%. Percentage change uses the original value as the denominator unless the question specifies another base.

Worked Practice

If a learner answers 36 of 45 questions correctly, the accuracy is 36/45 × 100 = 80%. If the same learner improves accuracy by 10 percentage points, the new accuracy is 90%; if accuracy improves by 10 percent, it is 88%. Percentage points and percent change are not interchangeable.

For every problem, write the base, estimate the direction of the answer, calculate, and check whether the result is plausible. This four-step habit prevents many avoidable errors.

Editorial note: practice questions should be checked against the stated assumptions. In data interpretation, read the table title, units, and time period before applying a percentage formula.