Expressing one quantity as a percentage of another

Learning outcome

Express one positive quantity as a percentage of another, choosing the correct reference and converting measurements to matching units.

Choosing the denominator

To express A as a percentage of B, calculate A/B, multiply the result by 100, and attach %. The denominator B is the reference quantity: it represents the whole against which A is compared. It must be nonzero.

Why divide first? The quotient tells us how many copies of B are represented by A. Multiplying that quotient by 100 expresses the same comparison per hundred.

Wording determines the base. “Of the total” uses the total; “of the other group” uses only that other group. A part-to-part comparison is not the same as a part-to-whole comparison. Label the reference before calculating.

For measurements of the same kind, convert both to a common unit before division. Their units then cancel. Dividing a numerical measurement in grams directly by one in kilograms produces the wrong comparison.

Reversing the quantities changes the reference and usually changes the percentage. The results need not add to 100%. A value above 100% is valid when A exceeds B. Preserve exact fractions if the percentage recurs.

Worked example 1

A learner answers 27 of 36 attempted questions correctly. What percentage of attempted questions is correct?

The reference is the 36 attempted questions. Calculate 27/36 = 3/4, then (3/4) × 100 = 75. Therefore 75% of attempted questions are correct. The calculation concerns attempted questions, not any larger question bank.

Worked example 2

Express 420 g as a percentage of 1.75 kg.

Convert 1.75 kg to 1750 g. Now calculate (420/1750) × 100 = 24. Thus 420 g is 24% of the reference mass. Matching units is essential before treating the numerical quotient as a relative size.

Worked example 3

Notebook A has 108 pages and notebook B has 72 pages. Express each count as a percentage of the other.

A relative to B is (108/72) × 100% = 150%, using B as reference. B relative to A is (72/108) × 100% = (200/3)% ≈ 66.67%. Different denominators explain the different results.

Common traps

Do not automatically use the larger quantity as the denominator. Do not substitute the combined total unless requested. Keep the direction of comparison clear, and mark rounded recurring percentages as approximate.

Practice questions

1. Of 60 seedlings, 39 flowered. What percentage flowered?
2. Express 225 mL as a percentage of 1.5 L.
3. Express a 450 g mass as a percentage of a 300 g reference mass.
4. A box contains 18 red and 30 blue counters. Express red counters as a percentage of blue counters and of all counters.

Worked answers

1. The whole group contains 60 seedlings. Calculate (39/60) × 100% = 65%, using the total group as the reference.
2. Convert 1.5 L to 1500 mL. Then (225/1500) × 100% = 15%; the common units cancel in the quotient.
3. Use the specified reference: (450/300) × 100% = 150%. The result exceeds the whole because the compared mass is larger.
4. Relative to blue, (18/30) × 100% = 60%. The total is 18 + 30 = 48, so red counters are (18/48) × 100% = 37.5% of all counters.