Finding a percentage of a quantity

Learning outcome

Find a stated percentage of a positive quantity using fractions, decimals or convenient parts, and report the result with the correct unit.

Why the method works

To find p% of a quantity Q, imagine dividing Q into 100 equal parts. Each part is Q/100, so p such parts amount to (p/100) × Q. The calculation works even when p is not a whole number, because fractions of equal parts are meaningful.

The quantity Q is the reference amount. It remains unchanged throughout this calculation; we are finding a portion or multiple of it, not automatically increasing or decreasing it. The answer has the same measurement unit as Q because the percentage multiplier has no unit.

Choose a convenient equivalent form. For example, 50% = 1/2, 25% = 1/4 and 12.5% = 1/8. A decimal multiplier is often convenient for other percentages. These are different expressions of the same method, not unrelated shortcuts.

You may add percentage portions calculated from the same base. You cannot automatically add percentages applied one after another to changing bases. Also check the scale: below 100% gives less than Q, while above 100% gives more than Q.

Keep recurring percentages as exact fractions. For indivisible objects, a fractional calculated count indicates that the stated exact percentage and count may be incompatible; do not silently round the number of objects.

Worked example 1

Find 18% of a ₹2,750 materials budget.

First find 1%: 2750/100 = ₹27.5. Then 18% is 18 × 27.5 = ₹495. Equivalently, 0.18 × 2750 = 495. This is the allocated amount, not the budget remaining after allocation.

Worked example 2

Find 12.5% of 3.6 L of juice.

Since 12.5% = 1/8, calculate 3.6/8 = 0.45 L. Converting the answer gives 450 mL. Working entirely in litres first keeps the calculation consistent; the final unit conversion does not change the quantity.

Worked example 3

Find exactly (100/3)% of a 54 kg rice stock.

Convert the percentage to a multiplier: (100/3)/100 = 1/3. Therefore the required mass is 54 × 1/3 = 18 kg. A rounded decimal percentage would unnecessarily replace this exact result with an approximation.

Common traps

Do not multiply by the percentage number without dividing by 100. Distinguish the calculated portion from the remaining quantity. Keep measurement units, and avoid rounding exact fractional multipliers before calculation.

Practice questions

1. Find 16% of ₹1,875.
2. Find 2.4% of a 750 g packet.
3. Find 135% of an 80 km reference distance.
4. Find exactly (200/3)% of 72 L of water.

Worked answers

1. Convert 16% to 16/100. Then 1875 × 16/100 = ₹300. The multiplier is below the whole, so an answer smaller than the reference is appropriate.
2. Use 2.4/100 = 0.024. Multiplying gives 750 × 0.024 = 18 g. The answer retains grams because the percentage multiplier is unitless.
3. Convert 135% to 1.35. Then 80 × 1.35 = 108 km. This exceeds the reference because the requested percentage exceeds the whole.
4. The exact multiplier is (200/3)/100 = 2/3. Therefore 72 × 2/3 = 48 L. Fraction arithmetic avoids any rounding of the recurring percentage.