Percentages compare proportions

The idea

A percentage is a proportion of its own whole, so percentages of different wholes cannot be compared directly; the actual amounts must be worked out before comparing them.

Who ate more sugar?

Madhu ate biscuits that are 25% sugar. Madhav ate biscuits that are 35% sugar.

Who ate more sugar?

• Madhav, because 35% is more than 25%
• Madhu
• We cannot tell until we know how much each one ate

Change how much each one ate

Each bar shows the biscuits eaten, and the shaded part is the sugar. Both bars use the same gram scale. Change the weights and find the three situations in the goal.

A percentage belongs to its own whole

A percentage is a proportion of its own whole, so percentages of different wholes cannot be compared directly; the actual amounts must be worked out before comparing them.

If both ate 100 g, Madhav has 35 g of sugar and Madhu has 25 g. With 120 g for Madhu and 95 g for Madhav, the amounts of sugar are Madhu: 25/100 × 120 = 30 g Madhav: 35/100 × 95 = 33.25 g Madhav ate more sugar.

A lower percentage can still be the larger amount. If Madhu had eaten 140 g, his sugar would be 35 g, which is more than Madhav's 33.25 g. The bigger whole made up for the smaller percentage.

Notes

A percentage is a proportion of its own whole, so percentages of different wholes cannot be compared directly; the actual amounts must be worked out before comparing them.

Check yourself

Anil's drink is 40% juice and he has 100 mL. Bela's drink is 30% juice and she has 300 mL. Who has more juice?

Jar B holds 80 g of a mix that is 35% salt. How many grams of salt are in it?

Answer: 28 g

35/100 × 80 = 28, so jar B has 28 g of salt.

Jar A holds 150 g of a mix that is 20% salt. Jar B holds 80 g that is 35% salt. How many grams more salt does the jar with more salt have?

Answer: 2 g

Jar A: 20% of 150 = 30 g. Jar B: 35% of 80 = 28 g. Jar A has 30 − 28 = 2 g more, although its percentage is lower.

In Class X, 50% of the 40 students are girls. In Class Y, 80% of the 25 students are girls. Which class has more girls?

• Bela: 90 mL against 40 mL — correct. Yes! 30% of 300 = 90 mL and 40% of 100 = 40 mL. Her drink is a bigger whole.
• Anil, because 40% is more than 30%. The two percentages are of different wholes, so they cannot be compared directly. Work out the amounts: 40 mL for Anil and 90 mL for Bela.
• They have the same amount. Work out both amounts: 40% of 100 = 40 mL and 30% of 300 = 90 mL. They are different.
• Both have 20 girls — correct. Yes! 50% of 40 = 20 and 80% of 25 = 20. The percentages are different, but the amounts are equal.
• Class Y, because 80% is more than 50%. The percentages are of different wholes (25 students and 40 students). Work out the amounts: Class Y has 20 girls, and so does Class X.
• Class X, because it has more students. More students does not mean more girls. Class X has 50% of 40 = 20 girls, and Class Y has 80% of 25 = 20 girls.