Mathematics Year 7: Percentages: the part, the whole and the percentage
Mathematics, Year 7 (Portugal): Three numbers are linked — part, whole and percentage — and knowing any two lets you find the third.
One relation, three questions
In “35% of 240”, 240 is the whole, 35% says how many parts in every 100 you take, and the result is the part. The relation is always the same: part = percentage × whole, with the percentage written as a fraction with denominator 100 (35% = 35/100 = 0.35). Three questions come out of it: how big is the part? (multiply); what is the whole? (undo the multiplication: divide the part by the percentage); and what percentage of the whole is the part? (divide the part by the whole).
Why percentages are worth mastering
Percentages let you compare things of different sizes: 12 students out of 30 and 20 out of 80 are hard to compare, but 40% and 25% are easy. Because the whole always counts as 100, the same idea works for money, populations or game scores. And there are shortcuts that save work: 10% is dividing by 10, 1% is dividing by 100, 50% is half, 25% is a quarter — and other percentages are built from those pieces, with no calculator.
35% of 240, in all three directions
A club has 240 members and 35% are young people. Part: 10% of 240 = 24; 30% = 3 × 24 = 72; 5% = 12; so 35% = 72 + 12 = 84. Whole: if 84 is 35% of the total, then 5% = 84 ÷ 7 = 12 and 100% = 20 × 12 = 240. Percentage: 84 out of 240 gives 84 ÷ 240 = 0.35 = 35%. Notice that the three numbers 240, 84 and 35% never change; only the one you are looking for does.
Percentage of what?
You know 24 is 30% of the whole and you want the whole. Many people work out 30% of 24 and get 7.2. But 30% of the whole is a smaller piece than the whole, so the whole has to be bigger than 24, not smaller. The 30% applies to the whole, which is exactly what you do not know. Do this instead: 30% = 24, so 10% = 8 and 100% = 80. Check: 30% of 80 = 24.
On labels, test marks and surveys
On a food label, “12% fat” means that in every 100 g there are 12 g of fat: in 250 g there is 12% of 250 = 30 g. On a test, 17 right answers out of 20 give 17 ÷ 20 = 0.85, that is, 85%. In a survey, if 45 people are 15% of those who answered, then 300 answered. Always the same relation: part, whole and percentage, with one of them left to find.
Test what you learned
Which of these calculations gives 35% of 240?
- a) 240 ÷ 35
- b) 240 + 35
- c) 35/100 × 240
- d) 240 × 35
Correct answer: c) 35/100 × 240 — Yes: 35% = 35/100 = 0.35, and 0.35 × 240 = 84. The part is the percentage (as a fraction of 100) times the whole.
In a test, Rui answered 27 of the 60 questions correctly. What percentage of the questions did he get right?
- a) 27%
- b) 45%
- c) 55%
- d) 222%
Correct answer: b) 45% — Yes: 27 ÷ 60 = 0.45 = 45/100 = 45%. Part divided by whole gives the percentage.
12 is 40% of a number. What is that number?
- a) 4.8
- b) 52
- c) 480
- d) 30
Correct answer: d) 30 — Yes: 40% = 12, so 10% = 12 ÷ 4 = 3 and 100% = 10 × 3 = 30. Check: 40% of 30 = 12.
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