Mini Mathematics
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Percentages and parts of a whole

Name the reference whole before converting or finding a percentage part.

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Percent means per hundred

If 18 of 60 pupils walk to school, the fraction is 18/60 = 3/10 = 30/100, so 30% walk. The reference whole is all 60 pupils, not 18 pupils. A percentage compares a part with its stated whole.

1/4 = 25% = 0.25. These are three ways to express the same proportion. For 0.07, seven hundredths means 7%, not 0.07%.

To change a percentage to a fraction, write the number over 100 and simplify: 35% = 35/100 = 7/20. To change a fraction to a percentage, find an equivalent fraction with denominator 100, or multiply its value by 100: 3/8 = 37.5%. The % sign already means “divided by 100”, so 35% = 0.35, not 35. For the opposite direction, 0.35 = 35/100 = 35%.

Check your understanding 1 Keep the same proportion

Write 12% as a simplest fraction and as a decimal. Then write 3/5 and 0.045 as percentages. Keep the % sign on each percentage answer.

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12% = 12/100 = 3/25 = 0.12. Also 3/5 = 60/100 = 60% and 0.045 = 4.5/100 = 4.5%. The decimal 0.045 is smaller than 0.05, so its percentage must be smaller than 5%. Writing 45% would make the proportion ten times larger.

Find a part from its percentage

To find 15% of 80, find 10% and 5%. Ten percent is 80÷10 = 8; five percent is half that, 4. Together the part is 12.

Equivalently 15/100 × 80 = 12. The percentage is a scale factor; the answer has the original unit, such as 12 pupils or 12 dollars.

Check your understanding 2 Same percentage, different part

Find 20% of 35 and 20% of 60. Explain why the two answers differ even though the percentages match.

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The answers are 7 and 12. Both take one fifth, but one fifth of a larger whole is larger. The percentage alone does not determine the number in the part.

Change the reference whole

Twelve pupils travel by bus in a class of 40. In another class of 60, twelve pupils also travel by bus. Which class has the larger percentage travelling by bus? Name the whole in each comparison before calculating.

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The first whole is 40 pupils: 12/40 = 30/100 = 30%. The second whole is 60 pupils: 12/60 = 20/100 = 20%. The same part is a larger fraction of the smaller whole. Dividing by 12 would compare the whole with the part instead.

Transfer between representations

A tank is 0.6 full. Write its fullness as a percentage. If its capacity is 45 ℓ, how many litres are in it?

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0.6 = 60/100 = 60%. The amount is 0.6 × 45 = 27 ℓ. Sixty percent refers to the 45 ℓ capacity, not to 60 ℓ.

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