One Quantity as a Percentage of Another

(a) "Of 200" names the base: 200 is the whole and stands for $100\%$. The quantity is 50.

Write 50 as a fraction of 200 and multiply by $100\%$: $\frac{50}{200} \times 100\% = \frac{1}{4} \times 100\% = 25\%$

The fraction was simplified first: dividing 50 and 200 by 50 gives $\frac{1}{4}$, and a quarter of $100\%$ is $25\%$.

(b) "As a percentage of 80" names the base: 80 is the whole. The quantity is 18.

$\frac{18}{80} \times 100\% = \frac{1800}{80}\% = 22.5\%$

Check: 18 is a little less than a quarter of 80 (a quarter of 80 is 20), so the answer should be a little less than $25\%$. It is.

The base is the total marks of the test, 40 marks. It stands for $100\%$. Her score, 34 marks, is a part of it.

Percentage scored $= \frac{34}{40} \times 100\%$

One way, multiplying first: $\frac{34}{40} \times 100\% = \frac{3400}{40}\% = 85\%$

Another way, making the denominator 100: $\frac{34}{40} = \frac{17}{20} = \frac{85}{100}$ (dividing the top and the bottom by 2, then multiplying them by 5). That is 85 hundredths, which is $85\%$.

The base is all the mangoes in the box: 10 mangoes stand for $100\%$. The ripe and the unripe mangoes are the two parts of this whole.

Number of unripe mangoes $= 10 - 7 = 3$

Percentage of unripe mangoes $= \frac{\text{number of unripe mangoes}}{\text{total number of mangoes}} \times 100\% = \frac{3}{10} \times 100\% = 30\%$

Another way, from $100\%$: the ripe mangoes are $\frac{7}{10} \times 100\% = 70\%$ of the box, so the unripe mangoes are $100\% - 70\% = 30\%$ of it.

The base is all 160 students in Secondary 1. Every one of them is a girl or a boy, so the girls and the boys are the two parts of this whole.

(a) Percentage of girls $= \frac{90}{160} \times 100\% = \frac{9000}{160}\% = 56.25\%$

(b) Method 1, from the number of boys. Number of boys $= 160 - 90 = 70$, so the percentage of boys $= \frac{70}{160} \times 100\% = \frac{7000}{160}\% = 43.75\%$

Method 2, from $100\%$. The two percentages add up to $100\%$, so the percentage of boys $= 100\% - 56.25\% = 43.75\%$

Check: $56.25\% + 43.75\% = 100\%$

The base is his whole pocket money, $80. Food, transport and savings are the three parts of it.

(a) Food: $\frac{36}{80} \times 100\% = \frac{3600}{80}\% = 45\%$

(b) Transport: $\frac{20}{80} \times 100\% = \frac{2000}{80}\% = 25\%$

(c) His savings are the rest. The three percentages add up to $100\%$, so savings $= 100\% - 45\% - 25\% = 30\%$

Check by the amount: he saves $80 − $36 − $20 = $24, and $\frac{24}{80} \times 100\% = 30\%$.

(a) The base is 2 kg. Change it to grams: 2 kg $= 2 \times 1000 = 2000$ g.

$\frac{250}{2000} \times 100\% = \frac{25\,000}{2000}\% = 12.5\%$

(b) The base is 2 hours. Change it to minutes: 2 hours $= 2 \times 60 = 120$ minutes.

$\frac{45}{120} \times 100\% = \frac{4500}{120}\% = 37.5\%$

(c) The base is $2.50. Change it to cents: $2.50 $= 2.50 \times 100 = 250$ cents.

$\frac{95}{250} \times 100\% = \frac{9500}{250}\% = 38\%$

The question asks for a percentage of the drink, so the base is the whole drink, the syrup and the water together. It is not the water alone.

Change to the same unit: 1.02 litres $= 1.02 \times 1000 = 1020$ ml.

The whole drink $= 180 + 1020 = 1200$ ml

Percentage of rose syrup $= \frac{180}{1200} \times 100\% = \frac{18\,000}{1200}\% = 15\%$

The water is the rest of the drink: $100\% - 15\% = 85\%$. Check: $\frac{1020}{1200} \times 100\% = 85\%$.

The base is everyone at the gathering: $15 + 22 = 37$ people.

(a) Children: $\frac{15}{37} \times 100\% = 40.5405\ldots\%$

To 3 significant figures, keep 4, 0 and 5. The next digit is 4, which is less than 5, so round down: $40.5405\ldots\% \approx 40.5\%$

(b) Adults: $\frac{22}{37} \times 100\% = 59.4594\ldots\%$. Keep 5, 9 and 4. The next digit is 5, so round up: $59.4594\ldots\% \approx 59.5\%$

Or from $100\%$, using the full value of (a): $100\% - 40.5405\ldots\% = 59.4594\ldots\% \approx 59.5\%$

The base is all 120 students in the survey. Check that the parts make the whole: $74 + 39 + 7 = 120$.

Bus: $\frac{74}{120} \times 100\% = 61.666\ldots\% \approx 61.7\%$

MRT: $\frac{39}{120} \times 100\% = 32.5\%$, which is exact.

Walk: $\frac{7}{120} \times 100\% = 5.8333\ldots\% \approx 5.83\%$. The first significant figure is the 5, so the three figures kept are 5, 8 and 3.

The rounded percentages add up to $61.7\% + 32.5\% + 5.83\% = 100.03\%$, not exactly $100\%$. That is the effect of rounding, not a mistake. The exact percentages, $61\frac{2}{3}\%$, $32.5\%$ and $5\frac{5}{6}\%$, add up to exactly $100\%$.