Problems Involving Percentages

Name the bases. 60% is a percentage of all the students. 25% is a percentage of the girls. Part (a) asks for a percentage of all the students.

Choose the number of units for 100%. 25% is a quarter, so the girls must come to a number of units that can be cut into quarters. With 10 units for all the students, the girls would be 6 units, and a quarter of 6 is not a whole number. With 20 units, the girls are $0.6 \times 20 = 12$ units and the boys are the other 8 units.

Draw the students as a bar of 20 units. Redraw the girls' 12 units below and cut them into 4 equal parts. The girls who cycle are 1 of the 4 parts: $12 \div 4 = 3$ units.

(a) The girls who cycle are 3 units out of the 20 units of all the students: $\frac{3}{20} \times 100\% = 15\%$.

By multipliers: the girls who cycle are 25% of 60% of the students, and $0.25 \times 60\% = 15\%$. The two ways agree.

(b) 15% of 1240 is $0.15 \times 1240 = 186$.

Check by working forward: the girls are $0.6 \times 1240 = 744$, and a quarter of them is $744 \div 4 = 186$. It is a whole number of girls, as it must be.

Name the bases. 25% and 30% are percentages of all the sweets sold. "10% more than" is a percentage of the blue sweets. Both parts of the question ask for percentages of all the sweets sold.

"10% more than the blue sweets" means $100\% + 10\% = 110\%$ of the blue sweets. The blue sweets are 30% of all the sweets, so the yellow sweets are 110% of 30% of all the sweets: $1.1 \times 30\% = 33\%$.

(a) Red, blue and yellow are now all percentages of the same base, so they can be added: $25\% + 30\% + 33\% = 88\%$. The pink sweets are the rest: $100\% - 88\% = 12\%$.

(b) Of 25%, 30%, 33% and 12%, the largest is 33%, so yellow sold the most. The number of yellow sweets is 33% of 2500: $0.33 \times 2500 = 825$.

Check by working out every number from the question, as in the table. The four numbers add up to 2500, and the 825 yellow sweets are 75 more than the 750 blue sweets, which is 10% of 750.

It would be wrong to say that the yellow sweets are $30\% + 10\% = 40\%$ of all the sweets. The 10% is a percentage of the blue sweets, not of all the sweets: 10% of 750 is 75, which is only 3% of 2500.

Name the bases. 30% is a percentage of all her money. 40% is a percentage of the money left after the book. Part (a) asks for a percentage of all her money.

Draw her money as a bar of 10 equal units, each 10% of it. The book is 3 units and the rest is 7 units. Redraw the rest below and cut it into 5 equal parts, since 40% is $\frac{2}{5}$. The present is 2 of those parts, and what is left at the end is the other 3.

(a) By multipliers: after the book, 70% of her money is left. The present is 40% of that, so 60% of it is still left at the end: $0.6 \times 70\% = 42\%$ of her money.

In units: the rest is 7 units and has to be cut into fifths, so make the whole $10 \times 5 = 50$ units. The book is 15 units and the rest is 35 units. The present is $\frac{2}{5} \times 35 = 14$ units, and $35 - 14 = 21$ units are left: $\frac{21}{50} \times 100\% = 42\%$. The two ways agree.

(b) $63 is 42% of her money. Find 1%, then 100%: 1% of her money is $63 \div 42 = 1.5$, which is $1.50, and 100% is $1.5 \times 100 = 150$, which is $150.

(c) The money left after the book is 70% of $150: $0.7 \times 150 = 105$, which is $105. The present is 40% of $105: $0.4 \times 105 = 42$, which is $42.

Check by working forward: the book costs 30% of $150, which is $45, and leaves $150 - 45 = 105$, which is $105. The present costs $42 and leaves $105 - 42 = 63$, which is the $63 in the question.

She did not have $100\% - 30\% - 40\% = 30\%$ left. That would treat the 40% as a percentage of all her money, when it is a percentage of the money left after the book.

Name the bases. Both of Ravi's percentages are percentages of his pocket money. Ali's 20% is a percentage of his pocket money, but his 25% is a percentage of the rest, the money left after food.

Ravi: his two percentages have the same base, so they can be added. He spends $20\% + 25\% = 45\%$ of his pocket money and has $100\% - 45\% = 55\%$ of it left: $0.55 \times 80 = 44$, which is $44.

Ali: after food he has 80% of his pocket money, which is $64 ($0.8 \times 80 = 64$). Transport is 25% of $64, which is $16, so he has $64 - 16 = 48$, which is $48, left. As a percentage of his pocket money, he has 75% of 80% of it left: $0.75 \times 80\% = 60\%$, and $0.6 \times 80 = 48$.

Check: Ravi spends $16 on food and $20 on transport, and $80 - 16 - 20 = 44$. Ali spends $16 on food and $16 on transport, and $80 - 16 - 16 = 48$.

The words "of the rest" make Ali's 25% a smaller amount than Ravi's 25%, because 25% of $64 is less than 25% of $80.

Name the bases. 20% is a percentage of the boys, 40% is a percentage of the girls, and part (a) asks for a percentage of all the Secondary 1 students. The 20% and the 40% have different bases, so they cannot be added or averaged as they stand.

Choose the units. The ratio gives the boys 3 units and the girls 5 units. 20% is $\frac{1}{5}$ and 40% is $\frac{2}{5}$, so each group must be a number of units that can be cut into fifths. Multiply both terms by 5: the boys are 15 units and the girls 25 units, which is still in the ratio $3 : 5$, and there are 40 units in all.

Boys who wear glasses: 20% of 15 units, which is $0.2 \times 15 = 3$ units. Girls who wear glasses: 40% of 25 units, which is $0.4 \times 25 = 10$ units.

(a) The students who wear glasses are $3 + 10 = 13$ units out of 40: $\frac{13}{40} \times 100\% = 32.5\%$.

By multipliers: the boys are $\frac{3}{8}$ of the students and the girls $\frac{5}{8}$. The boys who wear glasses are $20\% \times \frac{3}{8} = 7.5\%$ of all the students, and the girls who wear glasses are $40\% \times \frac{5}{8} = 25\%$ of all the students. These two percentages have the same base, so they can be added: $7.5\% + 25\% = 32.5\%$.

(b) 13 units = 52 students, so 1 unit = 4 students ($52 \div 13 = 4$), and 40 units $= 40 \times 4 = 160$ students.

Check: there are $15 \times 4 = 60$ boys and $25 \times 4 = 100$ girls, and $60 : 100 = 3 : 5$. 20% of 60 is 12 and 40% of 100 is 40, and $12 + 40 = 52$. Also $\frac{52}{160} \times 100\% = 32.5\%$.

The answer to (a) is not 30%, the average of 20% and 40%. There are more girls than boys, so the girls' 40% counts for more.

(a) More students of Class 1A passed, 30 against 23, but Class 1A also has more students, so compare percentages. Class 1A: $\frac{30}{40} \times 100\% = 75\%$ of the class passed. Class 1B: $\frac{23}{25} \times 100\% = 92\%$ of the class passed.

(b) The base is now both classes together: $40 + 25 = 65$ students, of whom $30 + 23 = 53$ passed.

$\frac{53}{65} \times 100\% = 81.538\ldots\% \approx 81.5\%$

(c) The average of the two percentages is $\frac{75\% + 92\%}{2} = 83.5\%$, which is not the answer. The average treats the two classes as if they were the same size. Class 1A has 40 students and Class 1B only 25, so Class 1A's 75% counts for more, and the combined percentage, 81.5%, is nearer 75% than 92%.

Check: 81.5% lies between 75% and 92%, as a combined percentage always does, and 81.5% of 65 students is $0.815 \times 65 = 52.975$, which is about 53 students.

Name the bases. 45% is a percentage of all the students. 40% is a percentage of the students who live in Punggol, and so is "the rest", which is $100\% - 40\% = 60\%$ of them.

Draw all the students as a bar of 20 equal units, each 5%. The Punggol students are 9 units ($0.45 \times 20 = 9$). Redraw them below and cut them into 5 equal parts, since 40% is $\frac{2}{5}$: 2 parts take a bus and 3 parts walk, and the 3 parts are the 324 students.

(a) The students who walk are 60% of 45% of all the students: $0.6 \times 45\% = 27\%$ of the school.

27% of the school is 324 students. 1% is $324 \div 27 = 12$ students, and 100% is $12 \times 100 = 1200$ students.

(b) The Punggol students are 45% of 1200: $0.45 \times 1200 = 540$. 40% of them take a bus: $0.4 \times 540 = 216$.

Check by working forward: $540 - 216 = 324$ students walk, as the question says, and every number is a whole number of students.

Dividing 324 by 0.6 gives 540, which is the number of students who live in Punggol, not the number in the school. Stopping there would answer a different question.

Name the bases. "100% more than" is a percentage of an officer's salary. "25% less than" is a percentage of a manager's salary. The $$45\,000$ is the total of all three salaries.

Turn each comparison into a multiplier. "100% more than" means $100\% + 100\% = 200\%$ of, so a manager is paid twice as much as an officer. "25% less than" means $100\% - 25\% = 75\%$ of, which is $\frac{3}{4}$ of a manager's salary.

Choose the units so that every salary is a whole number of units. A manager's salary has to be cut into quarters, so give it 4 units. Then an officer is paid half of that, 2 units, and a senior officer $\frac{3}{4}$ of it, 3 units.

The three salaries together are $2 + 3 + 4 = 9$ units, and 9 units = $$45\,000$. So 1 unit = $5000, since $45\,000 \div 9 = 5000$.

A senior officer is paid 3 units: $3 \times 5000 = 15\,000$, which is $$15\,000$.

Check: an officer is paid $$10\,000$ and a manager $$20\,000$, which is $$10\,000$ more, so 100% more. 75% of $$20\,000$ is $$15\,000$. And $10\,000 + 15\,000 + 20\,000 = 45\,000$.