Percentages Greater than 100%

(a) Per cent means out of 100, so $175\% = \frac{175}{100}$. The HCF of 175 and 100 is 25. Dividing the numerator and the denominator by 25: $\frac{175}{100} = \frac{7}{4} = 1\frac{3}{4}$.

As a decimal: $175 \div 100 = 1.75$.

(b) To change a number to a percentage, multiply it by 100%: $3.2 \times 100\% = 320\%$.

$\frac{9}{8} \times 100\% = \frac{900}{8}\% = 112.5\%$

Check: 1.75, 3.2 and $\frac{9}{8}$ are all greater than 1, and each percentage is greater than 100%, as it should be.

The base is 40, which is 100%. Draw it as a bar of 10 units of 10%. 150% is a bar of 15 of the same units, marked "?".

10 units = 40, so 1 unit = 4, since $40 \div 10 = 4$.

150% is 15 units: $15 \times 4 = 60$.

The same by the fraction: $150\% \text{ of } 40 = \frac{150}{100} \times 40 = \frac{3}{2} \times 40 = 60$.

The same by the decimal: $1.5 \times 40 = 60$.

Check: 150% is the base and half of it again, and $40 + 20 = 60$. The answer is greater than 40, as it must be.

The base is the number sold on Monday, 80 plates, which is 100%. 225% is more than 200%, so Saturday's number is more than twice Monday's.

Draw Monday's number as a bar of 4 units of 25%. Since $225 \div 25 = 9$, Saturday's number is 9 of the same units.

4 units = 80 plates, so 1 unit = 20 plates, since $80 \div 4 = 20$.

Saturday is 9 units: $9 \times 20 = 180$ plates.

The same in one line: $225\% \text{ of } 80 = 2.25 \times 80 = 180$.

Check: twice 80 is 160, and a quarter of 80 more is 20. $160 + 20 = 180$.

The base is the price in 2018, $32, which is 100%.

$115\% = \frac{115}{100} = 1.15$

Price now $= 1.15 \times 32 = 36.8$

Check: 100% of $32 is $32, 10% of $32 is $3.20 and 5% of $32 is $1.60. So 115% of $32 is $32 + 3.20 + 1.60 = 36.80$.

(a) The base is 2 kg. Both masses are in kilograms: $\frac{3}{2} \times 100\% = 150\%$.

3 kg is the 2 kg and half of it again, so 150% makes sense.

(b) The base is 50 min. Put both times in minutes first: 1 h 30 min = 90 min.

$\frac{90}{50} \times 100\% = 180\%$

Check: 1 h 30 min is longer than 50 min, so the answer must be more than 100%. Without changing to minutes, $\frac{1.5}{50} \times 100\% = 3\%$, which is far too small and so must be wrong.

"As a percentage of the price in 2019": the base is $1.50, which is 100%.

Draw the price in 2019 as a bar of 5 units of 20%. Each unit is $0.30, since $1.50 \div 5 = 0.30$. The price now, $1.80, is $1.80 \div 0.30 = 6$ of the same units.

6 units of 20% each: $6 \times 20\% = 120\%$.

The same by the rule: $\frac{1.80}{1.50} \times 100\% = 120\%$.

Check: $1.80 is more than $1.50, so the answer must be more than 100%. It is $1.50 and a fifth of $1.50 again.

"As a percentage of the number of males": the base is 968.

$\frac{1000}{968} \times 100\% = 103.305\ldots\%$

Correct to 3 significant figures, this is 103%.

Check: there were a few more females than males, so the answer is a little more than 100%.

The base is the quantity after "than": Ali's 150 stamps, which is 100%.

20% more than Ali means $100\% + 20\% = 120\%$ of Ali's number of stamps.

On the bar model, Ali's 150 stamps are 10 units of 10%, so 1 unit is $150 \div 10 = 15$ stamps. Ravi has 12 units: $12 \times 15 = 180$ stamps.

The same in one line: $120\% \text{ of } 150 = 1.2 \times 150 = 180$.

Check: 20% of 150 is 30, and Ravi has 30 stamps more than Ali: $150 + 30 = 180$.

The base is the quantity after "than": the 750 blue files, which is 100%.

10% more than the blue files means $100\% + 10\% = 110\%$ of the number of blue files.

$110\% \text{ of } 750 = \frac{110}{100} \times 750 = 825$

Check: 10% of 750 is 75, and $750 + 75 = 825$.

(a) The base is last year's price. 108% is 100% and 8% more, since $108\% - 100\% = 8\%$.

(b) The base is the number of people the canteen holds. 60% more than means $100\% + 60\% = 160\%$ of.

(c) The base is the number of visitors on Monday. $230\% - 100\% = 130\%$, so Saturday had 130% more visitors than Monday. That is more than twice Monday's number, since 230% is more than 200%.

The base is the quantity after "than": last year's $3500, which is 100%.

Hui Min has found 100% of $3500, which is the same amount as last year. But the question says 100% more than last year: all of last year's amount, and 100% of it again.

100% more than means $100\% + 100\% = 200\%$ of last year's amount.

$200\% \text{ of } 3500 = 2 \times 3500 = 7000$

Check: 100% of $3500 is $3500, so 100% more is $3500 more: $3500 + 3500 = 7000$.

The base is the quantity after "than": the 480 passengers between 6 a.m. and 7 a.m., which is 100%.

200% more than means $100\% + 200\% = 300\%$ of the base, which is three times the base.

$300\% \text{ of } 480 = 3 \times 480 = 1440$

Check on the bar model: 4 units = 480 passengers, so 1 unit = 120 passengers, since $480 \div 4 = 120$. The hour from 7 a.m. is 12 units: $12 \times 120 = 1440$ passengers.

Taking 200% of 480 would give 960, which is only 100% more than 480.

(a) Siti's marks are a part of the full marks. A part cannot be more than the whole, so her marks are at most 100% of the full marks.

(b) The girls are not a part of the boys. They are a separate group, and there can be more girls than boys. For example, 23 girls and 20 boys give $\frac{23}{20} \times 100\% = 115\%$.

(c) The students who went on the trip are a part of Class 1A. At most all of them, 100%, can go.

(d) This month's sales are compared with last month's sales, which are a separate amount. Sales can grow, so this month's sales can be more than last month's.

Ali's 42 marks are a part of the 48 full marks, so his score is at most 100%. An answer of about 114% must be wrong.

He divided the wrong way round. The base is the full marks, 48, so 48 goes in the denominator.

$\frac{42}{48} \times 100\% = 87.5\%$

Check: 42 is less than 48, so the answer is less than 100%.