Percentages, Fractions and Decimals

What "per cent" means, how to change a percentage into a fraction or a decimal and back again, how to order a mix of them, and how to find a percentage of a quantity.

Before you start

A fraction is in its simplest form when its numerator and denominator have no common factor other than 1. To simplify, divide both by their HCF: $\frac{18}{24} = \frac{3}{4}$ , dividing both by 6.

Dividing by 100 moves the decimal point two places to the left: $37 \div 100 = 0.37$ and $4.5 \div 100 = 0.045$ . Multiplying by 100 moves it two places to the right: $0.035 \times 100 = 3.5$ .

A fraction is changed to a decimal by dividing its numerator by its denominator. Some decimals stop, such as $\frac{3}{8} = 0.375$ , and some recur, such as $\frac{1}{3} = 0.333\ldots = 0.\dot{3}$ .

What "Per Cent" Means

The words per cent mean "out of a hundred". They come from the Latin words "per centum", "for each hundred".

The hundred-square below is one whole square cut into 100 equal small squares. 35 of the 100 small squares are shaded, so 35 out of every 100 parts of the whole are shaded. We say that 35 per cent of the square is shaded, and we write it as 35%.

35 of the 100 squares are shaded: 35% of the whole square.

The shaded part is 35 hundredths of the square. As a fraction it is $\frac{35}{100}$ of the square, and as a decimal it is 0.35 of the square. So $35\%$ , $\frac{35}{100}$ and 0.35 are three ways of writing the same number.

The other 65 small squares are not shaded, so 65% of the square is not shaded. The two parts together make the whole square: $35\% + 65\% = 100\%$ .

Definition: percentage

Per cent means "out of 100". Its symbol is %.

A percentage is a fraction whose denominator is 100, written with the symbol % in place of the denominator: $35\% = \frac{35}{100}$ and $7\% = \frac{7}{100}$ .

100% is the whole: $100\% = \frac{100}{100} = 1$ .

The whole is 100%

When a whole is split into parts, the percentages of the parts add up to 100%.

A percentage tells how many out of every 100. "56% of the students asked like online games" means 56 out of every 100 of them, whether 100 students were asked or 300.

A percentage can be less than 1%: 0.5% is half of one hundredth. It can also be more than 100% when a quantity is compared with a smaller one; percentages greater than 100% have a lesson of their own.

Example 1: The parts of a whole

At a CCA fair, every Secondary 1 student of a school signed up for one CCA. 45% of the students signed up for a sport, 32% for a uniformed group and the rest for a performing-arts group. What percentage of the students signed up for a performing-arts group?

Solution

All the students together are the whole, so they are 100%.

Every student signed up for exactly one of the three kinds of CCA, so their percentages add up to 100%.

Sport and uniformed group together: $45\% + 32\% = 77\%$

Performing arts: $100\% - 77\% = 23\%$

Check: $45\% + 32\% + 23\% = 100\%$

23% means 23 out of every 100 students signed up for a performing-arts group.

23% of the students signed up for a performing-arts group.

Changing a Percentage to a Fraction or a Decimal

A percentage is a fraction with denominator 100, so it can be written as a fraction straight away and then simplified. Dividing by 100 gives it as a decimal.

Rule: a percentage as a fraction and as a decimal

To a fraction: write the number over 100, then simplify: $75\% = \frac{75}{100} = \frac{3}{4}$ .

To a decimal: divide the number by 100, that is, move the decimal point two places to the left: $75\% = 75 \div 100 = 0.75$ .

Example 2: Percentages as fractions

Express each percentage as a fraction in its simplest form.
(a) 75%
(b) 8%
(c) 64%

Solution

(a) $75\% = \frac{75}{100}$ . The HCF of 75 and 100 is 25, so divide both by 25: $\frac{75}{100} = \frac{3}{4}$ .

(b) $8\% = \frac{8}{100}$ . The HCF of 8 and 100 is 4, so divide both by 4: $\frac{8}{100} = \frac{2}{25}$ .

Check: 3 and 4, 2 and 25, and 16 and 25 have no common factor other than 1, so each fraction is in its simplest form.

(a) $75\% = \frac{3}{4}$
(b) $8\% = \frac{2}{25}$
(c) $64\% = \frac{16}{25}$

Example 3: Percentages as decimals

Express each percentage as a decimal.
(a) 75%
(b) 8%
(c) 55.7%

Solution

(a) $75\% = 75 \div 100 = 0.75$

(b) $8\% = 8 \div 100$ . Write 8 as 8.0 and move the decimal point two places to the left. There is only one digit in front of the point, so a 0 is put before the 8: $8\% = 0.08$ . It is not 0.8, which is 80%.

Check (a) against Example 2: $\frac{3}{4} = 3 \div 4 = 0.75$ , the same number.

(a) $75\% = 0.75$
(b) $8\% = 0.08$
(c) $55.7\% = 0.557$

Changing a Fraction or a Decimal to a Percentage

To go the other way, multiply by 100%. This does not change the value of the number, because $100\% = \frac{100}{100} = 1$ , and multiplying a number by 1 leaves it as it is. It only writes the number as so many hundredths.

Rule: a fraction or a decimal as a percentage

Fraction to percentage: multiply by 100%: $\frac{3}{8} = \frac{3}{8} \times 100\% = \frac{300}{8}\% = 37.5\%$ . When 100 is a multiple of the denominator, an equivalent fraction with denominator 100 does the same: $\frac{7}{20} = \frac{35}{100} = 35\%$ .

Decimal to percentage: multiply by 100%, that is, move the decimal point two places to the right: $0.65 = 0.65 \times 100\% = 65\%$ .

Write the % sign every time: $\frac{3}{8} \times 100 = 37.5$ is a number, and $\frac{3}{8} = 37.5\%$ .

Example 4: Fractions as percentages

Express each fraction as a percentage.
(a) $\frac{1}{25}$
(b) $\frac{7}{20}$
(c) $\frac{3}{8}$

Solution

(a) $25 \times 4 = 100$ , so multiply the numerator and the denominator by 4: $\frac{1}{25} = \frac{4}{100} = 4\%$ .

(b) $20 \times 5 = 100$ , so multiply both by 5: $\frac{7}{20} = \frac{35}{100} = 35\%$ .

(c) 100 is not a multiple of 8, so multiply by 100%: $\frac{3}{8} \times 100\% = \frac{300}{8}\% = 37.5\%$ , since $300 \div 8 = 37.5$ .

Check (a) the other way: $\frac{1}{25} \times 100\% = \frac{100}{25}\% = 4\%$ .

(a) $\frac{1}{25} = 4\%$
(b) $\frac{7}{20} = 35\%$
(c) $\frac{3}{8} = 37.5\%$

Example 5: Decimals as percentages

Express each decimal as a percentage.
(a) 0.65
(b) 0.7
(c) 0.035

Solution

(a) $0.65 \times 100\% = 65\%$

(b) $0.7 \times 100\% = 70\%$ . To move the decimal point two places to the right, write 0.7 as 0.70: it is 70 hundredths. It is not 7%, which is 0.07.

(a) $0.65 = 65\%$
(b) $0.7 = 70\%$
(c) $0.035 = 3.5\%$

Percentages That Are Not Whole Numbers

A percentage need not be a whole number. One eighth of the hundred-square is $100 \div 8 = 12.5$ small squares: 12 whole squares and half of the next one. So $\frac{1}{8} = 12.5\%$ . In the same way, one third of the hundred-square is $100 \div 3 = 33\frac{1}{3}$ small squares, so $\frac{1}{3} = 33\frac{1}{3}\%$ . A percentage less than 1%, such as 0.5%, is less than one small square: here, half of one square out of the hundred.

$12\frac{1}{2}$ of the 100 squares: $\frac{1}{8}$ of the whole, or 12.5%

$33\frac{1}{3}$ of the 100 squares: $\frac{1}{3}$ of the whole, or $33\frac{1}{3}\%$

To write a percentage such as 12.5% as a fraction, put it over 100 as before, then multiply the numerator and the denominator by 10 (or by 100) to make the numerator a whole number, and simplify.

Example 6: A percentage with a decimal point

Express each percentage as a fraction in its simplest form and as a decimal.
(a) 12.5%
(b) 0.5%

Solution

(a) $12.5\% = \frac{12.5}{100}$ . Multiply the numerator and the denominator by 10 to make the numerator a whole number: $\frac{12.5}{100} = \frac{125}{1000}$ .

The HCF of 125 and 1000 is 125, so divide both by 125: $\frac{125}{1000} = \frac{1}{8}$ .

As a decimal: $12.5\% = 12.5 \div 100 = 0.125$ . Check: $1 \div 8 = 0.125$ .

12.5% is also written $12\frac{1}{2}\%$ . Working with the mixed number gives the same fraction: $12\frac{1}{2} = \frac{25}{2}$ , so $12\frac{1}{2}\% = \frac{25}{2} \div 100 = \frac{25}{200} = \frac{1}{8}$ .

(b) $0.5\% = \frac{0.5}{100} = \frac{5}{1000}$ , multiplying both by 10. The HCF of 5 and 1000 is 5, so $\frac{5}{1000} = \frac{1}{200}$ .

As a decimal: $0.5\% = 0.5 \div 100 = 0.005$ . It is less than 0.01, which is 1%.

(a) $12.5\% = \frac{1}{8} = 0.125$
(b) $0.5\% = \frac{1}{200} = 0.005$

Some fractions have decimals that recur, and then their percentages do not stop either. One third is $0.\dot{3} = 0.333\ldots$ , so as a percentage it is $33.333\ldots\%$ , which is $33\frac{1}{3}\%$ . The hundred-square above shows it: 33 whole squares and a third of the next one.

Rule: percentages from fractions whose decimals recur

For the exact percentage, write it as a mixed number: $\frac{1}{3} \times 100\% = \frac{100}{3}\% = 33\frac{1}{3}\%$ .

Or round it, and say so: $\frac{1}{3} \approx 33.3\%$ (to 3 significant figures). Write $\approx$ , not $=$ , before a rounded value.

Example 7: Fractions whose decimals recur

Express each fraction as a percentage, exactly and correct to 3 significant figures.
(a) $\frac{1}{3}$
(b) $\frac{2}{3}$
(c) $\frac{5}{6}$

Solution

(a) $\frac{1}{3} \times 100\% = \frac{100}{3}\%$ . $100 \div 3 = 33$ remainder 1, so $\frac{100}{3} = 33\frac{1}{3}$ and $\frac{1}{3} = 33\frac{1}{3}\%$ .

As a decimal, $33\frac{1}{3} = 33.333\ldots$ , which is 33.3 to 3 significant figures, so $\frac{1}{3} \approx 33.3\%$ .

(b) $\frac{2}{3} \times 100\% = \frac{200}{3}\%$ . $200 \div 3 = 66$ remainder 2, so $\frac{2}{3} = 66\frac{2}{3}\%$ .

As a decimal, $66\frac{2}{3} = 66.666\ldots$ . The digit after the third significant figure is 6, so round up: $\frac{2}{3} \approx 66.7\%$ .

(c) $\frac{5}{6} \times 100\% = \frac{500}{6}\% = \frac{250}{3}\%$ , dividing both by 2. $250 \div 3 = 83$ remainder 1, so $\frac{5}{6} = 83\frac{1}{3}\%$ .

As a decimal, $83\frac{1}{3} = 83.333\ldots$ , so $\frac{5}{6} \approx 83.3\%$ .

Check (a) by the decimal: $\frac{1}{3} = 0.333\ldots$ , and $0.333\ldots \times 100\% = 33.333\ldots\%$ , the same.

(a) $\frac{1}{3} = 33\frac{1}{3}\% \approx 33.3\%$
(b) $\frac{2}{3} = 66\frac{2}{3}\% \approx 66.7\%$
(c) $\frac{5}{6} = 83\frac{1}{3}\% \approx 83.3\%$

Example 8: Back from a mixed-number percentage

(a) Express $33\frac{1}{3}\%$ as a fraction in its simplest form.
(b) Express $16\frac{2}{3}\%$ as a fraction in its simplest form.
(c) Is 33% equal to $\frac{1}{3}$ ?

Solution

(a) Write the mixed number as an improper fraction: $33\frac{1}{3} = \frac{100}{3}$ . Dividing by 100 is the same as multiplying by $\frac{1}{100}$ , so $33\frac{1}{3}\% = \frac{100}{3} \times \frac{1}{100} = \frac{100}{300} = \frac{1}{3}$ .

(b) $16\frac{2}{3} = \frac{50}{3}$ , so $16\frac{2}{3}\% = \frac{50}{3} \times \frac{1}{100} = \frac{50}{300} = \frac{1}{6}$ .

(c) $33\% = \frac{33}{100}$ . To compare it with $\frac{1}{3}$ , use the common denominator 300: $\frac{33}{100} = \frac{99}{300}$ and $\frac{1}{3} = \frac{100}{300}$ .

They differ by $\frac{1}{300}$ , so 33% is close to $\frac{1}{3}$ but not equal to it.

(a) $33\frac{1}{3}\% = \frac{1}{3}$
(b) $16\frac{2}{3}\% = \frac{1}{6}$
(c) No. 33% is $\frac{33}{100}$ , which is $\frac{1}{300}$ less than $\frac{1}{3}$ .

Percentages worth knowing

These come up again and again. Each row is one number written in three ways.

Fraction	Decimal	Percentage
$\frac{1}{100}$	$0.01$	$1\%$
$\frac{1}{20}$	$0.05$	$5\%$
$\frac{1}{10}$	$0.1$	$10\%$
$\frac{1}{8}$	$0.125$	$12.5\%$
$\frac{1}{5}$	$0.2$	$20\%$
$\frac{1}{4}$	$0.25$	$25\%$
$\frac{1}{3}$	$0.\dot{3}$	$33\frac{1}{3}\%$
$\frac{1}{2}$	$0.5$	$50\%$
$\frac{2}{3}$	$0.\dot{6}$	$66\frac{2}{3}\%$
$\frac{3}{4}$	$0.75$	$75\%$
$1$	$1$	$100\%$

Ordering Fractions, Decimals and Percentages

To compare numbers written in different forms, write them all in the same form first. Percentages are often the easiest, because they are all out of 100; decimals work just as well. Comparing the numbers as they are written means nothing: 65% is not more than 0.7 just because 65 is more than 0.7.

Rule: ordering a mix of fractions, decimals and percentages

Write every number as a percentage (or every number as a decimal).
Compare them as you compare ordinary numbers.
Give the answer with the numbers in their original forms, in the order asked for: ascending means from the smallest to the largest, and descending from the largest to the smallest.

Example 9: Three numbers in ascending order

Arrange in ascending order: $0.7$ , $\frac{3}{5}$ , $65\%$

Solution

$0.7 = 0.7 \times 100\% = 70\%$

$\frac{3}{5} = \frac{60}{100} = 60\%$ , multiplying both by 20.

$65\%$ is already a percentage.

In ascending order: $60\% < 65\% < 70\%$

In their original forms: $\frac{3}{5} < 65\% < 0.7$

In ascending order: $\frac{3}{5}$ , $65\%$ , $0.7$

Example 10: Numbers that are close together

Arrange in descending order: $\frac{1}{3}$ , $0.33$ , $33.5\%$ , $0.3$

Solution

$\frac{1}{3} = 33\frac{1}{3}\%$ (Example 7)

$0.33 = 33\%$

$33.5\%$ is already a percentage.

$0.3 = 30\%$

$33\frac{1}{3} = 33.333\ldots$ , which lies between 33 and 33.5. So $33.5 > 33\frac{1}{3} > 33 > 30$ .

In descending order: $33.5\% > 33\frac{1}{3}\% > 33\% > 30\%$ , which in their original forms is $33.5\% > \frac{1}{3} > 0.33 > 0.3$ .

Check with decimals: $0.335 > 0.333\ldots > 0.33 > 0.3$ , the same order.

Had $\frac{1}{3}$ been rounded to 33%, it would have seemed equal to 0.33. Keep the exact value, or enough figures to tell the numbers apart.

In descending order: $33.5\%$ , $\frac{1}{3}$ , $0.33$ , $0.3$

Finding a Percentage of a Quantity

35% of $240 means 35 hundredths of $240. The $240 is the whole, so it is 100%. Draw a bar for the $240 and mark it from 0% to 100%. The part that is 35% of it is a little more than a third of the bar, because a third is $33\frac{1}{3}\%$ .

The word "of" means "multiply": 35% of $240 is $35\% \times 240$ . Write the percentage as a fraction or as a decimal, then multiply.

Rule: a percentage of a quantity

By the fraction: $35\% \text{ of } 240 = \frac{35}{100} \times 240$

By the decimal: $35\% \text{ of } 240 = 0.35 \times 240$

The two ways give the same answer, because $\frac{35}{100}$ and $0.35$ are the same number. The answer has the unit of the quantity: here, dollars.

Example 11: 35% of $240

Find 35% of $240
(a) by writing the percentage as a fraction,
(b) by writing the percentage as a decimal.

Solution

(a) $35\% = \frac{35}{100} = \frac{7}{20}$ , dividing both by 5.

$\frac{7}{20} \times 240 = 7 \times (240 \div 20) = 7 \times 12 = 84$

(b) $35\% = 0.35$ , and $0.35 \times 240 = 84$

Check by tens and fives: 10% of 240 is $240 \div 10 = 24$ , so 30% is $3 \times 24 = 72$ . 5% is half of 10%, which is 12. So 35% is $72 + 12 = 84$ .

$84 is a little more than a third of $240 (which is $80), as the bar showed.

35% of $240 is $84.

Example 12: Phone storage

Mei Ling's phone has 64 GB of storage, and 68% of it is used.
(a) How much storage is used?
(b) How much storage is free?

Solution

(a) The whole is the 64 GB of storage, so it is 100%.

$68\% = 0.68$ , so the storage used is $0.68 \times 64 = 43.52$ , which is 43.52 GB.

(b) The used part and the free part make up the whole, so the free part is $100\% - 68\% = 32\%$ of the storage.

$32\% = 0.32$ , and $0.32 \times 64 = 20.48$ , which is 20.48 GB.

Check: $43.52 + 20.48 = 64$

(a) 43.52 GB of storage is used.
(b) 20.48 GB of storage is free.

Example 13: Choosing the fraction or the decimal

(a) $33\frac{1}{3}\%$ of the 36 students in Class 1C wear spectacles. How many students wear spectacles?
(b) 0.5% of a 5 kg bag of rice is broken grains. What mass of the rice, in grams, is broken grains?

Solution

(a) The whole is the 36 students. $33\frac{1}{3}\% = \frac{1}{3}$ exactly (Example 8), so use the fraction: $\frac{1}{3} \times 36 = 12$ .

As a decimal, $33\frac{1}{3}\%$ is $0.333\ldots$ , which never stops. Cut short to 0.333, it gives $0.333 \times 36 = 11.988$ , which is not quite right, and is not a whole number of students. When the decimal of a percentage recurs, use its fraction.

(b) The answer is wanted in grams, so write the whole in grams: 5 kg = 5000 g.

$0.5\% = \frac{1}{200}$ (Example 6), so $\frac{1}{200} \times 5000 = 5000 \div 200 = 25$ , which is 25 g.

By the decimal, as a check: $0.5\% = 0.005$ , and $0.005 \times 5000 = 25$ .

(a) 12 students wear spectacles.
(b) 25 g of the rice is broken grains.

Finding what percentage one quantity is of another, such as 18 marks out of 25, is the subject of the next lesson.

Common Mistakes

Moving the decimal point one place instead of two. $8\% = 0.08$ , not 0.8. $0.7 = 70\%$ , not 7%. $0.5\% = 0.005$ , not 0.5 or 0.05.
Leaving out the % sign. $\frac{3}{8} \times 100 = 37.5$ , so $\frac{3}{8} = 37.5\%$ . Written without the sign, " $\frac{3}{8} = 37.5$ " is false.
Taking 33% for one third. $\frac{1}{3} = 33\frac{1}{3}\%$ exactly. 33% is $\frac{33}{100}$ , a little less than $\frac{1}{3}$ .
Leaving a decimal inside a fraction. $12.5\% = \frac{12.5}{100}$ is not finished. Multiply the numerator and the denominator by 10 and simplify: $\frac{125}{1000} = \frac{1}{8}$ .
Comparing numbers in different forms as they stand. 65% is less than 0.7, because $0.7 = 70\%$ . Write every number in the same form first.
Rounding a recurring decimal and then multiplying. $0.33 \times 36 = 11.88$ , but $33\frac{1}{3}\%$ of 36 is exactly 12. Use the fraction $\frac{1}{3}$ .

Practice

Work each question on paper before you open its solution.

Question 1

The label on a cushion cover says that its fabric is 50% cotton, 20% polyester, 20% nylon and the rest silk.
(a) What percentage of the fabric is silk?
(b) In every 100 g of the fabric, how many grams are cotton?
(c) Write the percentage of silk as a fraction in its simplest form.

ShowHide solution

(a) The whole fabric is 100%. $50\% + 20\% + 20\% = 90\%$ , so silk is $100\% - 90\% = 10\%$ .

(b) 50% means 50 out of every 100, so 50 g of every 100 g of the fabric is cotton.

(a) 10% of the fabric is silk.
(b) 50 g of every 100 g is cotton.
(c) $\frac{1}{10}$

Question 2

Express each percentage as a fraction in its simplest form.
(a) 45%
(b) 6%
(c) 17.5%
(d) 0.2%

ShowHide solution

(a) $45\% = \frac{45}{100} = \frac{9}{20}$ , dividing both by the HCF, 5.

(b) $6\% = \frac{6}{100} = \frac{3}{50}$ , dividing both by the HCF, 2.

(c) $17.5\% = \frac{17.5}{100} = \frac{175}{1000}$ , multiplying both by 10. The HCF of 175 and 1000 is 25, so $\frac{175}{1000} = \frac{7}{40}$ .

(d) $0.2\% = \frac{0.2}{100} = \frac{2}{1000}$ , multiplying both by 10. Dividing both by 2: $\frac{2}{1000} = \frac{1}{500}$ .

(a) $\frac{9}{20}$
(b) $\frac{3}{50}$
(c) $\frac{7}{40}$
(d) $\frac{1}{500}$

Question 3

Express each percentage as a decimal.
(a) 64%
(b) 3%
(c) 42.5%
(d) 0.25%

ShowHide solution

(a) $64\% = 64 \div 100 = 0.64$

(b) $3\% = 3 \div 100 = 0.03$ . There is one digit in front of the point, so a 0 goes before the 3.

(d) $0.25\% = 0.25 \div 100 = 0.0025$ . A percentage less than 1% gives a decimal less than 0.01.

(a) 0.64
(b) 0.03
(c) 0.425
(d) 0.0025

Question 4

Express each number as a percentage.
(a) $\frac{9}{25}$
(b) $\frac{5}{8}$
(c) 0.06
(d) 0.875
(e) $\frac{1}{40}$

ShowHide solution

(a) $25 \times 4 = 100$ , so $\frac{9}{25} = \frac{36}{100} = 36\%$ .

(b) 100 is not a multiple of 8: $\frac{5}{8} \times 100\% = \frac{500}{8}\% = 62.5\%$ .

(d) $0.875 \times 100\% = 87.5\%$

(e) $\frac{1}{40} \times 100\% = \frac{100}{40}\% = 2.5\%$

(a) 36%
(b) 62.5%
(c) 6%
(d) 87.5%
(e) 2.5%

Question 5

(a) Express $\frac{1}{6}$ as a percentage, exactly and correct to 3 significant figures.
(b) Express $\frac{2}{9}$ as a percentage, exactly and correct to 3 significant figures.
(c) Express $83\frac{1}{3}\%$ as a fraction in its simplest form.
(d) Siti says that 66% is the same as $\frac{2}{3}$ . Is she right? Explain.

ShowHide solution

(a) $\frac{1}{6} \times 100\% = \frac{100}{6}\% = \frac{50}{3}\%$ . $50 \div 3 = 16$ remainder 2, so $\frac{1}{6} = 16\frac{2}{3}\%$ . As a decimal, $16\frac{2}{3} = 16.666\ldots$ , so $\frac{1}{6} \approx 16.7\%$ .

(b) $\frac{2}{9} \times 100\% = \frac{200}{9}\%$ . $200 \div 9 = 22$ remainder 2, so $\frac{2}{9} = 22\frac{2}{9}\%$ . As a decimal, $22\frac{2}{9} = 22.222\ldots$ , so $\frac{2}{9} \approx 22.2\%$ .

(c) $83\frac{1}{3} = \frac{250}{3}$ , so $83\frac{1}{3}\% = \frac{250}{3} \times \frac{1}{100} = \frac{250}{300} = \frac{5}{6}$ .

(d) $\frac{2}{3} \times 100\% = 66\frac{2}{3}\%$ , but 66% is $\frac{66}{100}$ . Over the common denominator 300, $\frac{66}{100} = \frac{198}{300}$ and $\frac{2}{3} = \frac{200}{300}$ , so 66% is $\frac{2}{300} = \frac{1}{150}$ less than $\frac{2}{3}$ .

(a) $\frac{1}{6} = 16\frac{2}{3}\% \approx 16.7\%$
(b) $\frac{2}{9} = 22\frac{2}{9}\% \approx 22.2\%$
(c) $\frac{5}{6}$
(d) No. $\frac{2}{3} = 66\frac{2}{3}\%$ , so 66% is a little less than $\frac{2}{3}$ .

Question 6

Write each number in the two other forms. Give fractions in their simplest form and recurring decimals in dot notation.
(a) $\frac{3}{20}$ as a decimal and as a percentage
(b) 0.44 as a fraction and as a percentage
(c) 22.5% as a fraction and as a decimal
(d) $\frac{1}{12}$ as a decimal and as a percentage

ShowHide solution

(a) $20 \times 5 = 100$ , so $\frac{3}{20} = \frac{15}{100} = 0.15 = 15\%$ .

(b) $0.44 = \frac{44}{100} = \frac{11}{25}$ , dividing both by 4. As a percentage, $0.44 \times 100\% = 44\%$ .

(c) $22.5\% = \frac{22.5}{100} = \frac{225}{1000}$ . The HCF of 225 and 1000 is 25, so $\frac{225}{1000} = \frac{9}{40}$ . As a decimal, $22.5 \div 100 = 0.225$ .

(d) $1 \div 12 = 0.08333\ldots$ , in which only the 3 repeats: $\frac{1}{12} = 0.08\dot{3}$ .

As a percentage: $\frac{1}{12} \times 100\% = \frac{100}{12}\% = \frac{25}{3}\% = 8\frac{1}{3}\%$ .

(a) $0.15$ and $15\%$
(b) $\frac{11}{25}$ and $44\%$
(c) $\frac{9}{40}$ and $0.225$
(d) $0.08\dot{3}$ and $8\frac{1}{3}\%$

Question 7

Which is greater: $\frac{7}{12}$ or $58\%$ ?

ShowHide solution

Write $\frac{7}{12}$ as a percentage: $\frac{7}{12} \times 100\% = \frac{700}{12}\% = \frac{175}{3}\%$ .

$175 \div 3 = 58$ remainder 1, so $\frac{7}{12} = 58\frac{1}{3}\%$ .

$58\frac{1}{3}\% > 58\%$

Check with decimals: $\frac{7}{12} = 0.58333\ldots$ and $58\% = 0.58$ , and $0.58333\ldots > 0.58$ .

$\frac{7}{12}$ is greater than $58\%$ .

Question 8

Arrange in ascending order: $0.42$ , $\frac{2}{5}$ , $41\%$ , $\frac{3}{7}$

ShowHide solution

$0.42 = 42\%$

$\frac{2}{5} = \frac{40}{100} = 40\%$

$41\%$ is already a percentage.

$\frac{3}{7} \times 100\% = \frac{300}{7}\% = 42\frac{6}{7}\%$ , since $300 \div 7 = 42$ remainder 6. That is about 42.9%.

In ascending order: $40\% < 41\% < 42\% < 42\frac{6}{7}\%$

In their original forms: $\frac{2}{5} < 41\% < 0.42 < \frac{3}{7}$

In ascending order: $\frac{2}{5}$ , $41\%$ , $0.42$ , $\frac{3}{7}$

Question 9

Find
(a) 60% of 30,
(b) 12.5% of 56 kg,
(c) 0.5% of $3600,
(d) 7.5% of $38.90, correct to the nearest cent.

ShowHide solution

(a) $60\% = \frac{60}{100} = \frac{3}{5}$ , and $\frac{3}{5} \times 30 = 3 \times 6 = 18$ .

(b) $12.5\% = \frac{1}{8}$ , and $\frac{1}{8} \times 56 = 7$ , which is 7 kg.

(d) $7.5\% = 0.075$ , and $0.075 \times 38.90 = 2.9175$ . To the nearest cent means to 2 decimal places. The third decimal place is 7, so round up: $2.92.

(a) 18
(b) 7 kg
(c) $18
(d) $2.92

Question 10

Siti receives $120 of pocket money a month. She saves 15% of it and spends 40% of it on food.
(a) How much does she save each month?
(b) How much does she spend on food each month?
(c) How much is left for other things, and what percentage of her pocket money is that?

ShowHide solution

(a) The whole is the $120, which is 100%. $15\% = 0.15$ , and $0.15 \times 120 = 18$ , so she saves $18.

(b) $40\% = \frac{2}{5}$ , and $\frac{2}{5} \times 120 = 48$ , so she spends $48 on food.

As a percentage: $100\% - 15\% - 40\% = 45\%$ .

Check: $0.45 \times 120 = 54$

(a) She saves $18 each month.
(b) She spends $48 on food each month.
(c) $54 is left for other things, which is 45% of her pocket money.

Question 11

240 Secondary 1 students were asked how they travel to school. $66\frac{2}{3}\%$ of them take the bus or the MRT, 12.5% are driven by a parent, and the rest walk. How many students walk to school?

ShowHide solution

The whole is the 240 students.

Bus or MRT: $66\frac{2}{3}\% = \frac{2}{3}$ , and $\frac{2}{3} \times 240 = 160$ students. As a decimal, $66\frac{2}{3}\%$ is $0.666\ldots$ , which recurs, so the fraction is used.

Driven: $12.5\% = \frac{1}{8}$ , and $\frac{1}{8} \times 240 = 30$ students.

Walk: $240 - 160 - 30 = 50$ students.

Check: $160 + 30 + 50 = 240$

50 students walk to school.

Question 12

Hui Min spends $\frac{3}{8}$ of her savings on a bag, 0.3 of her savings on books and 25% of her savings on a birthday present for her brother.
(a) On which of the three does she spend the most?
(b) What percentage of her savings is left?
(c) Her savings were $160. How much money is left?

ShowHide solution

(a) Write each part as a percentage. Bag: $\frac{3}{8} \times 100\% = 37.5\%$ . Books: $0.3 = 30\%$ . Present: $25\%$ .

$37.5\% > 30\% > 25\%$ , so the bag takes the largest part.

(b) All her savings are 100%. $37.5\% + 30\% + 25\% = 92.5\%$ , so $100\% - 92.5\% = 7.5\%$ is left.

Check: the bag cost $60, since $\frac{3}{8} \times 160 = 60$ ; the books cost $48, since $0.3 \times 160 = 48$ ; the present cost $40, since $0.25 \times 160 = 40$ . $60 + 48 + 40 = 148$ , and $160 - 148 = 12$ .

(a) She spends the most on the bag.
(b) 7.5% of her savings is left.
(c) $12 is left.

Key Points

Per cent means "out of 100": $35\% = \frac{35}{100}$ . 100% is the whole, and the percentages of the parts of a whole add up to 100%.
A percentage to a fraction: write it over 100 and simplify. A percentage to a decimal: divide by 100, moving the decimal point two places to the left.
A fraction or a decimal to a percentage: multiply by 100%. This is multiplying by 1, so the value does not change.
A percentage may have a decimal point or be a mixed number: $12.5\% = \frac{1}{8}$ , $0.5\% = \frac{1}{200}$ and $33\frac{1}{3}\% = \frac{1}{3}$ .
When a fraction's decimal recurs, give its percentage exactly as a mixed number ( $\frac{1}{3} = 33\frac{1}{3}\%$ ) or rounded with $\approx$ ( $\frac{1}{3} \approx 33.3\%$ ). 33% is not $\frac{1}{3}$ .
To order fractions, decimals and percentages, write them all in one form first, then give the answer in their original forms.
To find a percentage of a quantity, multiply the quantity by the percentage written as a fraction or as a decimal. Use the fraction when the decimal recurs.