Maths · 1 Number

1.4 Fractions, decimals and percentages

Before you start

Times tables up to $12 \times 12$ and short division, without a calculator.
Place value in decimals: in $0.375$ the 3 is tenths, the 7 is hundredths and the 5 is thousandths.
Multiplying and dividing by $10$ , $100$ and $1000$ by moving the digits.
Finding the highest common factor (HCF) of two numbers, from subtopic 1.1.

Syllabus objectives covered (2)

Use the language and notation of the following in appropriate contexts: • proper fractions • improper fractions • mixed numbers • decimals • percentages.
Recognise equivalence and convert between these forms.

Part 1 of 6

Proper fractions, improper fractions and mixed numbers

In a fraction such as $\frac{3}{8}$ , the bottom number is the denominator. It tells you how many equal parts the whole is split into. The top number is the numerator. It tells you how many of those parts you have.
A proper fraction has a numerator smaller than its denominator, such as $\frac{3}{8}$ . Its value is less than 1, because you have fewer parts than make one whole.
An improper fraction has a numerator equal to or bigger than its denominator, such as $\frac{11}{4}$ or $\frac{4}{4}$ . Its value is 1 or more.
A mixed number is a whole number and a proper fraction written together, such as $2 \frac{3}{4}$ . It means $2 + \frac{3}{4}$ , and you read it as 'two and three-quarters'.
Mixed number to improper fraction: change the wholes into parts of the same size. In $2 \frac{3}{4}$ , each whole is 4 quarters, so 2 wholes are 8 quarters. Then add the 3 quarters you already have.
$2 \frac{3}{4} = \frac{2 \times 4 + 3}{4} = \frac{11}{4}$
The denominator does not change. You counted quarters before and after, so the answer is still in quarters.
Improper fraction to mixed number: divide the numerator by the denominator. The answer is the number of wholes. The remainder is the number of parts left over, so it goes over the same denominator.
$\frac{23}{5} : 23 \div 5 = 4 remainder 3, so \frac{23}{5} = 4 \frac{3}{5}$
Check by going back: $4 \times 5 + 3 = 23$ . If you do not get the numerator you started with, one of the steps is wrong.

Part 2 of 6

Equivalent fractions and simplest form

Equivalent fractions have the same value but different numbers. $\frac{1}{2}$ , $\frac{2}{4}$ and $\frac{5}{10}$ are all the same amount.
To make an equivalent fraction, multiply or divide the numerator and the denominator by the same number. This works because you are really multiplying by a fraction such as $\frac{5}{5}$ , which equals 1, and multiplying by 1 does not change a value.
$\frac{3}{4} = \frac{3 \times 5}{4 \times 5} = \frac{15}{20}$
Adding the same number to the top and the bottom does not give an equivalent fraction. Add 1 to both parts of $\frac{1}{2}$ and you get $\frac{2}{3}$ , which is a different amount.
A fraction is in its simplest form when the numerator and the denominator have no common factor except 1. The syllabus expects you to write fractions in their simplest form, and 'simplest form' in a question means you must cancel all the way.
The fastest way is to divide both numbers by their HCF. For $\frac{36}{48}$ , the HCF of 36 and 48 is 12.
$\frac{36}{48} = \frac{36 \div 12}{48 \div 12} = \frac{3}{4}$
You can also cancel in smaller steps, but then you must keep going until nothing else divides both numbers: $\frac{36}{48} = \frac{18}{24} = \frac{9}{12} = \frac{3}{4}$ . The trap is stopping at $\frac{9}{12}$ , because 9 and 12 still share the factor 3.
To test whether two fractions are equivalent, simplify both. $\frac{15}{35}$ and $\frac{6}{14}$ both simplify to $\frac{3}{7}$ , so they are equal. A quick second check is to multiply across: $15 \times 14 = 210$ and $35 \times 6 = 210$ . The products match, so the fractions are equal.

Part 3 of 6

Decimals to fractions

Each place after the decimal point is ten times smaller than the one before it: tenths, then hundredths, then thousandths. The place of the last digit tells you the denominator.
Count the decimal places. One place means tenths, two means hundredths, three means thousandths. Write the digits after the point over 10, 100 or 1000.
$0.7 = \frac{7}{10} 0.07 = \frac{7}{100} 0.375 = \frac{375}{1000}$
The number of zeros in the denominator equals the number of decimal places. $0.07$ has two decimal places, so it is $\frac{7}{100}$ , not $\frac{7}{10}$ . A zero straight after the point still counts as a place.
Then simplify. The HCF of 375 and 1000 is 125, so $0.375 = \frac{375}{1000} = \frac{3}{8}$ .
For a decimal bigger than 1, keep the whole number and change only the decimal part. That gives a mixed number. Change it to an improper fraction only if you need one.
$2.45 = 2 \frac{45}{100} = 2 \frac{9}{20} = \frac{49}{20}$

Part 4 of 6

Fractions to decimals: terminating and recurring

A fraction is a division: $\frac{3}{8}$ means $3 \div 8$ . So you can turn any fraction into a decimal by dividing the numerator by the denominator.
If the denominator goes exactly into 10, 100 or 1000, there is a quicker way. Make an equivalent fraction with that denominator, then read off the decimal.
$\frac{7}{20} = \frac{35}{100} = 0.35 \frac{3}{8} = \frac{375}{1000} = 0.375$
Otherwise, use short division. For $\frac{5}{8}$ : 8 into 5 goes 0 times, so write $0.$ and carry the 5. Then $50 \div 8 = 6$ remainder 2, $20 \div 8 = 2$ remainder 4, and $40 \div 8 = 5$ remainder 0. The remainder is 0, so the division stops: $\frac{5}{8} = 0.625$ .
A decimal that stops is a terminating decimal. Some divisions never stop. For $\frac{2}{3}$ : $20 \div 3 = 6$ remainder 2, and the same remainder 2 comes back at every step. So the digit 6 repeats for ever.
A decimal whose digits repeat for ever is a recurring decimal. Put a dot over the digit that repeats. If a block of digits repeats, put a dot over the first and the last digit of the block.
$\frac{2}{3} = 0.666 \dots = 0. \dot{6} \frac{5}{11} = 0.4545 \dots = 0. \dot{4} \dot{5}$
You can tell in advance which kind you will get. Write the fraction in its simplest form and find the prime factors of the denominator. If they are only 2s and 5s, the decimal terminates. Any other prime factor makes it recur. $\frac{7}{40}$ terminates because $40 = 2^{3} \times 5$ . $\frac{1}{6}$ recurs because $6 = 2 \times 3$ , and in fact $\frac{1}{6} = 0.1 \dot{6}$ .
When the decimal recurs, give it exactly with the dot notation, or round it to the accuracy the question asks for. If the question names no accuracy, give 3 significant figures: $\frac{2}{3} = 0.667$ . Cutting it off as $0.6$ or $0.66$ is neither exact nor correctly rounded.

Part 5 of 6

Percentages

Per cent means 'out of 100'. $37 %$ means 37 out of every 100, so $37 % = \frac{37}{100} = 0.37$ .
Percentage to decimal: divide by 100. $37 % = 0.37$ , $12.5 % = 0.125$ and $150 % = 1.5$ . The trap is a one-digit percentage: $4 % = 0.04$ , not $0.4$ . $0.4$ is $40 %$ .
Decimal to percentage: multiply by 100. $0.6 = 60 %$ , $0.065 = 6.5 %$ and $1.2 = 120 %$ .
Percentage to fraction: write it over 100, then simplify. If the percentage has a decimal point, multiply the top and the bottom by 10 to clear it, because a finished fraction has whole numbers on the top and the bottom.
$45 % = \frac{45}{100} = \frac{9}{20} 12.5 % = \frac{12.5}{100} = \frac{125}{1000} = \frac{1}{8}$
Fraction to percentage: make the denominator 100 if you can. If you cannot, change the fraction to a decimal and multiply by 100.
$\frac{7}{25} = \frac{28}{100} = 28 % \frac{3}{8} = 0.375 = 37.5 %$
A percentage can be more than 100, which means more than one whole: $100 % = 1$ and $250 % = 2.5 = 2 \frac{1}{2}$ . A percentage can also be less than 1: $0.5 % = 0.005 = \frac{1}{200}$ .
A recurring decimal gives a recurring percentage. $\frac{1}{3} = 0. \dot{3} = 33. \dot{3} %$ . You can also write this as $33 \frac{1}{3} %$ , or as $33.3 %$ to 3 significant figures. It is not $30 %$ .

Part 6 of 6

Equivalences worth knowing, and the right form for the context

Knowing these on sight saves time when you have no calculator.
$\frac{1}{2} = 0.5 = 50 % \frac{1}{4} = 0.25 = 25 % \frac{3}{4} = 0.75 = 75 % \frac{1}{5} = 0.2 = 20 %$
And these four:
$\frac{1}{10} = 0.1 = 10 % \frac{1}{8} = 0.125 = 12.5 % \frac{1}{100} = 0.01 = 1 % \frac{1}{3} = 0. \dot{3} = 33 \frac{1}{3} %$
Build others from them. $\frac{3}{8}$ is three lots of $\frac{1}{8}$ , so it is $3 \times 0.125 = 0.375$ . $\frac{2}{5}$ is two lots of $\frac{1}{5}$ , so it is $0.4 = 40 %$ .
To decide whether numbers in different forms are equal, change them all into the same form. Decimals are the easiest form to compare, because you can line them up digit by digit. $\frac{9}{20}$ , $0.45$ and $45 %$ all become $0.45$ , so they are equal.
In a context, the whole is the denominator and the part is the numerator. '12 out of 20 students' is $\frac{12}{20}$ of the students. In its simplest form that is $\frac{3}{5}$ , and as a percentage it is $60 %$ .
Read a decimal in context as a decimal, not as a clock time. A time of 2.5 hours is 2 hours 30 minutes, not 2 hours 50 minutes, because $0.5$ is a half and half of 60 minutes is 30 minutes.
Give the answer in the form the question names: 'as a fraction', 'as a decimal' or 'as a percentage'. Do not mix forms in one answer. $\frac{1.5}{4}$ has a decimal inside a fraction, so write it as $\frac{3}{8}$ or as $0.375$ .

Worked examples

Example 1. (a) Write $3 \frac{5}{8}$ as an improper fraction. (b) Write $\frac{47}{6}$ as a mixed number.

Show the working

(a) Each whole is 8 eighths, so 3 wholes are $3 \times 8 = 24$ eighths. Add the 5 eighths in the fraction part.
$3 \frac{5}{8} = \frac{3 \times 8 + 5}{8} = \frac{29}{8}$
Keep the denominator as 8. You counted eighths the whole way through.
(b) Divide the numerator by the denominator. $7 \times 6 = 42$ and $47 - 42 = 5$ , so $47 \div 6 = 7$ remainder 5.
The 7 is the number of wholes. The remainder is 5 sixths left over, so the 5 goes over 6.
$\frac{47}{6} = 7 \frac{5}{6}$
Check that the fraction part is proper and in its simplest form: $\frac{5}{6}$ is less than 1, and 5 and 6 share no factor except 1. Then check back: $7 \times 6 + 5 = 47$ .

Answer: (a) $\frac{29}{8}$ (b) $7 \frac{5}{6}$

Example 2. Write $0.875$ as a fraction in its simplest form. Then write $0.875$ as a percentage.

Show the working

Count the decimal places. $0.875$ has three, so the last digit is in the thousandths place. Write 875 over 1000.
$0.875 = \frac{875}{1000}$
Find a common factor. Both numbers end in 5 or 0, so both divide by 5. Keep dividing by 5 while you can.
$\frac{875}{1000} = \frac{175}{200} = \frac{35}{40} = \frac{7}{8}$
Stop when no number except 1 divides both. 7 is prime and does not divide 8, so $\frac{7}{8}$ is in its simplest form. In one step, you would divide by the HCF, which is $125$ .
For the percentage, multiply the decimal by 100: $0.875 \times 100 = 87.5$ .
Check with a known equivalence: $\frac{7}{8} = 7 \times \frac{1}{8} = 7 \times 0.125 = 0.875$ . It matches.

Answer: $\frac{7}{8}$ and $87.5 %$

Example 3. Without a calculator, write (a) $\frac{7}{40}$ and (b) $\frac{5}{6}$ as decimals. Then write each one as a percentage.

Show the working

(a) Look at the denominator first: $40 = 2^{3} \times 5$ . Its only prime factors are 2 and 5, so the decimal terminates, and you can make the denominator 1000.
$40 \times 25 = 1000$ , so multiply the top and the bottom by 25.
$\frac{7}{40} = \frac{7 \times 25}{40 \times 25} = \frac{175}{1000} = 0.175$
Multiply by 100 for the percentage: $0.175 \times 100 = 17.5$ , so it is $17.5 %$ .
(b) $6 = 2 \times 3$ . The prime factor 3 tells you the decimal will recur, so use short division of 5 by 6.
6 into 5 goes 0 times, so write $0.$ and carry the 5. $50 \div 6 = 8$ remainder 2. $20 \div 6 = 3$ remainder 2. The remainder is 2 again, so every later step is $20 \div 6$ and the 3 repeats for ever.
$\frac{5}{6} = 0.8333 \dots = 0.8 \dot{3}$
Only the 3 repeats, so only the 3 gets the dot. The 8 is not part of the repeating block.
For the percentage, multiply by 100 to get $83.333 \dots %$ . Write it exactly as $83. \dot{3} %$ or $83 \frac{1}{3} %$ , or round it to 3 significant figures: $83.3 %$ .

Answer: (a) $0.175 = 17.5 %$ (b) $0.8 \dot{3}$ , which is $83 \frac{1}{3} %$ ( $83.3 %$ to 3 significant figures)

Example 4. Here are six numbers: $\frac{9}{15}$ , $0.35$ , $60 %$ , $\frac{12}{20}$ , $0.06$ , $1 \frac{2}{3}$ . Write down all the numbers that are equal to $\frac{3}{5}$ .

Show the working

Change $\frac{3}{5}$ to a decimal first, so you have one target to compare against: $\frac{3}{5} = \frac{6}{10} = 0.6$ .
$\frac{9}{15}$ : the HCF of 9 and 15 is 3, so it simplifies to $\frac{3}{5}$ . Equal.
$0.35$ is 35 hundredths, and $0.6$ is 60 hundredths. Not equal. A 3 and a 5 in the digits do not make a number equal to $\frac{3}{5}$ .
$60 % = \frac{60}{100} = 0.6$ . Equal.
$\frac{12}{20}$ : divide the top and the bottom by 4 to get $\frac{3}{5}$ . Equal.
$0.06 = 6 %$ , which is ten times smaller than $0.6$ . Not equal.
$1 \frac{2}{3}$ is more than 1, and $\frac{3}{5}$ is less than 1, so they cannot be equal. As an improper fraction, $1 \frac{2}{3} = \frac{5}{3}$ . That is the reciprocal of $\frac{3}{5}$ , not the same number.

Answer: $\frac{9}{15}$ , $60 %$ and $\frac{12}{20}$

Where people go wrong

Mistake: Writing $4 %$ as $0.4$ .

Instead: $4 % = \frac{4}{100} = 0.04$ . Dividing by 100 moves the 4 two places, into the hundredths place.

$0.4$ is four tenths, which is $\frac{40}{100} = 40 %$ . Any whole-number percentage from $1 %$ to $9 %$ becomes a decimal with a zero straight after the point.

Mistake: Stopping a simplification too early, e.g. $\frac{18}{24} = \frac{9}{12}$ .

Instead: Divide by the HCF, 6, to get $\frac{3}{4}$ . Or keep cancelling until the numerator and the denominator share no factor except 1.

$\frac{9}{12}$ is equivalent and correct, but it is not in its simplest form, because 9 and 12 both divide by 3. 'Simplest form' asks for the end of the process.

Mistake: Writing $0.07$ as $\frac{7}{10}$ .

Instead: $0.07$ has two decimal places, so it is $\frac{7}{100}$ .

The 7 is in the hundredths place. The zero after the point holds the tenths place, and it counts as a decimal place.

Mistake: Turning $2 \frac{3}{5}$ into $\frac{6}{5}$ (multiplying the 2 by the 3) or into $\frac{23}{5}$ (writing the digits side by side).

Instead: $2 \frac{3}{5} = \frac{2 \times 5 + 3}{5} = \frac{13}{5}$ .

Each whole is 5 fifths, so 2 wholes are 10 fifths. Multiply the whole number by the denominator, not by the numerator, then add the numerator.

Mistake: Treating $\frac{1}{3}$ as $0.3$ or as $30 %$ .

Instead: $\frac{1}{3} = 0.333 \dots = 0. \dot{3} = 33 \frac{1}{3} %$ .

$0.3$ is $\frac{3}{10}$ , and $\frac{3}{10} = \frac{9}{30}$ is smaller than $\frac{1}{3} = \frac{10}{30}$ . The 3s in $\frac{1}{3}$ never stop, so cutting them off changes the value.

Check yourself

Write $\frac{19}{4}$ as a mixed number.
Answer
$19 \div 4 = 4$ remainder 3, so $\frac{19}{4} = 4 \frac{3}{4}$ .
Write $\frac{45}{60}$ in its simplest form.
Answer
The HCF of 45 and 60 is 15, so $\frac{45}{60} = \frac{3}{4}$ .
Write $0.65$ as a fraction in its simplest form.
Answer
$0.65 = \frac{65}{100} = \frac{13}{20}$ , dividing by the HCF, 5.
Without a calculator, write $\frac{3}{16}$ as a decimal and as a percentage.
Answer
$\frac{1}{8} = 0.125$ , so $\frac{1}{16} = 0.0625$ (half of it). Then $\frac{3}{16} = 3 \times 0.0625 = 0.1875 = 18.75 %$ .
Write $2.5 %$ as a decimal and as a fraction in its simplest form.
Answer
$2.5 \div 100 = 0.025$ . As a fraction, $0.025 = \frac{25}{1000} = \frac{1}{40}$ .
Write $1.35$ as a percentage, and as a mixed number in its simplest form.
Answer
$1.35 \times 100 = 135 %$ . $1.35 = 1 \frac{35}{100} = 1 \frac{7}{20}$ .
Which one of these is not equal to the other three: $\frac{18}{40}$ , $0.45$ , $45 %$ , $\frac{9}{25}$ ?
Answer
$\frac{9}{25} = \frac{36}{100} = 0.36$ . The other three are all $0.45$ , because $\frac{18}{40} = \frac{9}{20} = \frac{45}{100}$ .
In a group of 32 students, 20 have a pet. Write the fraction of the group who have a pet in its simplest form, then write it as a percentage.
Answer
$\frac{20}{32} = \frac{5}{8}$ , dividing by the HCF, 4. Then $\frac{5}{8} = 0.625 = 62.5 %$ .

Exam questions

Try each part first. Open the mark scheme after you have written your answer.

Question 1.

(a) Write $\frac{29}{6}$ as a mixed number. 1 mark No calculator

Mark scheme

B1 4 5/6 cao

(b) Write $3 \frac{3}{7}$ as an improper fraction. 1 mark No calculator

Mark scheme

B1 24/7 cao

(c) Write $\frac{42}{56}$ as a fraction in its simplest form. 2 marks No calculator

Mark scheme

M1 a correct partial cancel, e.g. 21/28 or 6/8, or HCF 14 seen
A1 3/4 cao

Question 2.

(a) Write $\frac{13}{40}$ as a decimal. 2 marks No calculator

Mark scheme

M1 325/1000 or 32.5/100 seen, or a correct short division of 13 by 40 started, e.g. 0.3 with remainder 10
A1 0.325 cao

(b) Write $0.28$ as a fraction in its simplest form. 2 marks No calculator

Mark scheme

M1 28/100 or 14/50 soi (implied by a correct answer)
A1 7/25 cao

(c) Write $3 %$ as a decimal. 1 mark No calculator

Mark scheme

B1 0.03 cao

(d) Write $2 \frac{3}{5}$ as a percentage. 1 mark No calculator

Mark scheme

B1 260 [%] cao

Question 3. A club has $40$ members. $14$ of the members are under $16$ years old.

(a) Write the fraction of the members who are under $16$ . Give your answer in its simplest form. 2 marks No calculator

Mark scheme

M1 14/40 soi (implied by a correct answer)
A1 7/20 cao

(b) Write your answer to part (a) as a percentage. 1 mark No calculator

Mark scheme

B1 35 [%]; FT their fraction from (a) correctly converted to a percentage

(c) Jo says, ' $\frac{1}{3}$ of the members is the same as $30 %$ of the members.' Explain why Jo is wrong. 2 marks No calculator

Mark scheme

B1 1/3 = 0.333… (or 0.3̇) or 33.3% (or 33⅓%) seen, or 30% = 3/10 (or 0.3) seen, or 1/3 of 40 = 13.3… and 30% of 40 = 12 seen
B1 dep on the first B1: a correct comparison using the values, e.g. 33⅓% is more than 30%, so they are not the same oe

Question 4. Here are five numbers: $\frac{11}{20}$ , $0.16$ , $55 %$ , $\frac{4}{25}$ , $5.5 %$ .

(a) Write down the two pairs of numbers that are equal. 2 marks

Mark scheme

B1 11/20 and 55% paired
B1 0.16 and 4/25 paired

(b) Write $\frac{17}{24}$ as a percentage. Give your answer correct to $3$ significant figures. 2 marks

Mark scheme

M1 17 ÷ 24 [× 100] soi, e.g. 0.7083… or 70.83… seen
A1 70.8 cao

How this is examined

Turning a decimal or a percentage into a fraction in its simplest form: A 2-mark Write $0.36$ (or $36 %$ ) as a fraction in its simplest form, answered without a calculator The unsimplified fraction over 10, 100 or 1000, such as $\frac{36}{100}$ , shows the method. The answer mark needs the fraction fully cancelled, $\frac{9}{25}$ , with no common factor left except 1. A fraction that still cancels, or a decimal inside the fraction, is not the final answer.
Turning a fraction into a decimal or a percentage by hand: A 1- or 2-mark Write $\frac{9}{40}$ as a decimal (or as a percentage), with no calculator allowed, as on Paper 2 Working that shows the method: an equivalent fraction over 100 or 1000 ( $\frac{225}{1000}$ ), or the short division. Then the exact decimal, $0.225$ , or the percentage, $22.5 %$ . For a recurring decimal, the exact value with dot notation, or the value rounded to the accuracy the question asks for.
Explaining why two numbers in different forms are not equal: A short Explain, such as why the claim ' $\frac{2}{3}$ is the same as $66 %$ ' is wrong Both numbers changed into the same form with the values written down, e.g. $\frac{2}{3} = 66. \dot{6} %$ (or $66.7 %$ to 3 significant figures) and $66 %$ , then a sentence that compares them: $66. \dot{6} %$ is more than $66 %$ . 'They are different' with no values written does not explain anything.
Picking out the equal values from a mixed list of fractions, decimals and percentages: A list of five or six numbers with a Write down part asking for the values equal to a given number, or for the pairs that match Exactly the right values and nothing extra, because an extra value is a wrong answer as well. Convert every item to a decimal in your working first, so each choice is checked and not guessed.
Changing between mixed numbers and improper fractions: A 1-mark Write $\frac{23}{6}$ as a mixed number, or the reverse, or the same change made as one step inside a longer fraction calculation To a mixed number: the whole number is how many times the denominator goes in, the remainder goes over the same denominator, and the fraction part is in its simplest form ( $3 \frac{5}{6}$ ). To an improper fraction: whole number × denominator + numerator, over the same denominator ( $2 \frac{4}{7} = \frac{18}{7}$ ).

Words to know

Numerator: The top number of a fraction: how many of the equal parts you have.
Denominator: The bottom number of a fraction: how many equal parts make one whole.
Proper fraction: A fraction whose numerator is smaller than its denominator, such as $\frac{3}{8}$ . Its value is less than 1.
Improper fraction: A fraction whose numerator is equal to or bigger than its denominator, such as $\frac{11}{4}$ . Its value is 1 or more.
Mixed number: A whole number and a proper fraction written together, such as $2 \frac{3}{4}$ , meaning $2 + \frac{3}{4}$ .
Equivalent fractions: Fractions with the same value, made by multiplying or dividing the numerator and denominator by the same number, e.g. $\frac{3}{4} = \frac{15}{20}$ .
Simplest form: A fraction whose numerator and denominator have no common factor except 1, e.g. $\frac{3}{4}$ but not $\frac{9}{12}$ .
Terminating decimal: A decimal that stops, such as $0.375$ .
Recurring decimal: A decimal whose digits repeat for ever, written with dots over the repeating digits, e.g. $0. \dot{6} = 0.666 \dots$ and $0. \dot{4} \dot{5} = 0.4545 \dots$
Percentage: A number out of 100, written with the % sign: $37 % = \frac{37}{100} = 0.37$ .

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