Fractions, decimals & percentages

Start here: three ways of saying the same thing

$\dfrac{1}{2}$ , $0.5$ , and $50\%$ are not three different numbers. They are three different ways of writing the same number.

As a fraction: $\dfrac{1}{2}$ .
As a decimal: $0.5$ .
As a percentage: $50\%$ (“half of every hundred”).

This topic is about moving smoothly between those three ways of writing. Once you can, you can pick whichever one makes a problem easiest.

What you will learn

write a fraction in its simplest form and recognise equivalent fractions,
place positive and negative rational numbers on a number line,
convert freely between a fraction, a decimal, and a percentage,
round decimals to a given accuracy,
add, subtract, multiply and divide positive fractions and decimals,
find a percentage of a quantity and do percentage increases and decreases.

Worked example 0 Real-world example: supermarket price comparison

A $400$ g box of cereal costs $5.00. A $750$ g box costs $8.25. Which is better value?

Small box: $5.00 $\div 400 = 0.0125$ per gram $= 1.25$ per $100$ g.
Large box: $8.25 $\div 750 = 0.011$ per gram $= 1.10$ per $100$ g.
The large box is cheaper per $100$ g — better value by $\dfrac{1.25 - 1.10}{1.25} = 12\%$ .

Key idea: converting to a common unit (price per 100 g) uses decimals; expressing the saving uses percentages. Each representation does its job.

0. Rational numbers on a number line

Positive and negative fractions and decimals are all rational numbers: they can be written as a ratio of integers. They take their place on the number line just as integers do.

-2,\; -\tfrac{3}{2},\; -1,\; -0.25,\; 0,\; \tfrac{1}{4},\; \tfrac{3}{4},\; 1.5,\; 2.

To place a fraction, divide the space between consecutive integers into equal parts given by the denominator.

1. Equivalent fractions and simplest form

Two fractions are equivalent when they represent the same amount. You can make an equivalent fraction by multiplying (or dividing) the top and bottom by the same non-zero number.

\frac{3}{4} \;=\; \frac{3 \times 2}{4 \times 2} \;=\; \frac{6}{8}, \qquad \frac{20}{30} \;=\; \frac{20 \div 10}{30 \div 10} \;=\; \frac{2}{3}.

A fraction is in simplest form when the top and bottom share no common factor other than $1$ .

Simplify a fraction

Divide the numerator and denominator by their greatest common factor (GCF).

\frac{a}{b} \;=\; \frac{a \div g}{b \div g}, \quad \text{where } g = \gcd(a,b).

Worked example E Very easy: simplify 6/8

$6$ and $8$ share the common factor $2$ . Divide top and bottom by $2$ :

\dfrac{6}{8} = \dfrac{6 \div 2}{8 \div 2} = \dfrac{3}{4}.

That’s it — keep dividing until there’s no common factor left.

Worked example 1 Simplify 24/36

$\gcd(24, 36) = 12$ . Divide top and bottom by $12$ :

\frac{24}{36} \;=\; \frac{24 \div 12}{36 \div 12} \;=\; \frac{2}{3}.

2. Moving between fractions, decimals and percentages

Think of them as three translations of the same value. Every fraction has a decimal; every decimal has a percentage; every percentage has a fraction.

Conversion rules

Fraction -> decimal

Divide the numerator by the denominator: $\displaystyle \frac{3}{8} = 3 \div 8 = 0.375$ .

Decimal -> percentage

Multiply by $100$ (move the decimal point two places to the right): $0.375 \to 37.5\%$ .

Percentage -> decimal

Divide by $100$ (move the decimal point two places to the left): $65\% \to 0.65$ .

Percentage -> fraction

Place the percentage over $100$ and simplify: $\displaystyle 65\% = \frac{65}{100} = \frac{13}{20}$ .

Key values worth memorising

$\displaystyle \frac{1}{2} = 0.5 = 50\%$ , $\quad \frac{1}{4} = 0.25 = 25\%$ , $\quad \frac{3}{4} = 0.75 = 75\%$ , $\quad \frac{1}{5} = 0.2 = 20\%$ , $\quad \frac{1}{10} = 0.1 = 10\%$ .

Worked example 2 Convert and compare

Place $\dfrac{3}{5}$ , $0.58$ , and $62\%$ in order from smallest to largest.

Convert everything to decimals: $\dfrac{3}{5} = 0.60$ ; $0.58$ stays; $62\% = 0.62$ .
Compare: $0.58 < 0.60 < 0.62$ .

0.58 \;<\; \frac{3}{5} \;<\; 62\%.

2a. Rounding decimals

Rounding a decimal to a certain place value:

Look at the digit immediately after the place you are keeping.
If it is $5$ or more, round up (add $1$ to the kept digit).
If it is less than $5$ , round down (leave the kept digit alone).

Worked example R Round 3.7486

To the nearest whole number: look at the tenths digit ( $7$ ). It is $\geq 5$ , so round up: $4$ .
To $1$ decimal place: look at the hundredths digit ( $4$ ). It is $< 5$ , so round down: $3.7$ .
To $2$ decimal places: look at the thousandths digit ( $8$ ). It is $\geq 5$ , so round up: $3.75$ .

3. Adding and subtracting fractions

To add or subtract, the fractions must have a common denominator.

Adding fractions

\frac{a}{b} + \frac{c}{d} \;=\; \frac{ad}{bd} + \frac{cb}{bd} \;=\; \frac{ad + cb}{bd}.

Worked example 3 Adding unlike fractions

Evaluate $\;\dfrac{2}{3} + \dfrac{1}{4}$ .

A common denominator of $3$ and $4$ is $12$ . Rewrite: $\dfrac{2}{3} = \dfrac{8}{12}$ , $\dfrac{1}{4} = \dfrac{3}{12}$ .
Add: $\dfrac{8}{12} + \dfrac{3}{12} = \dfrac{11}{12}$ .

\frac{2}{3} + \frac{1}{4} = \frac{11}{12}.

4. Multiplying and dividing fractions

Multiplying and dividing

Multiplying

\frac{a}{b} \times \frac{c}{d} \;=\; \frac{a \times c}{b \times d}.

Simplify before multiplying if you can - it is easier than simplifying a big number at the end.

Dividing

\frac{a}{b} \div \frac{c}{d} \;=\; \frac{a}{b} \times \frac{d}{c}.

“Keep, change, flip”: keep the first fraction, change division to multiplication, flip the second.

Worked example 4 Dividing fractions

Evaluate $\;\dfrac{3}{4} \div \dfrac{2}{5}$ .

Keep, change, flip:

\frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8} = 1\frac{7}{8}.

5. Percentages in context

Percentages of an amount

Percentage of a quantity

p\% \text{ of } A \;=\; \frac{p}{100} \times A.

Percentage increase

\text{new amount} \;=\; A + \frac{p}{100} \times A \;=\; A \times \left(1 + \frac{p}{100}\right).

Percentage decrease

\text{new amount} \;=\; A - \frac{p}{100} \times A \;=\; A \times \left(1 - \frac{p}{100}\right).

Worked example 5 Percentage of a price

A $48 pair of shoes is on sale with $25\%$ off. What is the sale price?

Find the discount: $25\%$ of $48$ is $\dfrac{25}{100} \times 48 = 12$ , so $12 off.
Subtract the discount: $48 - 12 = 36$ , so the sale price is $36.

Or in one step: $48 \times (1 - 0.25) = 48 \times 0.75 = 36$ , giving $36.

Practice

Fluency

Tier 1: basic skills

Simplify $\dfrac{18}{24}$ .
Simplify $\dfrac{36}{60}$ .
Write $\dfrac{3}{8}$ as a decimal.
Write $\dfrac{7}{20}$ as a decimal.
Write $0.45$ as a simplified fraction.
Write $1.25$ as a mixed number.
Convert $0.6$ to a percentage.
Convert $82\%$ to a decimal.
Convert $35\%$ to a simplified fraction.
Which is bigger: $\dfrac{3}{5}$ or $\dfrac{7}{10}$ ?
Evaluate $\dfrac{1}{2} + \dfrac{1}{3}$ .
Evaluate $\dfrac{5}{6} - \dfrac{1}{4}$ .
Evaluate $\dfrac{2}{3} \times \dfrac{9}{10}$ .
Evaluate $\dfrac{4}{5} \div \dfrac{2}{3}$ .
Find $20\%$ of $75.
Find $15\%$ of $60$ .
Find $7\%$ of $200.
Increase $80$ by $25\%$ .
Decrease $120 by $10\%$ .
Write $\dfrac{5}{8}$ as a percentage.

Fluency

Rounding and number line

Round $4.762$ to the nearest whole number.
Round $12.38$ to $1$ decimal place.
Round $0.0458$ to $2$ decimal places.
Round $18.739 to the nearest cent.
Which is smaller: $-\dfrac{2}{3}$ or $-\dfrac{1}{3}$ ?
Place these on a number line (in order): $-1.5,\ -\dfrac{3}{4},\ 0,\ 0.25,\ \dfrac{3}{2}$ .
Find two rational numbers between $-0.5$ and $0$ .

Reasoning

Tier 2: mixed practice

Order from smallest to largest: $\;0.7,\ \dfrac{2}{3},\ 68\%,\ \dfrac{7}{10}$ .
Evaluate $\dfrac{2}{3} + \dfrac{3}{4} - \dfrac{1}{2}$ .
Evaluate $\left(\dfrac{1}{2}\right)^2 + \dfrac{1}{4}$ .
A recipe uses $\dfrac{3}{4}$ cup of sugar. You want to make $\dfrac{2}{3}$ of the recipe. How much sugar do you need?
Fill in the missing number: $\dfrac{\square}{12} = \dfrac{3}{4}$ .
What percentage is $18$ out of $40$ ?
What percentage is $27$ out of $60$ ?
A jacket costs $85 and is reduced by $20\%$ . What is the new price?
A bike costs $320 and its price rises by $5\%$ . What is the new price?
A number increased by $40\%$ gives $84$ . What was the original number?
Work out $\dfrac{3}{4}$ of $\dfrac{2}{5}$ of $120$ .
Evaluate $1 - \dfrac{3}{8} - \dfrac{1}{4}$ .

Reasoning

Tier 3: explain and spot the mistake

Sam says " $\dfrac{1}{2} + \dfrac{1}{3} = \dfrac{2}{5}$ " because he added the tops and the bottoms. Is Sam correct? If not, what is the correct answer and what mistake has Sam made?
Explain why dividing by $\dfrac{1}{2}$ is the same as multiplying by $2$ .
A shop advertises ” $50\%$ off then another $20\%$ off”. Is this the same as $70\%$ off? Explain with a worked example using $100.
Without calculating exactly, decide whether $\dfrac{7}{13}$ is greater than $\dfrac{1}{2}$ . Explain your reasoning.
Is $0.3$ equal to $\dfrac{3}{10}$ or $\dfrac{1}{3}$ ? Explain the difference.

Problem solving

Tier 4: real-world problems

Zara’s phone bill is $65 per month. The company raises prices by $8\%$ . What will her new bill be?
A pizza is cut into $8$ equal slices. Tom eats $3$ slices, Mia eats $2$ slices. What fraction of the pizza is left?
In a class of $30$ students, $18$ walk to school. What percentage walk to school? What percentage do not?
A $240 pair of headphones is on sale for $180. What is the percentage discount?
Mia saves $\dfrac{1}{5}$ of her $25 pocket money each week. How much has she saved after $8$ weeks?
A water tank is $\dfrac{3}{4}$ full. $120$ litres are used, leaving the tank $\dfrac{1}{2}$ full. What is the capacity of the tank?
A shirt’s price was marked up by $20\%$ to $42. What was the original price?
Last year a school had $750$ students. This year enrolment has risen by $12\%$ . How many students are enrolled now?