1.4 Fractions

Quick Notes & Formulae

Essential concepts for solving fraction problems.

Equivalent Fractions

Multiply or divide both numerator and denominator by the same non-zero number.

\frac{a}{b} = \frac{a \times c}{b \times c} = \frac{a \div c}{b \div c}

Mixed ↔ Improper

To convert:

A \frac{b}{c} = \frac{A \times c + b}{c}

. Reverse: divide numerator by denominator for quotient and remainder.

Adding/Subtracting

Find LCM of denominators. Convert:

\frac{a}{b} + \frac{c}{d} = \frac{a d + b c}{b d}

. Simplify result.

Multiplying

Multiply numerators and denominators:

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

. Simplify before multiplying if possible.

Dividing

Multiply by reciprocal:

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

Past Paper Style Questions

Question 1: Equivalent Fractions + Simplifying

(a) Write an equivalent fraction to

\frac{3}{8}

with denominator 40. [1]

(b) Simplify

\frac{42}{56}

. [2]

\frac{7}{12}

as a fraction with denominator 60. [1]

(d) Explain why

\frac{42}{56}

and

\frac{3}{4}

are equivalent. [2]

(e) Order

\frac{3}{8}, \frac{7}{12}, \frac{3}{4}

in ascending order. [2]

Question 2: Improper ↔ Mixed Numbers

(a) Convert

\frac{29}{6}

to a mixed number. [2]

(b) Convert

4 \frac{5}{8}

to an improper fraction. [1]

2 \frac{3}{5}

to an improper fraction. [1]

(d) Arrange

\frac{29}{6}, 4 \frac{5}{8}, 2 \frac{3}{5}

in ascending order. [3]

(e) Write the difference between the largest and smallest as a mixed number. [3]

Question 3: Addition and Subtraction of Fractions

(a) Work out

\frac{5}{12} + \frac{7}{18}

. [2]

(b) Work out

\frac{3}{10} - \frac{7}{25}

. [2]

2 \frac{1}{6} + 1 \frac{5}{12}

. [3]

(d) Work out

4 \frac{3}{5} - 2 \frac{7}{10}

. [3]

(e) Give each answer in its simplest form. [2]

Question 4: Multiplication and Division of Fractions

(a) Work out

\frac{7}{9} \times \frac{3}{14}

. [2]

(b) Work out

\frac{5}{6} \div \frac{10}{9}

. [2]

2 \frac{1}{4} \times \frac{8}{15}

. [3]

(d) Work out

3 \frac{1}{3} \div 1 \frac{1}{2}

. [3]

(e) Give each answer in its simplest form. [2]

Question 5: Reverse - Find the Missing Fraction

(a) Find

x

\frac{3}{5} + x = \frac{7}{10}

. [2]

(b) Find

y

y - \frac{2}{3} = \frac{1}{6}

. [2]

k

\frac{4}{7} \times k = \frac{2}{7}

. [2]

(d) Find

m

m \div \frac{3}{8} = \frac{4}{9}

. [2]

(e) Write

x, y, k, m

in ascending order. [2]

Question 6: Mixed Numbers in a Real Context (Recipe)

A recipe uses

1 \frac{1}{4}

cups of flour for one batch.

(a) Write

1 \frac{1}{4}

as an improper fraction. [1]

(b) How many cups of flour are needed for 3 batches? Give your answer as a mixed number. [2]

5 \frac{1}{2}

cups of flour. How much is left after making 3 batches? [3]

(d) Is there enough flour to make one more batch? Show working. [2]

(e) If not, how much more flour is needed? [2]

Question 7: Fractions + Simplifying Using Common Factors

(a) Simplify

\frac{84}{126}

. [2]

(b) Simplify

\frac{72}{90}

. [2]

\frac{84}{126} + \frac{72}{90}

and simplify fully. [3]

(d) Write your answer to (c) as a mixed number. [1]

(e) Explain why simplifying before adding can reduce errors. [2]

Question 8: Reverse - Build Original from Simplified Form

A fraction

\frac{p}{q}

simplifies to

\frac{3}{7}

Also,

p + q = 80

(a) Write

p = 3 k

and

q = 7 k

. [1]

(b) Form an equation in

k

. [1]

k

, hence find

p

and

q

. [3]

(d) Write

\frac{p}{q}

and simplify to check. [2]

(e) Find the value of

\frac{q}{p}

in simplest form. [1]

Question 9: Reverse - Mixed Number Equation

A number

x

is a mixed number.

x - 1 \frac{3}{4} = 2 \frac{5}{8}

(a) Convert all mixed numbers to improper fractions. [2]

(b) Solve for

x

as an improper fraction. [2]

x

to a mixed number in simplest form. [2]

(d) Check your answer by substitution. [2]

(e) Find

\frac{1}{x}

in simplest form. [2]

Question 10: Challenge - Fraction Inequality + Interpretation

Let

x

be a positive integer and

A = \frac{x + 3}{x + 5}

(a) Show that

A < 1

for all positive integers

x

. [2]

(b) Find

A

when

x = 5

. [1]

x

such that

A > \frac{3}{4}

. [3]

(d) Explain why

A

gets closer to 1 as

x

increases. [2]

(e) State whether

A

can ever be equal to 1. Give a reason. [2]

Question 11: Equivalent Fractions + Ordering

(a) Write an equivalent fraction to

\frac{9}{20}

with denominator 100. [1]

(b) Simplify

\frac{63}{84}

. [2]

\frac{5}{6}

as a fraction with denominator 36. [1]

(d) Arrange

\frac{9}{20}, \frac{63}{84}, \frac{5}{6}

in ascending order. [2]

(e) Write the difference between the largest and smallest as a fraction in simplest form. [2]

Question 12: Improper ↔ Mixed + Operations

(a) Convert

\frac{41}{8}

to a mixed number. [1]

(b) Convert

5 \frac{3}{10}

to an improper fraction. [1]

\frac{41}{8} - 5 \frac{3}{10}

. Give your answer as a mixed number. [3]

(d) Work out

2 \frac{1}{4} + \frac{41}{8}

. Give your answer as an improper fraction. [2]

(e) Hence find the value of

(2 \frac{1}{4} + \frac{41}{8}) \div (\frac{41}{8} - 5 \frac{3}{10})

. [3]

Question 13: Add/Subtract Fractions (Exam Style)

(a) Work out

\frac{7}{12} + \frac{5}{18}

. [2]

(b) Work out

\frac{11}{15} - \frac{7}{20}

. [2]

3 \frac{5}{6} - 1 \frac{7}{12}

. [2]

(d) Work out

2 \frac{1}{3} + 4 \frac{5}{9}

. [2]

(e) Write all answers in their simplest form. [2]

Question 14: Multiply/Divide Fractions + Mixed Numbers

(a) Work out

\frac{15}{28} \times \frac{14}{25}

. [2]

(b) Work out

\frac{9}{10} \div \frac{3}{20}

. [2]

1 \frac{3}{5} \times 2 \frac{1}{4}

. [2]

(d) Work out

4 \frac{2}{3} \div 1 \frac{1}{6}

. [2]

(e) Give each answer as a mixed number in simplest form where appropriate. [2]

Question 15: Reverse - Missing Fraction in Equation

(a) Find

x

such that

\frac{5}{8} + x = \frac{11}{12}

. [2]

(b) Find

y

such that

y - \frac{3}{10} = \frac{7}{25}

. [2]

k

such that

\frac{7}{9} \times k = \frac{14}{27}

. [2]

(d) Find

m

such that

m \div \frac{5}{6} = 1 \frac{1}{4}

. [2]

(e) Arrange

x, y, k, m

in descending order. [2]

Question 16: Reverse - Reconstruct Numerator/Denominator

A fraction

\frac{p}{q}

simplifies to

\frac{4}{9}

Also,

p - q = 25

(a) Write

p = 4 k

and

q = 9 k

. [1]

(b) Form an equation in

k

. [1]

k

, hence find

p

and

q

. [3]

(d) Check by simplifying

\frac{p}{q}

. [2]

(e) Find

\frac{p + q}{p}

in simplest form. [1]

Question 17: Context - Measurement with Mixed Numbers

A ribbon is

7 \frac{1}{2}

m long.

It is cut into pieces each of length

\frac{3}{4}

(a) Convert

7 \frac{1}{2}

to an improper fraction. [1]

(b) Find the number of complete pieces that can be cut. [2]

(d) Write the leftover length as a fraction of the original ribbon length. [2]

(e) Give your answer to (d) as a percentage. [1]

Question 18: Combined Operations (One Expression)

Work out:

(\frac{5}{6} - \frac{1}{4}) \div (\frac{7}{9} \times \frac{3}{14})

(a) Work out

\frac{5}{6} - \frac{1}{4}

in simplest form. [2]

(b) Work out

\frac{7}{9} \times \frac{3}{14}

in simplest form. [2]

(d) Write the final answer as an improper fraction in simplest form. [1]

(e) Convert your final answer to a mixed number. [1]

Question 19: Reverse - Mixed Number Target

A mixed number

x

satisfies:

\frac{x}{2} + 1 \frac{1}{3} = 3 \frac{5}{6}

(a) Convert the mixed numbers to improper fractions. [2]

(b) Write an equation in

x

using fractions. [2]

x

as an improper fraction. [2]

(d) Write

x

as a mixed number in simplest form. [1]

(e) Find

x \div \frac{7}{12}

. [2]

Question 20: Challenge - Fraction Inequality

Let

n

be a positive integer and

B = \frac{2 n + 1}{3 n + 4}

(a) Show that

B < \frac{2}{3}

for all positive integers

n

. [2]

(b) Find

B

when

n = 2

. [1]

n

such that

B > \frac{1}{2}

. [3]

(d) Explain why

B

decreases as

n

increases. [2]

(e) State whether

B

can ever be equal to

\frac{2}{3}

. Give a reason. [2]