Fractions

Q: How do you compare two fractions?

Give them the same denominator and compare the numerators. \tfrac{3}{4} = \tfrac{9}{12} is bigger than \tfrac{2}{3} = \tfrac{8}{12}.

A fraction shows part of a whole: the top number (numerator) says how many parts you have and the bottom number (denominator) says how many equal parts the whole is split into. To add or subtract, the parts must be the same size.

Formulas

Same denominator

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

Different denominators (use a common denominator, usually the LCM)

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

Simplifying

\frac{a}{b} = \frac{a \div k}{b \div k} (k = HCF of a and b)

Worked examples

Example 1. Add

\frac{3}{8} + \frac{1}{4}

Write $\frac{1}{4}$ with denominator 8: $\frac{2}{8}$ .
$\frac{3}{8} + \frac{2}{8} = \frac{5}{8}$ .

Answer:

\frac{5}{8}

Example 2. Subtract

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

LCM of 6 and 4 is 12.
$\frac{10}{12} - \frac{3}{12} = \frac{7}{12}$ .

Answer:

\frac{7}{12}

Example 3. Add

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

Change to improper fractions: $\frac{7}{3} + \frac{7}{4}$ .
With denominator 12: $\frac{28}{12} + \frac{21}{12} = \frac{49}{12}$ .
$\frac{49}{12} = 4 \frac{1}{12}$ .

Answer:

4 \frac{1}{12}

Example 4. Simplify

\frac{36}{48}

The HCF of 36 and 48 is 12.
$\frac{36 \div 12}{48 \div 12} = \frac{3}{4}$ .

Answer:

\frac{3}{4}

Common mistakes

Adding the denominators as well: $\frac{1}{2} + \frac{1}{3}$ is not $\frac{2}{5}$ . Find a common denominator first.
Stopping before simplifying. Always check whether the numerator and denominator share a factor.
Adding whole numbers and fractions of mixed numbers inconsistently. Either change both to improper fractions, or add the whole parts and fraction parts separately.

Practice questions with answers

1. What fraction does 'two-sevenths' name?

A $\frac{7}{2}$
B $\frac{2}{7}$
C $\frac{7}{9}$
D $\frac{1}{7}$

Show answer

Answer: B)

\frac{2}{7}

'Two-sevenths' means 2 out of 7 equal parts — the numerator is 2 and the denominator is 7.

2. Subtract:

\frac{4}{7} - \frac{1}{7}

A $\frac{3}{7}$
B $\frac{3}{14}$
C $\frac{5}{7}$
D $\frac{3}{0}$

Show answer

Answer: A)

\frac{3}{7}

Like fractions: 4 - 1 = 3, denominator stays 7. So

\frac{4}{7} - \frac{1}{7} = \frac{3}{7}

3. Fill in the blank:

\frac{1}{3} = \frac{?}{9}

Show answer

Answer: A) 3

Denominator

3 \times 3 = 9

, so numerator

1 \times 3 = 3

. Hence

\frac{1}{3} = \frac{3}{9}

4. Which of these is an improper fraction?

A $\frac{2}{7}$
B $\frac{4}{9}$
C $\frac{8}{5}$
D $\frac{1}{2}$

Show answer

Answer: C)

\frac{8}{5}

An improper fraction has numerator greater than or equal to denominator.

\frac{8}{5}

has 8 > 5.

5. Assertion (A):

\frac{1}{3} + \frac{1}{6} = \frac{2}{9}

. Reason (R): Unlike fractions must first be rewritten with a common denominator before they can be added. Which is correct?

ABoth A and R are true, and R correctly explains A
BA is false but R is true
CA is true but R is false
DBoth A and R are false

Show answer

Answer: B) A is false but R is true

R is a correct rule. Applying it,

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

, so

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

— not

\frac{2}{9}

. The

\frac{2}{9}

in A came from adding the numerators and the denominators separately. So A is false while R is true.

6. A relay race in Chennai has three runners. The first runs

\frac{3}{8}

of the route,
the second runs

\frac{1}{4}

of the route, and the third runs the rest.
What fraction of the route does the third runner cover?

A $\frac{3}{8}$
B $\frac{5}{8}$
C $\frac{1}{2}$
D $\frac{1}{8}$

Show answer

Answer: A)

\frac{3}{8}

First + second =

\frac{3}{8} + \frac{1}{4} = \frac{3}{8} + \frac{2}{8} = \frac{5}{8}

. Third =

1 - \frac{5}{8} = \frac{3}{8}

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Frequently asked questions

How do you add fractions with different denominators?

Rewrite them with a common denominator (the LCM of the denominators is the neatest), add the numerators and keep the denominator.

How do you simplify a fraction?

Divide the numerator and denominator by their highest common factor until no common factor is left.

How do you compare two fractions?

Give them the same denominator and compare the numerators. $\frac{3}{4} = \frac{9}{12}$ is bigger than $\frac{2}{3} = \frac{8}{12}$ .

What is the difference between a proper and an improper fraction?

In a proper fraction the numerator is smaller than the denominator $(\frac{3}{5})$ . In an improper fraction it is equal or larger $(\frac{7}{4})$ , which can be written as a mixed number $(1 \frac{3}{4})$ .

Fractions

Formulas

Worked examples

Common mistakes

Practice questions with answers

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Frequently asked questions

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