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Basic Fractions

What is a fraction?

Definition

A fraction represents part of a whole. It is written using two integers:

  • Numerator – the number of parts you have
  • Denominator – the total number of equal parts the whole is divided into

Example: \( \frac{3}{4} \) means “three out of four equal parts”.

The top number (numerator) is divided (÷) by the bottom number (denominator).

\[ \frac{Numerator}{Denominator} \]

Example

What fraction of these counters is yellow?

Counters showing yellow and non-yellow pieces
One yellow counter out of eight

One eighth — or one in eight — of these counters is yellow.

Common fractions are written in the form:

\[ \frac{a}{b} \]
Fraction bar showing numerator and denominator

Types of fractions

Mixed number

A whole number combined with a proper fraction.

Example: \( 2\frac{1}{3} \)

Unit fraction

Numerator = 1.

Example: \( \frac{1}{8} \)

Equivalent fractions

Different fractions that represent the same value.

Example: \( \frac{1}{2} = \frac{2}{4} = \frac{50}{100} \)

Equivalent Fractions

Simplified fraction

A fraction where numerator and denominator have no common factors except 1.

Example: \( \frac{12}{18} = \frac{2}{3} \)

Proper Fractions

A proper fraction has a smaller numerator than denominator.

Numerator < denominator.

This represents a value less than 1.

Example of a proper fraction
\[ \frac{3}{4} \] \[ \frac{32}{37} \]

Improper Fractions

An improper fraction has a larger numerator than denominator.

Numerator ≥ denominator.

This represents a value equal to or greater than 1.

Example of an improper fraction
\[ \frac{13}{4} \] \[ \frac{39}{37} \]

Simplifying Fractions

To simplify (or cancel) a fraction, look for a whole number which divides exactly into both the top number (numerator) and the bottom number (denominator).

Keep repeating until you cannot continue — remembering that even numbers can always be divided by 2, and numbers ending in 5 or 0 can be divided by 5.

Examples

\[ \frac{12}{18} \] \[ = \frac{6}{9} \] \[ = \frac{2}{3} \]
\[ \frac{45}{60} \] \[ = \frac{15}{20} \] \[ = \frac{3}{4} \]
\[ \frac{84}{96} \] \[ = \frac{42}{48} \] \[ = \frac{21}{24} \] \[ = \frac{7}{8} \]
\[ \frac{72}{90} \] \[ = \frac{36}{45} \] \[ = \frac{12}{15} \] \[ = \frac{4}{5} \]

What if you don’t know your tables?

Try dividing top and bottom by 2, 3, 5, or 10.

Prime Factor Method

(This method uses prime factors.)

  • Divide by each prime number in turn until you cannot divide exactly.
  • Write the number as the product of its prime factors.
  • Cancel out matching factors.
  • Multiply what remains.
Example

Simplify \( \frac{36}{54} \)

\[ 36 = 2 \times 2 \times 3 \times 3 \] \[ 54 = 2 \times 3 \times 3 \times 3 \] \[ \frac{36}{54} = \frac{2 \times 2 \times 3 \times 3}{2 \times 3 \times 3 \times 3} \] \[ = \frac{2}{3} \]

Converting Improper Fractions to Mixed Numbers

Divide the numerator (top) by the denominator (bottom). Write the quotient, then put the remainder over the denominator. Simplify if possible.

Examples

\[ \frac{13}{4} \] \[ 13 \div 4 = 3 \text{ remainder } 1 \] \[ = 3 \frac{1}{4} \]
\[ \frac{14}{5} \] \[ 14 \div 5 = 2 \text{ remainder } 4 \] \[ = 2 \frac{4}{5} \]
\[ \frac{29}{8} \] \[ 29 \div 8 = 3 \text{ remainder } 5 \] \[ = 3 \frac{5}{8} \]
\[ \frac{16}{3} \] \[ 16 \div 3 = 5 \text{ remainder } 1 \] \[ = 5 \frac{1}{3} \]

Converting Mixed Numbers to Improper Fractions

Multiply the whole number by the denominator (bottom). Add the numerator (top). Write this answer over the denominator.

Examples

\[ 3 \frac{2}{5} \] \[ 3 \times 5 = 15 \] \[ 15 + 2 = 17 \] \[ = \frac{17}{5} \]
\[ 4 \frac{3}{7} \] \[ 4 \times 7 = 28 \] \[ 28 + 3 = 31 \] \[ = \frac{31}{7} \]

\( 2\tfrac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3} \)

\( 4\tfrac{5}{6} = \frac{4 \times 6 + 5}{6} = \frac{29}{6} \)

Calculating Fractions of Amounts

To calculate \( \frac{a}{b} \text{ of } C \)
Divide C by the denominator, then multiply by the numerator.
\( \frac{C}{b} \times a \)
" Divide by the bottom, times by the top"

Examples

\[ \frac{2}{5} \text{ of } 30 \] \[ = 30 \div 5 \] \[ = 6 \] \[ 6 \times 2 = 12 \]

Find \( \frac{2}{5} \) of £45:

\( \frac{2}{5} \times 45 = 2 \times \frac{45}{5} = 2 \times 9 = 18 \)

So the answer is £18.

Find \( \frac{3}{8} \) of 648:

\( \frac{3}{8} \times 648 = 3 \times \frac{648}{8} = 3 \times 81 = 243 \)

So the answer is 243.

Find \( \frac{1}{7} \) of 649:

\( \frac{1}{7} \times 649 = 649 \div 7 = 92.714285\ldots \)

So the answer is \( \displaystyle 92\tfrac{5}{7} \).

Find \( \frac{3}{5} \) of 650:

\( \frac{3}{5} \times 650 = 3 \times \frac{650}{5} = 3 \times 130 = 390 \)

So the answer is 390.

Find \( \frac{2}{3} \) of 650:

\( \frac{2}{3} \times 650 = 2 \times \frac{650}{3} = 2 \times 216.\overline{6} = 433.\overline{3} \)

So the answer is \( \displaystyle 433\tfrac{1}{3} \).

Learn

Fraction learning poster
Adding / Subtracting Fractions

Multiplying Fractions

Dividing Fractions

Algebraic Fractions

Fractions Drill Questions

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