Fractions Toolkit

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Notes

Basic Fractions

A fraction is written as $\frac{a}{b}$ where $a$ (numerator) and $b$ (denominator) are integers.
The denominator shows how many parts each whole is split into; the numerator shows how many parts you have.
A fraction is less than 1 if numerator < denominator; greater than 1 if numerator > denominator.
To find a fraction of an amount: divide by denominator, multiply by numerator (e.g., $\frac{2}{5}$ of $60: 60 \div 5 = 12, 12 \times 2 = 24)$ .
Equivalent fractions represent the same amount; multiply top and bottom by the same number to find them.
Simplify a fraction by dividing numerator and denominator by a common factor (e.g., $\frac{12}{18} = \frac{2}{3}$ ).

Mixed Numbers & Improper Fractions

A **mixed number** has an integer part and a fraction part (e.g., $3\frac{3}{4}$ ).
An **improper fraction** (top-heavy) has numerator ≥ denominator (e.g., $\frac{15}{4}$ ).
To convert a mixed number to an improper fraction: multiply the whole number by the denominator, add the numerator, write over the denominator.
To convert an improper fraction to a mixed number: divide numerator by denominator; the quotient is the whole number, remainder over denominator is the fraction.
Always convert mixed numbers to improper fractions before adding, subtracting, multiplying, or dividing.

Adding & Subtracting Fractions

Find the **lowest common denominator** (LCD) of the fractions.
Rewrite each fraction as an equivalent fraction with the LCD.
Add or subtract the numerators; keep the denominator the same.
Simplify the result by cancelling common factors.
If mixed numbers are involved, convert them to improper fractions first, then add/subtract, and convert back if needed.

Multiplying Fractions

Cancel any common factors between numerators and denominators before multiplying.
Multiply the numerators together and the denominators together.
Simplify the resulting fraction if possible.
If a mixed number is involved, convert it to an improper fraction first.
Example: $\frac{4}{15} \times \frac{25}{11} = \frac{4 \times 25}{15 \times 11} = \frac{100}{165} = \frac{20}{33}$ after cancelling.

Dividing Fractions

To divide, **flip the second fraction** (find its reciprocal) and change $\div to \times$ .
Then follow the multiplication steps: cancel, multiply numerators, multiply denominators.
If a mixed number is involved, convert it to an improper fraction first.
Example: $3\frac{1}{4} \div \frac{3}{8} = \frac{13}{4} \times \frac{8}{3} = \frac{13 \times 8}{4 \times 3} = \frac{104}{12} = \frac{26}{3} = 8\frac{2}{3}$ .

Working with Fractions of Amounts

To find a fraction of an amount, multiply the amount by the fraction.
Write the amount as a fraction over $1 (e.g., 60 =$ $\frac{60}{1}$ ).
Then multiply the fractions as usual.
Alternatively, divide by denominator then multiply by numerator.

Simplifying Fractions

A fraction is in its **simplest form** when numerator and denominator have no common factors other than 1.
Divide both numerator and denominator by their **highest common factor (HCF)**.
Your calculator can simplify fractions automatically, but show the factor you divided by if working is required.

Common Mistakes & Tips

When adding/subtracting, do **not** add denominators.
Always simplify final answers unless stated otherwise.
Convert mixed numbers to improper fractions before any operation.
Remember **"flip'n'times"** for division: flip the second fraction and multiply.

Fraction of an Amount (Bar Model)

Converting Mixed Number to Improper Fraction

Adding Fractions (Common Denominator)

Dividing Fractions (Flip'n'Times)

Practice questions

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1.What is a fraction?
Easy
- AA number written as a/b where a and b are integers
- BA number with a decimal point
- CA number less than 1
- DA number greater than 1
2.What is the numerator in the fraction $\frac{3}{5}$ ?
Easy
- A3
- B5
- C8
- D2
3.Which of the following is an improper fraction?
Easy
- A $\frac{7}{4}$
- B $\frac{3}{5}$
- C $\frac{2}{3}$
- D $\frac{1}{2}$
4.Convert $5 \frac{2}{7}$ into an improper fraction.
Easy
- A $\frac{37}{7}$
- B $\frac{35}{7}$
- C $\frac{12}{7}$
- D $\frac{33}{7}$
5.Convert $\frac{22}{3}$ into a mixed number.
Easy
- A $7 \frac{1}{3}$
- B $7 \frac{2}{3}$
- C $6 \frac{1}{3}$
- D $7 \frac{1}{2}$
6.Simplify $\frac{12}{18}$ to its lowest terms.
Easy
- A $\frac{2}{3}$
- B $\frac{3}{4}$
- C $\frac{1}{2}$
- D $\frac{4}{6}$
7.Work out $\frac{2}{5}$ of 60.
Easy
- A24
- B12
- C30
- D150
8.Which fraction is equivalent to 3.875?
Easy
- A $\frac{31}{8}$
- B $\frac{15}{4}$
- C $\frac{29}{8}$
- D $\frac{15}{8}$

Summary

## Fractions Toolkit - Quick Revision ### Basic Fractions - **Fraction**: numerator/denominator (e.g., 2/3). - **Finding a fraction of an amount**: Divide by denominator, multiply by numerator. E.g., 2/5 of 60 = 60÷5×2 = 24. - **Equivalent fractions**: Multiply top and bottom by same number. - **Simplifying**: Divide top and bottom by common factor. ### Mixed Numbers & Improper Fractions - **Mixed → Improper**: Multiply whole by denominator, add numerator. E.g., 5 2/7 = (5×7+2)/7 = 37/7. - **Improper → Mixed**: Divide numerator by denominator; whole part is quotient, remainder over denominator. E.g., 22/3 = 7 1/3. ### Adding & Subtracting Fractions - Find lowest common denominator. - Convert to equivalent fractions. - Add/subtract numerators only. - Simplify. - For mixed numbers, convert to improper first. ### Multiplying & Dividing Fractions - **Multiply**: Cancel common factors, then multiply numerators and denominators. - **Divide**: Flip the second fraction and multiply ("flip'n'times"). - For mixed numbers, convert to improper first. ### Key Tips - Use calculator to simplify, but show working if asked. - Always simplify final answer. - Convert answers to mixed numbers if required.

Key terms

Fraction: A number written as a/b where a and b are integers; a is numerator, b is denominator.
Equivalent fractions: Fractions that represent the same amount, obtained by multiplying numerator and denominator by the same number.
Simplifying (cancelling) a fraction: Dividing numerator and denominator by a common factor to get the smallest possible integer values.
Mixed number: A number consisting of an integer part and a fraction part, e.g., 3 3/4.
Improper fraction: A fraction where the numerator is greater than the denominator, e.g., 15/4.
Lowest common denominator: The smallest number that can be divided by all denominators in a set of fractions.
Reciprocal: The flipped version of a fraction; used when dividing fractions.

Slides

Basic Fractions

A fraction is written as a/b where a is the numerator and b is the denominator.
The denominator tells how many parts each whole is split into; the numerator tells how many parts you have.
A fraction is less than 1 if numerator < denominator, greater than 1 if numerator > denominator.

What does the denominator in a fraction represent?

The number of parts you have
The total number of parts each whole is divided into
The value of the fraction
The number of wholes

Finding a Fraction of an Amount

Method 1: Divide by the denominator, then multiply by the numerator. $E.g., \frac{2}{5}$ of $60: 60 \div 5 = 12, 12 \times 2 = 24$ .
Method 2: Convert the fraction to a decimal and multiply. $E.g., \frac{9}{10}$ of $15 = 0.9 \times 15 = 13.5$ .
Method 3: Write the amount as a fraction $(e.g., 8 = \frac{8}{1})$ and multiply the two fractions.

To find $\frac{3}{4}$ of 20, first divide 20 $by ____$ then multiply $by ____$ .

Equivalent Fractions & Simplifying

Equivalent fractions represent the same amount. Multiply top and bottom by the same number to get an equivalent fraction.
Simplifying (cancelling) means dividing numerator and denominator by a common factor to get the smallest possible integers.
Example: $\frac{12}{18} = (12\div 6)/(18\div 6) = \frac{2}{3}$ .

Which of these are true?

$\frac{2}{3}$ is equivalent to $\frac{4}{6}$
Simplifying $\frac{8}{12}$ gives $\frac{2}{3}$
$\frac{5}{7}$ is equivalent to $\frac{10}{14}$
Simplifying $\frac{9}{15}$ gives $\frac{3}{5}$

Mixed Numbers & Improper Fractions

A mixed number has an integer part and a fraction part, e.g., 3 3/4.
An improper fraction has numerator ≥ denominator, e.g., 15/4.
Convert mixed to improper: multiply whole number by denominator, add numerator, write over denominator. $E.g., 5 \frac{2}{7} = (5\times 7+2)/7 = \frac{37}{7}$ .
Convert improper to mixed: divide numerator by denominator; quotient is whole number, remainder over denominator. $E.g., \frac{22}{3} = 7 \frac{1}{3}$ .

What is $4 \frac{2}{5}$ as an improper fraction?

$\frac{22}{5}$
$\frac{20}{5}$
$\frac{13}{5}$
$\frac{18}{5}$

Adding & Subtracting Fractions

Find the lowest common denominator (LCD).
Rewrite each fraction as an equivalent fraction with the LCD.
Add or subtract the numerators, keep the denominator the same.
Simplify if possible. Convert mixed numbers to improper fractions first.

To add $\frac{2}{3}$ and $\frac{1}{5}$ , the LCD $is ____$ . Then $\frac{2}{3} = ____/15$ and $\frac{1}{5} = ____/15$ , so the sum $is ____/15$ .

Multiplying Fractions

Cancel any common factors between numerators and denominators before multiplying.
Multiply the numerators together and the denominators together.
Convert mixed numbers to improper fractions first.
Simplify the result if possible.

Work out $\frac{3}{4} \times \frac{7}{9}$ .

$\frac{21}{36}$
$\frac{7}{12}$
$\frac{10}{13}$
$\frac{21}{36}$

Dividing Fractions

Flip the second fraction (find its reciprocal) and change $\div to \times$ .
Cancel common factors, then multiply.
Convert mixed numbers to improper fractions first.
Convert the answer back to a mixed number if required.

To divide $\frac{1}{3}$ by $\frac{4}{5}$ , flip $\frac{4}{5}$ to $get ____ and$ change $\div to \times$ , giving $\frac{1}{3} \times ____$ .

Working with Mixed Numbers in Operations

Always convert mixed numbers to improper fractions before adding, subtracting, multiplying, or dividing.
After the operation, convert the result back to a mixed number if needed.
Example: $3 \frac{3}{4} - \frac{5}{8} = \frac{15}{4} - \frac{5}{8} = \frac{30}{8} - \frac{5}{8} = \frac{25}{8} = 3 \frac{1}{8}$ .

Work out $2 \frac{1}{4} \div 1 \frac{5}{6}$ . Give your answer as a mixed number.

$1 \frac{7}{22}$
$1 \frac{1}{2}$
$2 \frac{1}{3}$
$1 \frac{5}{22}$

Which of these are correct?

$\frac{1}{2} + \frac{1}{3} = \frac{5}{6}$
$\frac{3}{5} - \frac{1}{3} = \frac{4}{15}$
$\frac{2}{3} \times \frac{3}{4} = \frac{1}{2}$
$\frac{1}{4} \div \frac{2}{3} = \frac{3}{8}$

Exam Tips

Your calculator can simplify fractions automatically, but show your working if required.
Remember 'flip'n'times' for division: multiply by the reciprocal.
Always simplify your final answer.

Work out $\frac{7}{11}$ of 198 kg.

126 kg
128 kg
130 kg
124 kg

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