Operations With Fractions

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Notes

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** only; keep the denominator the same.
Simplify the result by **cancelling common factors**.
If a fraction is a **mixed number**, convert it to an **improper fraction** first.
After calculation, convert back to a mixed number if required.

Multiplying Fractions

**Cancel** any common factors between numerators and denominators before multiplying.
Multiply the **numerators** together and the **denominators** together.
If a fraction is a **mixed number**, convert it to an **improper fraction** first.
Simplify the final fraction by cancelling any remaining common factors.
Convert improper fractions back to mixed numbers if needed.

Dividing Fractions

**Flip** the second fraction (find its reciprocal) and change $\div to \times$ .
This is often remembered as **'flip and times'**.
**Cancel** common factors before multiplying.
Multiply the numerators together and the denominators together.
If a fraction is a **mixed number**, convert it to an **improper fraction** first.
Simplify and convert back to a mixed number if required.

Working with Mixed Numbers

Convert mixed numbers to improper fractions: multiply the whole part by the denominator, add the numerator, write over the denominator.
Perform the operation (add, subtract, multiply, divide) using the improper fractions.
Convert the final answer back to a mixed number if the question asks for it or if it's an improper fraction.

Simplifying Fractions

Always give your final answer in **simplest form**.
Cancel common factors by dividing numerator and denominator by the same number.
Check for common factors at every step to keep numbers small.
A fraction is in simplest form when the numerator and denominator have no common factors other than 1.

Common Mistakes to Avoid

Do **not** add or subtract denominators; only numerators.
Always find a common denominator before adding or subtracting.
Remember to convert mixed numbers before performing operations.
When dividing, flip only the second fraction, not the first.

Adding Fractions with Different Denominators

Multiplying Fractions with Cancelling

Dividing Fractions: Flip and Times

Converting Mixed Number to Improper Fraction

Practice questions

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1.Work out $\frac{5}{6} + \frac{2}{3}$ . Give your answer as a mixed number in its simplest form.
Easy
- A $1\frac{1}{2}$
- B $1\frac{1}{3}$
- C $\frac{7}{9}$
- D $1\frac{1}{6}$
2.Giulio's reaction times are $\frac{1}{3}$ second and $\frac{1}{8}$ second. Find the difference between the two reaction times.
Easy
- A $\frac{5}{24}$ s
- B $\frac{11}{24}$ s
- C $\frac{1}{5}$ s
- D $\frac{1}{24}$ s
3.Work out $\frac{12}{35} \times \frac{7}{9}$ . Give your answer as a fraction in its simplest form.
Easy
- A $\frac{4}{15}$
- B $\frac{84}{315}$
- C $\frac{12}{45}$
- D $\frac{4}{45}$
4.Write down a fraction that completes $\frac{13}{11} \times \ldots = 1$ .
Easy
- A $\frac{11}{13}$
- B $\frac{13}{11}$
- C $\frac{1}{11}$
- D $\frac{1}{13}$
5.Work out $\frac{7}{8} + \frac{1}{6}$ . Give your answer as a mixed number in its simplest form.
Easy
- A $1\frac{1}{24}$
- B $\frac{25}{24}$
- C $1\frac{1}{12}$
- D $\frac{8}{14}$
6.Work out $1\frac{3}{4} - \frac{11}{12}$ . Give your answer as a fraction in its simplest form.
Medium
- A $\frac{5}{6}$
- B $\frac{7}{12}$
- C $\frac{1}{2}$
- D $\frac{11}{12}$
7.Work out $\frac{15}{28} \div \frac{4}{7}$ . Give your answer as a fraction in its simplest form.
Medium
- A $\frac{15}{16}$
- B $\frac{105}{112}$
- C $\frac{15}{49}$
- D $\frac{60}{28}$
8.Work out $1\frac{7}{12} + \frac{13}{20}$ . Give your answer as a mixed number in its simplest form.
Medium
- A $2\frac{7}{30}$
- B $2\frac{1}{15}$
- C $1\frac{59}{60}$
- D $2\frac{1}{2}$

Summary

## Operations with Fractions ### Adding & Subtracting Fractions - Find the **lowest common denominator** (LCD). - Write each fraction as an equivalent fraction over the LCD. - Add or subtract the numerators; keep the denominator the same. - Simplify by cancelling common factors. - If mixed numbers: convert to improper fractions first, then convert back at the end. **Example:** $\frac{2}{3} + \frac{1}{5} = \frac{10}{15} + \frac{3}{15} = \frac{13}{15}$ **Example:** $3\frac{3}{4} - \frac{5}{8} = \frac{15}{4} - \frac{5}{8} = \frac{30}{8} - \frac{5}{8} = \frac{25}{8} = 3\frac{1}{8}$ ### Multiplying Fractions - Cancel any common factors between numerators and denominators (cross-cancel). - Multiply numerators together, multiply denominators together. - Simplify if possible. - For mixed numbers: convert to improper fractions first. **Example:** $\frac{4}{15} \times \frac{25}{11} = \frac{4}{3} \times \frac{5}{11} = \frac{20}{33}$ ### Dividing Fractions - **Flip the second fraction** (find its reciprocal) and change ÷ to ×. - Then follow multiplication steps. - Remember: “flip’n’times”. **Example:** $3\frac{1}{4} \div \frac{3}{8} = \frac{13}{4} \times \frac{8}{3} = \frac{13}{1} \times \frac{2}{3} = \frac{26}{3} = 8\frac{2}{3}$ ### Key Tips - Always show working. - Simplify answers to simplest form. - For mixed number answers, ensure the fraction part is proper. - No calculators allowed in exam.

Key terms

Lowest common denominator (LCD): The smallest number that is a multiple of both denominators; used to write fractions with a common denominator for addition/subtraction.
Improper fraction: A fraction where the numerator is greater than or equal to the denominator, e.g., 13/4.
Mixed number: A number consisting of an integer part and a proper fraction part, e.g., 3 1/4.
Reciprocal: The flipped version of a fraction; the reciprocal of a/b is b/a. Used in division: dividing by a fraction is multiplying by its reciprocal.
Simplify: Cancel common factors from numerator and denominator to write the fraction in its simplest form.

Slides

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, keeping the denominator the same.
Simplify the result by cancelling common factors.

What is the first step when adding $\frac{2}{3}$ and $\frac{1}{5}$ ?

Add the numerators and denominators directly.
Find the lowest common denominator of 3 and 5.
Convert to decimals.
Multiply the fractions together.

Mixed Numbers in Addition/Subtraction

If a fraction is a mixed number, convert it to an improper fraction first.
Then follow the same steps: find LCD, rewrite, add/subtract numerators.
Convert the final answer back to a mixed number if required.

To add $3 \frac{3}{4}$ and $\frac{5}{8}$ , first convert $3 \frac{3}{4}$ to the improper fraction ____.

Multiplying Fractions

Cancel any common factors between numerators and denominators before multiplying.
Multiply the numerators together and the denominators together.
Simplify the result if possible.

Which of these are true about multiplying fractions?

You can cancel common factors before multiplying.
You multiply the numerators and add the denominators.
Mixed numbers must be converted to improper fractions first.

Dividing Fractions

Flip the second fraction (find its reciprocal) and change $\div to \times$ .
Cancel common factors if possible.
Multiply the numerators and denominators.
Simplify and convert back to a mixed number if needed.

What is $\frac{1}{3} \div \frac{4}{5}$ ?

$\frac{1}{3} \times \frac{5}{4}$
$\frac{1}{3} \times \frac{4}{5}$
$\frac{3}{1} \times \frac{4}{5}$
$\frac{1}{3} \div \frac{5}{4}$

Working with Mixed Numbers in Division

Convert any mixed numbers to improper fractions before dividing.
Then apply the 'flip and multiply' rule.
Convert the final answer back to a mixed number if required.

To divide $3 \frac{1}{4}$ by $\frac{3}{8}$ , first write $3 \frac{1}{4}$ $as ____$ .

Example: Adding Fractions

Find $\frac{2}{3} + \frac{1}{5}$ .
LCD of 3 and 5 is 15.
Rewrite: $\frac{2}{3} = \frac{10}{15}, \frac{1}{5} = \frac{3}{15}$ .
Add: $\frac{10}{15} + \frac{3}{15} = \frac{13}{15}$ .

What is $\frac{2}{3} + \frac{1}{5}$ ?

$\frac{3}{8}$
$\frac{13}{15}$
$\frac{3}{15}$
$\frac{10}{15}$

Example: Subtracting Mixed Numbers

Find $3 \frac{3}{4} - \frac{5}{8}$ .
Convert $3 \frac{3}{4}$ to 15/4.
LCD of 4 and 8 is 8.
Rewrite: $\frac{15}{4} = \frac{30}{8}$ , subtract $\frac{5}{8} = \frac{25}{8}$ .
Convert to mixed number: 3 1/8.

$3 \frac{3}{4} - \frac{5}{8} = ____ (as a$ mixed number).

Example: Multiplying Fractions

Find $\frac{4}{15} \times \frac{25}{11}$ .
Cancel common factor $5: \frac{4}{3} \times \frac{5}{11}$ .
Multiply: $(4\times 5)/(3\times 11) = \frac{20}{33}$ .

What is $\frac{4}{15} \times \frac{25}{11}$ ?

$\frac{20}{33}$
$\frac{100}{165}$
$\frac{4}{33}$
$\frac{20}{165}$

Example: Dividing Mixed Numbers

Find $3 \frac{1}{4} \div \frac{3}{8}$ .
Convert $3 \frac{1}{4}$ to 13/4.
Flip and multiply: $\frac{13}{4} \times \frac{8}{3}$ .
Cancel $4: \frac{13}{1} \times \frac{2}{3} = \frac{26}{3}$ .
Convert to mixed number: 8 2/3.

$3 \frac{1}{4} \div \frac{3}{8} = ____ (as a$ mixed number).

Flashcards

How do you add or subtract fractions?

Find the lowest common denominator, write each fraction as an equivalent fraction over this denominator, then add or subtract the numerators. Do not add the denominators. Simplify if possible.

What should you do if any fractions are mixed numbers when adding or subtracting?

Convert any mixed numbers into improper fractions first, then add or subtract as usual. Convert the answer back to a mixed number if required.

How do you multiply two fractions?

Cancel any common factors between numerators and denominators, then multiply the numerators together and the denominators together. Simplify if possible.

What should you do if a fraction is a mixed number when multiplying?

Convert the mixed number into an improper fraction before multiplying. Convert the answer back to a mixed number if required.

How do you divide two fractions?

Flip the second fraction (find its reciprocal) and change $\div to \times$ . Cancel any common factors, then multiply the fractions. Simplify if possible.

What should you do if a fraction is a mixed number when dividing?

Convert the mixed number into an improper fraction before dividing. Convert the answer back to a mixed number if required.

Find the lowest common denominator of 3 and 5.

Convert $3 \frac{3}{4}$ to an improper fraction.

$\frac{15}{4}$

Convert $3 \frac{1}{4}$ to an improper fraction.

$\frac{13}{4}$

What is the reciprocal of $\frac{4}{5}$ ?

$\frac{5}{4}$

Simplify: $\frac{2}{3} + \frac{1}{5}$

$\frac{13}{15}$

Simplify: $3 \frac{3}{4} - \frac{5}{8} (answer$ as mixed number)

$3 \frac{1}{8}$

Simplify: $\frac{4}{15} \times \frac{25}{11}$

$\frac{20}{33}$

Simplify: $3 \frac{1}{4} \div \frac{3}{8} (answer$ as mixed number)

$8 \frac{2}{3}$

Without a calculator, work out $\frac{5}{6} + \frac{2}{3}$ . Give your answer as a mixed number in simplest form.

$1 \frac{1}{2}$

Find the difference between $\frac{1}{3}$ and 1/8.

$\frac{5}{24}$

Without a calculator, work out $\frac{12}{35} \times \frac{7}{9}$ . Give your answer as a fraction in simplest form.

$\frac{4}{15}$

Find a fraction that completes $\frac{13}{11} \times$ ? $= 1$ .

$\frac{11}{13}$

Without a calculator, work out $\frac{7}{8} + \frac{1}{6}$ . Give your answer as a mixed number in simplest form.

$1 \frac{1}{24}$

Without a calculator, work out $1 \frac{3}{4} - \frac{11}{12}$ . Give your answer as a fraction in simplest form.

$\frac{5}{6}$

Without a calculator, work out $\frac{15}{28} \div \frac{4}{7}$ . Give your answer as a fraction in simplest form.

$\frac{15}{16}$

Without a calculator, work out $1 \frac{7}{12} + \frac{13}{20}$ . Give your answer as a mixed number in simplest form.

$2 \frac{7}{30}$

Without a calculator, work out $\frac{5}{16} \times 1 \frac{1}{7}$ . Give your answer as a fraction in simplest form.

$\frac{5}{14}$

Without a calculator, work out $2 \frac{1}{3} + \frac{4}{5}$ . Give your answer as a mixed number in simplest form.

$3 \frac{2}{15}$

Without a calculator, work out $2 \frac{1}{4} \div \frac{3}{7}$ . Give your answer as a mixed number in simplest form.

$5 \frac{1}{4}$

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