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Making fraction calculations easier

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visibility 51update 6 months agobookmarkshare

🎯 In this topic you will

  • Simplify calculations containing fractions
 

🧠 Key Words

  • factor
  • strategies
Show Definitions
  • factor: A number that divides another number exactly, with no remainder.
  • strategies: Different methods or approaches used to solve a mathematical problem.
 

When you are calculating with fractions, there are methods that you can use to make a calculation easier, such as:

  • Break a fraction into parts using factors.
  • Use equivalent fractions.
  • Find factors that are the same in the numerator and the denominator.
  • Instead of working out a large fraction of a number, work out the corresponding small fraction and subtract it from the number.

For all calculations, you must always remember to use the correct order of operations.

 
Worked example

a. Work out: $\tfrac{1}{4} \times 600$

b. Work out: $\tfrac{3}{5} \times 120$

c. Work out: $\tfrac{7}{8} \times \tfrac{4}{5}$

Answer:

a. $\tfrac{1}{4} = \tfrac{1}{2} \times \tfrac{1}{2}$

$600 \div 2 = 300$

$300 \div 2 = 150$

b. $\tfrac{3}{5} = \tfrac{6}{10}$

$120 \div 10 = 12$

$12 \times 6 = 72$

c. $\tfrac{7}{8} \times \tfrac{4}{5} = \tfrac{7 \times 4}{8 \times 5}$

$= \tfrac{7 \times 4}{2 \times 5 \times 4}$

$= \tfrac{7}{10} \times \tfrac{4}{4}$

$= \tfrac{7}{10} \times 1 = \tfrac{7}{10}$

For a. Use factors to change the $4$ into $2 \times 2$. Finding $\tfrac{1}{4}$ of a number is the same as halving the number and then halving again. Half of $600$ is $300$, and half of $300$ is $150$.

For b. Rewrite $\tfrac{3}{5}$ as $\tfrac{6}{10}$ because dividing by $10$ is easier. First work out $\tfrac{1}{10}$ of $120$ ($120 \div 10 = 12$). Then multiply by $6$ to get $72$.

For c. Multiply the numerators and denominators as usual. Replace $8$ with $2 \times 4$ since $4$ is in the numerator. Rearrange so that $\tfrac{4}{4}$ appears, which equals $1$. So $\tfrac{7}{10} \times 1 = \tfrac{7}{10}$.

 

🧠 PROBLEM-SOLVING Strategy

Simplifying Calculations with Fractions

Fractions can often be simplified by using factors or by rewriting them as equivalent fractions before multiplying.

  1. Use factors: Break the denominator into factors of 2, 3, 5, etc. This helps you divide in smaller, easier steps.
    Example: $\tfrac{1}{8}\times120 = \tfrac{1}{2}\times\tfrac{1}{2}\times\tfrac{1}{2}\times120$.
  2. Cancel common factors before multiplying fractions together. This avoids large numbers.
    Example: $\tfrac{3}{5}\times\tfrac{10}{17} = \tfrac{3}{\cancel{5}}\times\tfrac{\cancel{10}}{17} = \tfrac{6}{17}$.
  3. Use equivalent fractions to simplify the calculation.
    Example: $\tfrac{2}{5}\times160 = \tfrac{1}{5}\times320 = 64$.
  4. Partition when needed: Break down numbers into easier chunks.
    Example: $6\times21 = 6\times20 + 6\times1 = 120+6 = 126$.
  5. Choose the order of steps wisely: For multiplications like $\tfrac{1}{14}\times700$, decide whether it is easier to divide by 7 first or by 2 first.

Quick examples:

  • $\tfrac{1}{2}\times28=14$
  • $\tfrac{1}{4}\times520 = 520\div4=130$
  • $\tfrac{1}{14}\times700 = (700\div7)\div2=50$
  • $\tfrac{6}{7}\times280 = 280\div7\times6=240$
 

❓ EXERCISES

1. Copy and complete the workings to make these calculations easier. Use factors to change the fractions.

a. $\dfrac{1}{2}\times 28$

b. $\dfrac{1}{4}\times 520=\dfrac{1}{2}\times\dfrac{1}{2}\times 520$

c. $\dfrac{1}{8}\times 120=\dfrac{1}{2}\times\dfrac{1}{2}\times\dfrac{1}{2}\times 120$

d. $\dfrac{1}{14}\times 700=\dfrac{1}{7}\times\dfrac{1}{2}\times 700$

🔎 Reasoning Tip

In part b, to work out $520 \div 2$, first work out $52 \div 2 = 26$. Then multiply by 10, which gives $26 \times 10 = 260$.

👀 Show answers

1a. $28\div 2=14$

1b. $520\div 2=260$,   $260\div 2=130$

1c. $120\div 2=60$,   $60\div 2=30$,   $30\div 2=15$

1d. $700\div 7=100$,   $100\div 2=50$

 

🧠 Think like a Mathematician

Task: Explore whether the order of division steps matters, and decide which method is easier.

Questions:

a) In Question 1d, does it matter if you do $700 \div 7$ or $700 \div 2$ first?
b) Which is the easier method? - $700 \div 7$ then $100 \div 2$ - $700 \div 2$ then $350 \div 7$
c) How can you decide which division to do first?
👀 show answer
  • a) It does not matter — both ways give the same result, because division is being applied in steps that correspond to factors of the divisor.
  • b) - Method 1: $700 \div 7 = 100$, then $100 \div 2 = 50$. - Method 2: $700 \div 2 = 350$, then $350 \div 7 = 50$. Both give 50, but Method 1 is easier because $700 \div 7$ gives a clean whole number straight away.
  • c) Choose the division that produces a simpler number to work with (fewer remainders or smaller numbers). Looking for multiples that divide exactly helps make the calculation easier.
 

❓ EXERCISES

3. Work out the following. Use factors to change the fractions, showing all your working.

a. $\dfrac{1}{4}\times 108$

b. $\dfrac{1}{6}\times 150$

c. $\dfrac{1}{8}\times 280$

d. $\dfrac{1}{15}\times 180$

👀 Show answers

3a. $108\div 4=27$

3b. $150\div 6=25$

3c. $280\div 8=35$

3d. $180\div 15=12$

4. Copy and complete the workings to make these calculations easier. Use equivalent fractions.

a. $\dfrac{1}{5}\times 340$

b. $\dfrac{2}{5}\times 160$

🔎 Reasoning Tip

In part b, remember that multiplying by 4 is the same as multiplying by 2 and then by 2 again, i.e. $\times 4 = \times 2 \times 2$.

👀 Show answers

4a. $340\div 10=34,\; 34\times 2=68$

4b. $160\div 10=16,\; 16\times 4=64$

5. Work out the following. Use equivalent fractions, showing all your working.

a. $\dfrac{1}{5}\times 270$

b. $\dfrac{4}{5}\times 80$

c. $\dfrac{3}{5}\times 210$

d. $\dfrac{2}{5}\times 320$

🔎 Reasoning Tip

In part c, you can use partitioning to calculate $6 \times 21$.
So, $6 \times 21 = 6 \times 20 + 6 \times 1$.

👀 Show answers

5a. $270\div 5=54$

5b. $80\div 5=16,\; 16\times 4=64$

5c. $210\div 5=42,\; 42\times 3=126$

5d. $320\div 5=64,\; 64\times 2=128$

6. Use Emyr’s method to work out:

a. $\dfrac{1}{20}\times 1100$

b. $\dfrac{3}{20}\times 1900$

c. $\dfrac{7}{20}\times 900$

d. $\dfrac{11}{20}\times 7000$

👀 Show answers

6a. $\dfrac{1}{20}\times 1100 = 55$

6b. $\dfrac{3}{20}\times 1900 = 285$

6c. $\dfrac{7}{20}\times 900 = 315$

6d. $\dfrac{11}{20}\times 7000 = 3850$

7. Copy and complete the workings to make these calculations easier.

a. $\dfrac{5}{6}\times\dfrac{2}{3}$

b. $\dfrac{3}{5}\times\dfrac{10}{17}$

👀 Show answers

7a. $\dfrac{5}{6}\times\dfrac{2}{3} = \dfrac{10}{18} = \dfrac{5}{9}$

7b. $\dfrac{3}{5}\times\dfrac{10}{17} = \dfrac{30}{85} = \dfrac{6}{17}$

8. Work out the following. Use the method from Q7.

a. $\dfrac{3}{8}\times\dfrac{5}{6}$

b. $\dfrac{13}{14}\times\dfrac{2}{5}$

c. $\dfrac{7}{9}\times\dfrac{18}{25}$

d. $\dfrac{12}{19}\times\dfrac{2}{3}$

👀 Show answers

8a. $\dfrac{3}{8}\times\dfrac{5}{6} = \dfrac{15}{48} = \dfrac{5}{16}$

8b. $\dfrac{13}{14}\times\dfrac{2}{5} = \dfrac{26}{70} = \dfrac{13}{35}$

8c. $\dfrac{7}{9}\times\dfrac{18}{25} = \dfrac{126}{225} = \dfrac{14}{25}$

8d. $\dfrac{12}{19}\times\dfrac{2}{3} = \dfrac{24}{57} = \dfrac{8}{19}$

9. This is part of Iqra’s homework.

Check every step. Is Iqra’s homework correct? Explain your answer.

👀 Show answer

Step check

  • Rewrite: $\dfrac{3}{20}=\dfrac{15}{100}$ ✅ correct.
  • Step 1: $4500 \div 100 = 45$ — this is only part of the work; she still must multiply by $15$. The step is incomplete.
  • Step 2: $15 \times 45 = 675$ ✅ correct.

Correct working, clearly:
$\dfrac{3}{20}\times 4500=\dfrac{15}{100}\times 4500=(4500\div 100)\times 15=45\times 15=675.$

Conclusion: The final answer is correct (675), but Iqra’s explanation should show that after dividing by $100$ she then multiplies by $15$.

Quick check: $4500\div 20=225$ and $225\times 3=675$ ✅

10. This is how Arun works out $\tfrac{5}{6}\times 180$.

First, I work out $180 \div 6$, which is $30$. Then I work out $30 \times 5$, which is $150$.

a. Explain how Arun’s method works.

This is how Sofia works out $\tfrac{5}{6}\times 180$.

First, I work out $180 \div 6$, which is $30$. Then I work out $180-30$, which is $150$.

b. Explain how Sofia’s method works.

c. Use both Arun’s method and Sofia’s method to work out:

i. $\tfrac{6}{7}\times 280$    ii. $\tfrac{14}{15}\times 900$

d. Whose method did you find it easiest to use to work out the answers to part c? Explain why.

e. Critique each method by explaining the advantages and disadvantages of each method.

f. What do you think is the best method to use to work out questions such as $\tfrac{19}{20}\times 1800$, $\tfrac{24}{25}\times 800$ and $\tfrac{13}{14}\times 2240$? Explain why.

g. Use your favourite method to work out the answers to the following.

i. $\tfrac{19}{20}\times 1800$    ii. $\tfrac{24}{25}\times 800$    iii. $\tfrac{13}{14}\times 2240$

👀 Show answer

a.

Divide the whole by the denominator, then multiply by the numerator: $180 \div 6 = 30$, then $30 \times 5 = 150$.

b.

Find one sixth, then subtract it from the whole to get five sixths: $180 \div 6 = 30$, then $180-30 = 150$.

c.

i. Arun: $280 \div 7 = 40$, then $40 \times 6 = 240$. Sofia: $280 - 40 = 240$. So $\tfrac{6}{7}\times 280 = 240$.

ii. Arun: $900 \div 15 = 60$, then $60 \times 14 = 840$. Sofia: $900 - 60 = 840$. So $\tfrac{14}{15}\times 900 = 840$.

d.

Answers will vary. Many find Arun’s “divide then multiply” consistent, while others prefer Sofia’s subtraction when the fraction is “one part less than the whole”.

e.

Arun’s method is general and always works; it may involve bigger multiplications. Sofia’s is fast when the numerator is the whole minus one part (e.g., $\tfrac{n-1}{n}$), but is less convenient for other fractions.

f.

Sofia’s subtraction approach is efficient here because each fraction is “one part less than the whole”: subtract $\tfrac{1}{20}$, $\tfrac{1}{25}$, or $\tfrac{1}{14}$ of the whole once. Arun’s method also works reliably.

g.

i. $1800-\tfrac{1}{20}\times 1800 = 1800-90 = 1620$.

ii. $800-\tfrac{1}{25}\times 800 = 800-32 = 768$.

iii. $2240-\tfrac{1}{14}\times 2240 = 2240-160 = 2080$.

11. Both Sofia and Zara work out the answer to $ \tfrac{1}{5} + \tfrac{2}{5} \times \tfrac{3}{4} $.

a. Who is correct?

b. Explain the mistake that the other person has made.

 
👀 Show answer
Zara is correct: $ \tfrac{2}{5} \times \tfrac{3}{4} = \tfrac{6}{20} = \tfrac{3}{10} $ Then $ \tfrac{1}{5} + \tfrac{3}{10} = \tfrac{2}{10} + \tfrac{3}{10} = \tfrac{5}{10} = \tfrac{1}{2} $. Sofia’s mistake is adding $ \tfrac{1}{5} + \tfrac{2}{5} $ before multiplying, which is not the correct order of operations.

12. Work out the following. Give each answer in its simplest form, showing all your working.

a. $ \tfrac{5}{6} + \tfrac{7}{9} \times \tfrac{3}{8} $

b. $ \tfrac{9}{10} - \tfrac{2}{3} \times \tfrac{1}{4} $

c. $ \tfrac{1}{4} \times \tfrac{7}{8} + \tfrac{3}{4} \times \tfrac{1}{6} $

👀 Show answer
a. $ \tfrac{7}{9} \times \tfrac{3}{8} = \tfrac{21}{72} = \tfrac{7}{24} $ $ \tfrac{5}{6} + \tfrac{7}{24} = \tfrac{20}{24} + \tfrac{7}{24} = \tfrac{27}{24} = 1 \tfrac{1}{8} $ b. $ \tfrac{2}{3} \times \tfrac{1}{4} = \tfrac{2}{12} = \tfrac{1}{6} $ $ \tfrac{9}{10} - \tfrac{1}{6} = \tfrac{54}{60} - \tfrac{10}{60} = \tfrac{44}{60} = \tfrac{11}{15} $ c. $ \tfrac{1}{4} \times \tfrac{7}{8} = \tfrac{7}{32} $ $ \tfrac{3}{4} \times \tfrac{1}{6} = \tfrac{3}{24} = \tfrac{1}{8} = \tfrac{4}{32} $ $ \tfrac{7}{32} + \tfrac{4}{32} = \tfrac{11}{32} $
 

🔗 Learning Bridge

You’ve just practised simplifying fraction calculations (using factors, equivalent fractions, and cancelling before multiplying). Next, you’ll push those ideas into fast, mostly mental work that mixes +, −, ×, ÷ and brackets.

  • Reuse your shortcuts mentally: cancel across, swap to friendlier equivalents (e.g. 3/5 = 6/10), and split numbers with partitioning.
  • Think “operation order” first: do brackets → × and ÷ → + and −. This keeps mental steps tidy.
  • Invert & multiply on sight: for divisions by fractions (e.g. 6 ÷ 3/4 → 6 × 4/3).
  • Estimate to sense-check: round fractions to nearby easy benchmarks (½, ⅓, ¼, 1) to see if your exact answer is reasonable.

Mini examples

  • 3/4 + 3/8 = 6/8 + 3/8 = 9/8 = 1 1/8
  • 4/5 − 3/4 = (16−15)/20 = 1/20 (use cross-multiply method)
  • 6 ÷ 3/4 = 6 × 4/3 = 8
  • (2/3 + 1/2) × 2/5 = 7/6 × 2/5 = 14/30 = 7/15

Goal: keep calculations small and accurate by choosing the friendliest form first, then apply the order of operations with confidence.

 

When you are calculating using fractions, you can often make a calculation easier by using different strategies. These strategies will help you to work with fractions mentally. This means you should be able to do simple additions, subtractions, multiplications and divisions ‘in your head’. You should also be able to solve word problems mentally. This section will help you to practise the skills you need.

$\tfrac{3}{4} + \tfrac{3}{8} = ?$

For harder questions, it may help you to write down some of the steps in the working. These workings will help you to remember what you have done so far, and what you still need to do.

With all calculations, you must remember the correct order of operations.

 
Worked example

a. Work out: $\tfrac{3}{4} + \tfrac{3}{8}$

b. Work out: $\tfrac{4}{5} - \tfrac{3}{4}$

c. Work out: $6 \div \tfrac{3}{4}$

d. Work out: $\tfrac{2}{5} \times \left(\tfrac{2}{3} + \tfrac{1}{2}\right)$

Answer:

a. $\tfrac{6}{8} + \tfrac{3}{8} = \tfrac{9}{8}$

$= 1 \tfrac{1}{8}$

b. $\tfrac{4}{5} - \tfrac{3}{4} = \tfrac{4 \times 4 - 3 \times 5}{5 \times 4}$

$= \tfrac{16 - 15}{20}$

$= \tfrac{1}{20}$

c. $6 \div \tfrac{3}{4}$

$= 6 \times 4 = 24$

$24 \div 3 = 8$

d. $\tfrac{2}{3} + \tfrac{1}{2} = \tfrac{2 \times 2 + 3 \times 1}{3 \times 2}$

$= \tfrac{4 + 3}{6}$

$= \tfrac{7}{6}$

$\tfrac{2}{5} \times \tfrac{7}{6} = \tfrac{14}{30}$

$= \tfrac{7}{15}$

For a. Rewrite $\tfrac{3}{4}$ as $\tfrac{6}{8}$ so it can be added to $\tfrac{3}{8}$. Then convert $\tfrac{9}{8}$ into the mixed number $1\tfrac{1}{8}$.

For b. Multiply diagonally across the fractions: $4 \times 4$ and $3 \times 5$. Denominator is $5 \times 4$. Subtract numerators: $16 - 15 = 1$. Final result $\tfrac{1}{20}$.

For c. Dividing by $\tfrac{3}{4}$ is the same as multiplying by $\tfrac{4}{3}$. So $6 \times \tfrac{4}{3} = 8$.

For d. Brackets first: add $\tfrac{2}{3} + \tfrac{1}{2} = \tfrac{7}{6}$. Then multiply by $\tfrac{2}{5}$. Multiply numerators and denominators: $\tfrac{2}{5} \times \tfrac{7}{6} = \tfrac{14}{30} = \tfrac{7}{15}$.

 

🧠 PROBLEM-SOLVING Strategy

Mental Strategies with Fractions

Use quick strategies to simplify fractions so you can add, subtract, multiply, and divide mentally.

  1. Find common denominators quickly when adding or subtracting fractions. E.g. $\tfrac{3}{4}+\tfrac{3}{8}=\tfrac{6}{8}+\tfrac{3}{8}=\tfrac{9}{8}$.
  2. Use cross-multiplication for subtraction/addition with unlike denominators. E.g. $\tfrac{4}{5}-\tfrac{3}{4}=\tfrac{16-15}{20}=\tfrac{1}{20}$.
  3. Use reciprocals for division. Dividing by a fraction = multiply by its reciprocal. E.g. $6\div\tfrac{3}{4}=6\times\tfrac{4}{3}=8$.
  4. Work brackets first and then apply multiplication/division before addition/subtraction (BIDMAS/BODMAS rule).
  5. Estimate before working: round fractions to simple values to check if your final answer makes sense.

Quick examples:

  • $\tfrac{3}{4}+\tfrac{3}{8}=1\tfrac{1}{8}$
  • $\tfrac{4}{5}-\tfrac{3}{4}=\tfrac{1}{20}$
  • $6\div\tfrac{3}{4}=8$
  • $\tfrac{2}{5}\times\big(\tfrac{2}{3}+\tfrac{1}{2}\big)=\tfrac{7}{15}$
 

❓ EXERCISES

In this exercise, work out as many of the answers as you can mentally. Write each answer in its simplest form and as a mixed number when appropriate.

 

1. Work out these additions and subtractions. Some working has been shown to help you.

a. $ \tfrac{1}{3} + \tfrac{1}{6} $

b. $ \tfrac{1}{8} + \tfrac{1}{4} $

c. $ \tfrac{4}{5} - \tfrac{1}{10} $

d. $ \tfrac{5}{6} - \tfrac{1}{3} $

Fraction addition and subtraction working example

👀 Show answer
a. $ \tfrac{1}{3} + \tfrac{1}{6} = \tfrac{2}{6} + \tfrac{1}{6} = \tfrac{3}{6} = \tfrac{1}{2} $
b. $ \tfrac{1}{8} + \tfrac{1}{4} = \tfrac{1}{8} + \tfrac{2}{8} = \tfrac{3}{8} $
c. $ \tfrac{4}{5} - \tfrac{1}{10} = \tfrac{8}{10} - \tfrac{1}{10} = \tfrac{7}{10} $
d. $ \tfrac{5}{6} - \tfrac{1}{3} = \tfrac{5}{6} - \tfrac{2}{6} = \tfrac{3}{6} = \tfrac{1}{2} $

2. Work out these additions and subtractions. Use the same method as in part a of the worked example.

a. $ \tfrac{1}{2} + \tfrac{1}{6} $

b. $ \tfrac{3}{4} + \tfrac{1}{8} $

c. $ \tfrac{3}{5} + \tfrac{1}{10} $

d. $ \tfrac{1}{2} + \tfrac{3}{8} $

e. $ \tfrac{3}{4} + \tfrac{5}{12} $

f. $ \tfrac{7}{15} + \tfrac{4}{5} $

g. $ \tfrac{1}{3} - \tfrac{1}{9} $

h. $ \tfrac{1}{4} - \tfrac{1}{8} $

i. $ \tfrac{1}{5} - \tfrac{1}{15} $

j. $ \tfrac{2}{3} - \tfrac{1}{6} $

k. $ \tfrac{4}{5} - \tfrac{1}{10} $

l. $ \tfrac{11}{20} - \tfrac{2}{5} $

👀 Show answer
a. $ \tfrac{1}{2} + \tfrac{1}{6} = \tfrac{3}{6} + \tfrac{1}{6} = \tfrac{4}{6} = \tfrac{2}{3} $
b. $ \tfrac{3}{4} + \tfrac{1}{8} = \tfrac{6}{8} + \tfrac{1}{8} = \tfrac{7}{8} $
c. $ \tfrac{3}{5} + \tfrac{1}{10} = \tfrac{6}{10} + \tfrac{1}{10} = \tfrac{7}{10} $
d. $ \tfrac{1}{2} + \tfrac{3}{8} = \tfrac{4}{8} + \tfrac{3}{8} = \tfrac{7}{8} $
e. $ \tfrac{3}{4} + \tfrac{5}{12} = \tfrac{9}{12} + \tfrac{5}{12} = \tfrac{14}{12} = 1 \tfrac{1}{6} $
f. $ \tfrac{7}{15} + \tfrac{4}{5} = \tfrac{7}{15} + \tfrac{12}{15} = \tfrac{19}{15} = 1 \tfrac{4}{15} $
g. $ \tfrac{1}{3} - \tfrac{1}{9} = \tfrac{3}{9} - \tfrac{1}{9} = \tfrac{2}{9} $
h. $ \tfrac{1}{4} - \tfrac{1}{8} = \tfrac{2}{8} - \tfrac{1}{8} = \tfrac{1}{8} $
i. $ \tfrac{1}{5} - \tfrac{1}{15} = \tfrac{3}{15} - \tfrac{1}{15} = \tfrac{2}{15} $
j. $ \tfrac{2}{3} - \tfrac{1}{6} = \tfrac{4}{6} - \tfrac{1}{6} = \tfrac{3}{6} = \tfrac{1}{2} $
k. $ \tfrac{4}{5} - \tfrac{1}{10} = \tfrac{8}{10} - \tfrac{1}{10} = \tfrac{7}{10} $
l. $ \tfrac{11}{20} - \tfrac{2}{5} = \tfrac{11}{20} - \tfrac{8}{20} = \tfrac{3}{20} $

3. Work out these additions and subtractions. Use the same method as in part b of the worked example.

a. $ \tfrac{1}{3} + \tfrac{1}{5} $

b. $ \tfrac{1}{4} + \tfrac{1}{7} $

c. $ \tfrac{2}{9} + \tfrac{1}{5} $

d. $ \tfrac{3}{4} + \tfrac{2}{3} $

e. $ \tfrac{5}{8} + \tfrac{1}{5} $

f. $ \tfrac{1}{4} + \tfrac{5}{6} $

g. $ \tfrac{1}{2} - \tfrac{1}{3} $

h. $ \tfrac{4}{5} - \tfrac{1}{4} $

i. $ \tfrac{5}{7} - \tfrac{1}{2} $

j. $ \tfrac{3}{4} - \tfrac{2}{7} $

k. $ \tfrac{7}{12} - \tfrac{3}{8} $

l. $ \tfrac{8}{9} - \tfrac{3}{4} $

 
👀 Show answer
a. $ \tfrac{1}{3} + \tfrac{1}{5} = \tfrac{5}{15} + \tfrac{3}{15} = \tfrac{8}{15} $
b. $ \tfrac{1}{4} + \tfrac{1}{7} = \tfrac{7}{28} + \tfrac{4}{28} = \tfrac{11}{28} $
c. $ \tfrac{2}{9} + \tfrac{1}{5} = \tfrac{10}{45} + \tfrac{9}{45} = \tfrac{19}{45} $
d. $ \tfrac{3}{4} + \tfrac{2}{3} = \tfrac{9}{12} + \tfrac{8}{12} = \tfrac{17}{12} = 1 \tfrac{5}{12} $
e. $ \tfrac{5}{8} + \tfrac{1}{5} = \tfrac{25}{40} + \tfrac{8}{40} = \tfrac{33}{40} $
f. $ \tfrac{1}{4} + \tfrac{5}{6} = \tfrac{3}{12} + \tfrac{10}{12} = \tfrac{13}{12} = 1 \tfrac{1}{12} $
g. $ \tfrac{1}{2} - \tfrac{1}{3} = \tfrac{3}{6} - \tfrac{2}{6} = \tfrac{1}{6} $
h. $ \tfrac{4}{5} - \tfrac{1}{4} = \tfrac{16}{20} - \tfrac{5}{20} = \tfrac{11}{20} $
i. $ \tfrac{5}{7} - \tfrac{1}{2} = \tfrac{10}{14} - \tfrac{7}{14} = \tfrac{3}{14} $
j. $ \tfrac{3}{4} - \tfrac{2}{7} = \tfrac{21}{28} - \tfrac{8}{28} = \tfrac{13}{28} $
k. $ \tfrac{7}{12} - \tfrac{3}{8} = \tfrac{14}{24} - \tfrac{9}{24} = \tfrac{5}{24} $
l. $ \tfrac{8}{9} - \tfrac{3}{4} = \tfrac{32}{36} - \tfrac{27}{36} = \tfrac{5}{36} $
 

🧠 Think like a Mathematician

Task: Work out the fraction of dark chocolates in the box and reflect on the best method to use.

Questions:

a) In a box of chocolates, $\dfrac{1}{5}$ are white chocolate, $\dfrac{1}{2}$ are milk chocolate, and the rest are dark chocolate. What fraction of the chocolates are dark chocolate?
b) Reflect on your method. Do you think it was the best method to use?
👀 show answer
  • a) Total = 1. White + milk = $\dfrac{1}{5} + \dfrac{1}{2} = \dfrac{2}{10} + \dfrac{5}{10} = \dfrac{7}{10}$. Dark = $1 - \dfrac{7}{10} = \dfrac{3}{10}$. So, $\dfrac{3}{10}$ of the chocolates are dark.
  • b) The subtraction method (adding white and milk, then subtracting from 1) is straightforward and efficient. Another method is to find a common denominator and check remaining parts directly, but subtracting from 1 is usually simplest.
 

❓ EXERCISES

5. In a hockey squad, $ \tfrac{1}{3} $ of the players are short, $ \tfrac{1}{4} $ of the players are medium height and the rest are tall.
What fraction of the squad are tall?

6. In a box of fruit, $ \tfrac{2}{5} $ are apples, $ \tfrac{1}{6} $ are guavas and the rest are coconuts.
What fraction of the fruit in the box are coconuts?

7. Work out these calculations. Use the same method as in part c of the worked example.
Some working has been shown to help you with the first two.

a. $ 4 \div \tfrac{2}{3} $

b. $ 8 \div \tfrac{4}{5} $

c. $ 9 \div \tfrac{1}{2} $

d. $ 6 \div \tfrac{2}{5} $

e. $ 9 \div \tfrac{3}{4} $

f. $ 10 \div \tfrac{5}{6} $

Fraction division and word problems

👀 Show answer
5. $ \tfrac{1}{3} + \tfrac{1}{4} = \tfrac{4}{12} + \tfrac{3}{12} = \tfrac{7}{12} $. Fraction tall $ = 1 - \tfrac{7}{12} = \tfrac{5}{12} $.

6. $ \tfrac{2}{5} + \tfrac{1}{6} = \tfrac{12}{30} + \tfrac{5}{30} = \tfrac{17}{30} $. Fraction coconuts $ = 1 - \tfrac{17}{30} = \tfrac{13}{30} $.

7a. $ 4 \div \tfrac{2}{3} = 4 \times \tfrac{3}{2} = \tfrac{12}{2} = 6 $
7b. $ 8 \div \tfrac{4}{5} = 8 \times \tfrac{5}{4} = \tfrac{40}{4} = 10 $
7c. $ 9 \div \tfrac{1}{2} = 9 \times 2 = 18 $
7d. $ 6 \div \tfrac{2}{5} = 6 \times \tfrac{5}{2} = \tfrac{30}{2} = 15 $
7e. $ 9 \div \tfrac{3}{4} = 9 \times \tfrac{4}{3} = \tfrac{36}{3} = 12 $
7f. $ 10 \div \tfrac{5}{6} = 10 \times \tfrac{6}{5} = \tfrac{60}{5} = 12 $
 

🧠 Think like a Mathematician

Task: Practise using the calculator’s fraction and S ⇔ D buttons to convert between improper fractions and mixed numbers.

Questions:

a) Work out how to use the fraction button on a calculator.
b) Work out the answer to $18 \div \dfrac{5}{7}$. Write your answer first as an improper fraction.
c) Use the calculator to turn the improper fraction into a mixed number. You will need to use the S ⇔ D button.
👀 show answer
  • a) The fraction button allows you to enter numerators and denominators directly in fraction form instead of typing division.
  • b)$18 \div \dfrac{5}{7} = 18 \times \dfrac{7}{5} = \dfrac{126}{5}$.
  • c) Using the S ⇔ D button converts $\dfrac{126}{5}$ into the mixed number $25\dfrac{1}{5}$.
 

❓ EXERCISES

9.

a. Work out mentally

i. $ 9 \div \tfrac{4}{5} $

ii. $ 7 \div \tfrac{3}{5} $

iii. $ 11 \div \tfrac{2}{3} $

iv. $ 8 \div \tfrac{5}{7} $

b. Use a calculator to check your answers to part a.

c. Did you get your answers to part a correct? If not, what mistakes did you make?

👀 Show answer
i. $ 9 \div \tfrac{4}{5} = 9 \times \tfrac{5}{4} = \tfrac{45}{4} = 11 \tfrac{1}{4} $
ii. $ 7 \div \tfrac{3}{5} = 7 \times \tfrac{5}{3} = \tfrac{35}{3} = 11 \tfrac{2}{3} $
iii. $ 11 \div \tfrac{2}{3} = 11 \times \tfrac{3}{2} = \tfrac{33}{2} = 16 \tfrac{1}{2} $
iv. $ 8 \div \tfrac{5}{7} = 8 \times \tfrac{7}{5} = \tfrac{56}{5} = 11 \tfrac{1}{5} $

10. The diagram shows a path. The area of the path is $10 \, m^2$. The width of the path is $ \tfrac{3}{4} \, m $.
What is the length of the path?

Path area diagram

👀 Show answer
Area $ = \text{length} \times \text{width} $
$ 10 = \text{length} \times \tfrac{3}{4} $
$ \text{length} = 10 \div \tfrac{3}{4} = 10 \times \tfrac{4}{3} = \tfrac{40}{3} = 13 \tfrac{1}{3} \, m $

11. This is how Marcus mentally works out $ \tfrac{1}{3} \times ( \tfrac{5}{6} - \tfrac{1}{2} ) $.

a. Explain the mistake Marcus has made.

b. Work out the correct answer.

👀 Show answer
Marcus’s mistake: He simplified $ \tfrac{5}{6} - \tfrac{1}{2} $ incorrectly, writing it as $ \tfrac{4}{4} = 1 $.
Correct: $ \tfrac{5}{6} - \tfrac{1}{2} = \tfrac{5}{6} - \tfrac{3}{6} = \tfrac{2}{6} = \tfrac{1}{3} $.
Then $ \tfrac{1}{3} \times \tfrac{1}{3} = \tfrac{1}{9} $.

12. Work out these calculations. If you cannot do them mentally, write down some workings to help you.

a. $ 6 \times \left( \tfrac{5}{6} - \tfrac{1}{6} \right) $

b. $ 4 \div \left( \tfrac{1}{3} + \tfrac{1}{3} \right) $

c. $ \tfrac{11}{12} - \left( \tfrac{3}{4} - \tfrac{1}{2} \right) $

d. $ \tfrac{11}{12} - \left( \tfrac{1}{2} + \tfrac{1}{3} \right) $

e. $ \left( \tfrac{1}{4} + \tfrac{1}{2} \right) \times \left( \tfrac{5}{9} - \tfrac{2}{9} \right) $

f. $ \left( 4 \tfrac{1}{3} + 9 \tfrac{2}{3} \right) \div \left( \tfrac{2}{5} + \tfrac{3}{10} \right) $

 
👀 Show answer
a. $ 6 \times \left( \tfrac{5}{6} - \tfrac{1}{6} \right) = 6 \times \tfrac{4}{6} = 6 \times \tfrac{2}{3} = 4 $
b. $ 4 \div \left( \tfrac{1}{3} + \tfrac{1}{3} \right) = 4 \div \tfrac{2}{3} = 4 \times \tfrac{3}{2} = 6 $
c. $ \tfrac{11}{12} - \left( \tfrac{3}{4} - \tfrac{1}{2} \right) = \tfrac{11}{12} - \tfrac{1}{4} = \tfrac{11}{12} - \tfrac{3}{12} = \tfrac{8}{12} = \tfrac{2}{3} $
d. $ \tfrac{11}{12} - \left( \tfrac{1}{2} + \tfrac{1}{3} \right) = \tfrac{11}{12} - \tfrac{5}{6} = \tfrac{11}{12} - \tfrac{10}{12} = \tfrac{1}{12} $
e. $ \left( \tfrac{1}{4} + \tfrac{1}{2} \right) \times \left( \tfrac{5}{9} - \tfrac{2}{9} \right) = \tfrac{3}{4} \times \tfrac{3}{9} = \tfrac{3}{4} \times \tfrac{1}{3} = \tfrac{1}{4} $
f. $ \left( 4 \tfrac{1}{3} + 9 \tfrac{2}{3} \right) = \tfrac{13}{3} + \tfrac{29}{3} = \tfrac{42}{3} = 14 $
Denominator: $ \tfrac{2}{5} + \tfrac{3}{10} = \tfrac{4}{10} + \tfrac{3}{10} = \tfrac{7}{10} $
$ 14 \div \tfrac{7}{10} = 14 \times \tfrac{10}{7} = \tfrac{140}{7} = 20 $

13. Zara works out the answers to these calculation cards.

A. $ 3 \times \left( \tfrac{3}{4} + \tfrac{3}{4} \right) $

B. $ 2 - \left( \tfrac{7}{10} - \tfrac{1}{5} \right) $

C. $ 4 \times \left( 2 \tfrac{2}{3} - 1 \tfrac{1}{6} \right) $

D. $ 6 \div \left( \tfrac{34}{9} - 1 \tfrac{7}{9} \right) $

Read what Zara says.
Is Zara correct?
Write the first term and the term-to-term rule of the sequences you can find.

Zara explaining sequences from fraction calculation cards

👀 Show answer
A. $ 3 \times \left( \tfrac{3}{4} + \tfrac{3}{4} \right) = 3 \times \tfrac{6}{4} = 3 \times \tfrac{3}{2} = \tfrac{9}{2} = 4 \tfrac{1}{2} $
B. $ 2 - \left( \tfrac{7}{10} - \tfrac{1}{5} \right) = 2 - \left( \tfrac{7}{10} - \tfrac{2}{10} \right) = 2 - \tfrac{5}{10} = 2 - \tfrac{1}{2} = \tfrac{3}{2} $
C. $ 4 \times \left( 2 \tfrac{2}{3} - 1 \tfrac{1}{6} \right) = 4 \times \left( \tfrac{8}{3} - \tfrac{7}{6} \right) = 4 \times \left( \tfrac{16}{6} - \tfrac{7}{6} \right) = 4 \times \tfrac{9}{6} = 4 \times \tfrac{3}{2} = 6 $
D. $ 6 \div \left( \tfrac{34}{9} - 1 \tfrac{7}{9} \right) = 6 \div \left( \tfrac{34}{9} - \tfrac{16}{9} \right) = 6 \div \tfrac{18}{9} = 6 \div 2 = 3 $

The answers are: $ 4 \tfrac{1}{2}, \tfrac{3}{2}, 6, 3 $. These can be rearranged into two sequences:
  • Sequence 1: $ \tfrac{3}{2}, 3, 4 \tfrac{1}{2}, 6 $ with term-to-term rule “add $ \tfrac{3}{2} $”.
  • Sequence 2: $ 6, 4 \tfrac{1}{2}, 3, \tfrac{3}{2} $ with term-to-term rule “subtract $ \tfrac{3}{2} $”.
So yes, Zara is correct.
 

⚠️ Be careful! Simplifying Fraction Calculations

  • Don’t add or subtract denominators. E.g. $\tfrac{1}{4}+\tfrac{1}{2}\neq\tfrac{2}{6}$ — always find a common denominator first.
  • Cancel common factors before multiplying. It is easier and avoids large numbers: $\tfrac{7}{8}\times\tfrac{4}{5}=\tfrac{7}{10}$, not $\tfrac{28}{40}$.
  • Break down fractions into factors. For example, $\tfrac{1}{8}\times120$ is easier as halving three times, not as $120\div8$ done in one step.
  • Watch the order of operations (BODMAS/BIDMAS). Do brackets first, then multiplication/division, then addition/subtraction.
  • Equivalent fractions must represent the same value. $\tfrac{3}{5}$ can be written as $\tfrac{6}{10}$, but not as $\tfrac{6}{15}$.
  • Estimate first to check. For $\tfrac{3}{5}\times120$, estimate $\tfrac{1}{2}\times120=60$ — so the exact answer $72$ is sensible.
 

📘 What we've learned — Simplifying Fraction Calculations

  • Fractions can be made easier by breaking denominators into factors (e.g. $ \tfrac{1}{8}\times120 = (120\div2)\div2\div2$).
  • Use equivalent fractions to simplify calculations (e.g. $ \tfrac{3}{5}\times120 = \tfrac{6}{10}\times120 = 72$).
  • Cancel common factors before multiplying fractions to avoid large numbers.
  • Partitioning is useful when multiplying with whole numbers (e.g. $6\times21 = 6\times20 + 6\times1$).
  • Apply the order of operations (BIDMAS) carefully when fractions are mixed with +, −, ×, ÷, and brackets.
  • Dividing by a fraction means multiplying by its reciprocal (e.g. $6\div\tfrac{3}{4}=6\times\tfrac{4}{3}=8$).
  • When adding or subtracting fractions, find a common denominator (e.g. $ \tfrac{3}{4}+\tfrac{3}{8}=\tfrac{9}{8}=1\tfrac{1}{8}$).
  • Estimate first by rounding fractions to friendly values (½, ¼, ⅓, 1) to check the reasonableness of answers.
  • Mini examples:
    • $ \tfrac{1}{14}\times700 = (700\div7)\div2=50$
    • $ \tfrac{4}{5}-\tfrac{3}{4} = \tfrac{1}{20}$
    • $ ( \tfrac{2}{3} + \tfrac{1}{2})\times\tfrac{2}{5} = \tfrac{7}{15}$
  • Always simplify the final answer, writing it as a mixed number when necessary.

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