Dividing fractions
Dividing fractions
- Unit 1: Integers
- Unit 2: Place value & Rounding
- Unit 3: Decimals
- Unit 4: Fractions
- Unit 5: Percentages
- Unit 6: Ratio & Proportion
🎯 In this topic you will
- Divide two proper fractions
- Cancel common factors before dividing fractions
- Estimate the answers to calculations
🧠 Key Words
- reciprocal
- upside down
Show Definitions
- reciprocal: The value you get by dividing 1 by a number; for a fraction, swapping numerator and denominator.
- upside down: An informal way of describing a reciprocal, where the numerator and denominator of a fraction are flipped.
The diagram shows a rectangle.
$\tfrac{2}{3}$ of the rectangle is yellow.
$\tfrac{1}{6}$ of the rectangle is also shown.
🔎 Reasoning Tip
Reciprocal fractions: When you turn a fraction upside down you get a reciprocal fraction. For example, the reciprocal of \( \tfrac{2}{3} \) is \( \tfrac{3}{2} \).
Solving $\tfrac{2}{3} \div \tfrac{1}{6}$ is the same as asking ‘How many $\tfrac{1}{6}$ are there in $\tfrac{2}{3}$?’
You can see that the answer is $4$, so $\tfrac{2}{3} \div \tfrac{1}{6} = 4$.
The calculation is $\tfrac{2}{3} \div \tfrac{1}{6} = \tfrac{2}{3} \times \tfrac{6}{1} = \tfrac{2 \times 6}{3 \times 1} = \tfrac{12}{3} = 4$.
Here is a method for dividing a fraction by a fraction:
- 1 Turn the second fraction upside down.
- 2 Multiply the fractions together, as usual.
- 3 Write the answer in its simplest form and as a mixed number when possible.
❓ EXERCISES
1. Copy and complete:
a. $\tfrac{1}{5} \div \tfrac{3}{4}$
b. $\tfrac{2}{3} \div \tfrac{6}{7}$
2. Work out:
🔎 Reasoning Tip
Mixed numbers: In parts d, e, and f, write your answer as a mixed number.
a. $\tfrac{1}{4} \div \tfrac{2}{3}$
b. $\tfrac{1}{2} \div \tfrac{3}{5}$
c. $\tfrac{3}{8} \div \tfrac{4}{7}$
d. $\tfrac{4}{5} \div \tfrac{1}{9}$
e. $\tfrac{3}{5} \div \tfrac{2}{11}$
f. $\tfrac{9}{10} \div \tfrac{1}{3}$
3. Work out the following. Write each answer in its simplest form and as a mixed number when possible.
a. $\tfrac{3}{4} \div \tfrac{1}{2}$
b. $\tfrac{4}{5} \div \tfrac{3}{10}$
c. $\tfrac{5}{6} \div \tfrac{2}{3}$
d. $\tfrac{4}{9} \div \tfrac{1}{3}$
e. $\tfrac{6}{7} \div \tfrac{3}{7}$
f. $\tfrac{7}{8} \div \tfrac{3}{4}$
4. This is part of Sofia’s homework. She has made a mistake in her solution.
Question: Work out $\tfrac{8}{9} \div \tfrac{4}{5}$
Solution shown: $\tfrac{8}{9} \div \tfrac{4}{5} = \tfrac{9}{8} \times \tfrac{4}{5} = \tfrac{36}{40} = \tfrac{9}{10}$
a. Explain Isaac’s mistake.
b. Work out the correct answer.
👀 Show answer
Q1
- a. $\tfrac{1}{5} \div \tfrac{3}{4} = \tfrac{1}{5} \times \tfrac{4}{3} = \tfrac{4}{15}$
- b. $\tfrac{2}{3} \div \tfrac{6}{7} = \tfrac{2}{3} \times \tfrac{7}{6} = \tfrac{14}{18} = \tfrac{7}{9}$
Q2
- a. $\tfrac{1}{4} \div \tfrac{2}{3} = \tfrac{1}{4} \times \tfrac{3}{2} = \tfrac{3}{8}$
- b. $\tfrac{1}{2} \div \tfrac{3}{5} = \tfrac{1}{2} \times \tfrac{5}{3} = \tfrac{5}{6}$
- c. $\tfrac{3}{8} \div \tfrac{4}{7} = \tfrac{3}{8} \times \tfrac{7}{4} = \tfrac{21}{32}$
- d. $\tfrac{4}{5} \div \tfrac{1}{9} = \tfrac{4}{5} \times 9 = \tfrac{36}{5} = 7 \tfrac{1}{5}$
- e. $\tfrac{3}{5} \div \tfrac{2}{11} = \tfrac{3}{5} \times \tfrac{11}{2} = \tfrac{33}{10} = 3 \tfrac{3}{10}$
- f. $\tfrac{9}{10} \div \tfrac{1}{3} = \tfrac{9}{10} \times 3 = \tfrac{27}{10} = 2 \tfrac{7}{10}$
Q3
- a. $\tfrac{3}{4} \div \tfrac{1}{2} = \tfrac{3}{4} \times 2 = \tfrac{6}{4} = \tfrac{3}{2} = 1 \tfrac{1}{2}$
- b. $\tfrac{4}{5} \div \tfrac{3}{10} = \tfrac{4}{5} \times \tfrac{10}{3} = \tfrac{40}{15} = \tfrac{8}{3} = 2 \tfrac{2}{3}$
- c. $\tfrac{5}{6} \div \tfrac{2}{3} = \tfrac{5}{6} \times \tfrac{3}{2} = \tfrac{15}{12} = \tfrac{5}{4} = 1 \tfrac{1}{4}$
- d. $\tfrac{4}{9} \div \tfrac{1}{3} = \tfrac{4}{9} \times 3 = \tfrac{12}{9} = \tfrac{4}{3} = 1 \tfrac{1}{3}$
- e. $\tfrac{6}{7} \div \tfrac{3}{7} = \tfrac{6}{7} \times \tfrac{7}{3} = \tfrac{6}{3} = 2$
- f. $\tfrac{7}{8} \div \tfrac{3}{4} = \tfrac{7}{8} \times \tfrac{4}{3} = \tfrac{28}{24} = \tfrac{7}{6} = 1 \tfrac{1}{6}$
Q4
- a. Isaac’s mistake: he inverted the wrong fraction. He wrote $\tfrac{8}{9} \div \tfrac{4}{5} = \tfrac{9}{8} \times \tfrac{4}{5}$ instead of keeping $\tfrac{8}{9}$ and inverting $\tfrac{4}{5}$.
- b. Correct calculation: $\tfrac{8}{9} \div \tfrac{4}{5} = \tfrac{8}{9} \times \tfrac{5}{4} = \tfrac{40}{36} = \tfrac{10}{9} = 1 \tfrac{1}{9}$
5. The area of this rectangle is $\tfrac{2}{15}$ m$^2$. The width is $\tfrac{3}{10}$ m. Work out the length of the rectangle.
🔎 Reasoning Tip
Area of a rectangle: The formula for the area of a rectangle is \(\text{Area} = \text{length} \times \text{width}\). Rearranging gives \(\text{Length} = \dfrac{\text{Area}}{\text{width}}\).
👀 Show answer
Area = length × width
$\tfrac{2}{15} = \text{length} \times \tfrac{3}{10}$
So $\text{length} = \tfrac{2}{15} \div \tfrac{3}{10} = \tfrac{2}{15} \times \tfrac{10}{3} = \tfrac{20}{45} = \tfrac{4}{9}$ m
6. Cheng is using fraction cards to make correct calculations.
$\tfrac{7}{12} \div \; ? \; = \tfrac{7}{10}$
Which of these four fraction cards is the correct card for the missing fraction in the division?
Options: $\tfrac{3}{4}, \; \tfrac{3}{10}, \; \tfrac{5}{6}, \; \tfrac{7}{15}$
👀 Show answer
We need $ \tfrac{7}{12} \div x = \tfrac{7}{10}$. Rearranging: $x = \tfrac{7}{12} \div \tfrac{7}{10} = \tfrac{7}{12} \times \tfrac{10}{7} = \tfrac{10}{12} = \tfrac{5}{6}$.
✅ The missing card is $\tfrac{5}{6}$.
🧠 Think like a Mathematician
Question 7:
Look again at Question 6.
a) As a class, discuss the different methods that you used to answer the question.
b) Critique each method by explaining the advantages and disadvantages of each method.
c) Which is the best method that was used? Can you improve this method?
👀 show answer
a) Possible methods might include: converting to decimals, simplifying fractions, or using common denominators.
b) • Converting to decimals: Advantage – easy to compare; Disadvantage – rounding errors may occur.
• Simplifying fractions: Advantage – keeps values exact; Disadvantage – may take extra steps.
• Using common denominators: Advantage – systematic and always works; Disadvantage – can involve large numbers.
c) The best method often depends on the fractions given. For simple comparisons, converting to decimals is fast; for accuracy, using common denominators or simplifying is best. ✅ An improved method is to first simplify fractions, then compare using either decimals or common denominators to balance speed and accuracy.
❓ EXERCISES
8. Arun is looking for general patterns in the fraction division questions. He thinks of two ideas. Are Arun’s ideas correct? Explain your answers. Look back at the questions you have completed in this exercise to help you to explain.
When you divide two proper fractions:
- If the first fraction is bigger than the second fraction, then the answer will be smaller than $1$.
- If the first fraction is smaller than the second fraction, then the answer will be bigger than $1$.
👀 Show answer
Answer (Q8):
Arun’s ideas are correct:
- If the first fraction is bigger, dividing by a smaller fraction makes the answer less than $1$.
- If the first fraction is smaller, dividing by a larger fraction makes the answer greater than $1$.
Examples from earlier exercises support this pattern.
9. Look at this fractions pattern:
| Pattern | Working | Answer |
|---|---|---|
| $\tfrac{1}{2}\times\tfrac{1}{3}$ | $\tfrac{1}{2}\times\tfrac{1}{3}=\tfrac{1}{6}$ | $\tfrac{1}{6}$ |
| $\tfrac{1}{2}\times\tfrac{1}{3}\times\tfrac{1}{4}$ | $\tfrac{1}{6}\times\tfrac{1}{4}=\tfrac{1}{24}$ | $\tfrac{1}{24}$ |
| $\tfrac{1}{2}\times\tfrac{1}{3}\times\tfrac{1}{4}\times\tfrac{1}{5}$ | $\tfrac{1}{24}\times\tfrac{1}{5}=\tfrac{1}{120}$ | $\tfrac{1}{120}$ |
| $\tfrac{1}{2}\times\tfrac{1}{3}\times\tfrac{1}{4}\times\tfrac{1}{5}\times\tfrac{1}{6}$ | $\tfrac{1}{120}\times\tfrac{1}{6}=\tfrac{1}{720}$ | $\tfrac{1}{720}$ |
| $\tfrac{1}{2}\times\tfrac{1}{3}\times\tfrac{1}{4}\times\tfrac{1}{5}\times\tfrac{1}{6}\times\tfrac{1}{7}$ | $\tfrac{1}{720}\times\tfrac{1}{7}=\tfrac{1}{5040}$ | $\tfrac{1}{5040}$ |
At which pattern does the answer become greater than $1$? Write down this answer.
👀 Show answer
Answer (Q9):
The answers decrease as more fractions are multiplied, so they never become greater than $1$ in this sequence. Each multiplication reduces the product further.
🍬 Learning Bridge
Now that you can divide fractions by flipping the second fraction (the reciprocal), multiplying, simplifying, and writing mixed numbers, you’re ready to make the process faster. Next, you’ll apply the same reciprocal idea to divisions like whole number ÷ fraction and fraction ÷ fraction, cancel common factors before multiplying to cut down the working, and use estimation and inverse checks to keep answers sensible and in simplest form.
You already know how to divide an integer by a fraction and also a fraction by a fraction. In both cases, you turn the fraction you are dividing by upside down, and then multiply instead. This is called multiplying by the reciprocal of the fraction. Just as you did in Section 8.3, you can cancel common factors before you multiply, to make the calculations easier.
❓ EXERCISES
10. Copy and complete these divisions. Write each answer in its simplest form and as a mixed number when appropriate.
a. $16 \div \tfrac{4}{7}$
b. $21 \div \tfrac{3}{5}$
c. $14 \div \tfrac{2}{9}$
d. $8 \div \tfrac{4}{11}$
👀 Show answer
Answers:
- a. $16 \div \tfrac{4}{7}=16 \times \tfrac{7}{4}= \tfrac{112}{4}=28$
- b. $21 \div \tfrac{3}{5}=21 \times \tfrac{5}{3}= \tfrac{105}{3}=35$
- c. $14 \div \tfrac{2}{9}=14 \times \tfrac{9}{2}= \tfrac{126}{2}=63$
- d. $8 \div \tfrac{4}{11}=8 \times \tfrac{11}{4}= \tfrac{88}{4}=22$
11. Match each question card (A to E) with the correct answer card (i to v).
A $25 \div \tfrac{5}{8}$ B $22 \div \tfrac{2}{3}$ C $6 \div \tfrac{4}{9}$ D $32 \div \tfrac{6}{13}$ E $42 \div \tfrac{4}{7}$
i $33$ ii $73 \tfrac{1}{2}$ iii $40$ iv $13 \tfrac{1}{2}$ v $69 \tfrac{1}{3}$
👀 Show answer
Answers:
- A $\rightarrow$ iii, since $25 \div \tfrac{5}{8}=25 \times \tfrac{8}{5}=40$
- B $\rightarrow$ i, since $22 \div \tfrac{2}{3}=22 \times \tfrac{3}{2}=33$
- C $\rightarrow$ iv, since $6 \div \tfrac{4}{9}=6 \times \tfrac{9}{4}= \tfrac{27}{2}=13 \tfrac{1}{2}$
- D $\rightarrow$ v, since $32 \div \tfrac{6}{13}=32 \times \tfrac{13}{6}= \tfrac{208}{3}=69 \tfrac{1}{3}$
- E $\rightarrow$ ii, since $42 \div \tfrac{4}{7}=42 \times \tfrac{7}{4}= \tfrac{147}{2}=73 \tfrac{1}{2}$
12. Copy and complete these divisions. Write each answer in its lowest terms and as a mixed number when appropriate.
a. $\tfrac{8}{9} \div \tfrac{4}{7}$
b. $\tfrac{7}{9} \div \tfrac{2}{5}$
c. $\tfrac{6}{7} \div \tfrac{3}{14}$
d. $\tfrac{5}{6} \div \tfrac{15}{24}$
👀 Show answer
Answers:
- a. $\tfrac{8}{9} \div \tfrac{4}{7}= \tfrac{8}{9} \times \tfrac{7}{4}= \tfrac{56}{36}= \tfrac{14}{9}= 1 \tfrac{5}{9}$
- b. $\tfrac{7}{9} \div \tfrac{2}{5}= \tfrac{7}{9} \times \tfrac{5}{2}= \tfrac{35}{18}= 1 \tfrac{17}{18}$
- c. $\tfrac{6}{7} \div \tfrac{3}{14}= \tfrac{6}{7} \times \tfrac{14}{3}= \tfrac{84}{21}=4$
- d. $\tfrac{5}{6} \div \tfrac{15}{24}= \tfrac{5}{6} \times \tfrac{24}{15}= \tfrac{20}{15}= \tfrac{4}{3}= 1 \tfrac{1}{3}$
13. Write these cards in order of answer size, starting with the smallest.
A $25 \div \tfrac{5}{8}$ B $\tfrac{8}{15} \div \tfrac{12}{25}$ C $\tfrac{9}{28} \div \tfrac{15}{42}$ D $\tfrac{6}{7} \div \tfrac{9}{10}$
👀 Show answer
Answers:
- A: $25 \div \tfrac{5}{8} = 25 \times \tfrac{8}{5} = 40$
- B: $\tfrac{8}{15} \div \tfrac{12}{25} = \tfrac{8}{15} \times \tfrac{25}{12} = \tfrac{200}{180} = \tfrac{10}{9} \approx 1.11$
- C: $\tfrac{9}{28} \div \tfrac{15}{42} = \tfrac{9}{28} \times \tfrac{42}{15} = \tfrac{378}{420} = \tfrac{9}{10} = 0.9$
- D: $\tfrac{6}{7} \div \tfrac{9}{10} = \tfrac{6}{7} \times \tfrac{10}{9} = \tfrac{60}{63} = \tfrac{20}{21} \approx 0.95$
Order (smallest to largest): C $(0.9)$, D $(0.95)$, B $(1.11)$, A $(40)$
14. This is part of Jake’s homework.
Example: $2 \tfrac{1}{2} \div 3 \tfrac{4}{7}$
- Change to improper fractions: $\tfrac{5}{2} \div \tfrac{25}{7}$
- Invert and multiply: $\tfrac{5}{2} \times \tfrac{7}{25}$
- Cancel: $\tfrac{1}{2} \times \tfrac{7}{5}$
- Multiply: $\tfrac{7}{10}$
- Check estimate: $\tfrac{7}{10} \approx \tfrac{3}{4}$
Jake’s method is to round each fraction to the nearest whole number to estimate. Use Jake’s method to estimate and work out these divisions. Write each answer in simplest form and as a mixed number when appropriate.
a. $1 \tfrac{1}{2} \div 1 \tfrac{4}{5}$
b. $2 \tfrac{1}{4} \div 1 \tfrac{2}{3}$
c. $4 \tfrac{1}{8} \div 5 \tfrac{1}{6}$
d. $2 \tfrac{2}{3} \div 3 \tfrac{1}{4}$
e. $5 \tfrac{1}{2} \div 2 \tfrac{3}{4}$
f. $4 \tfrac{4}{5} \div 2 \tfrac{2}{3}$
g. $1 \tfrac{1}{4} \div \tfrac{10}{11}$
h. $\tfrac{3}{5} \div 2 \tfrac{1}{10}$
👀 Show answer
Answers:
- a. $1 \tfrac{1}{2} \div 1 \tfrac{4}{5} = \tfrac{3}{2} \div \tfrac{9}{5} = \tfrac{3}{2} \times \tfrac{5}{9} = \tfrac{15}{18} = \tfrac{5}{6}$
- b. $2 \tfrac{1}{4} \div 1 \tfrac{2}{3} = \tfrac{9}{4} \div \tfrac{5}{3} = \tfrac{9}{4} \times \tfrac{3}{5} = \tfrac{27}{20} = 1 \tfrac{7}{20}$
- c. $4 \tfrac{1}{8} \div 5 \tfrac{1}{6} = \tfrac{33}{8} \div \tfrac{31}{6} = \tfrac{33}{8} \times \tfrac{6}{31} = \tfrac{198}{248} = \tfrac{99}{124}$
- d. $2 \tfrac{2}{3} \div 3 \tfrac{1}{4} = \tfrac{8}{3} \div \tfrac{13}{4} = \tfrac{8}{3} \times \tfrac{4}{13} = \tfrac{32}{39}$
- e. $5 \tfrac{1}{2} \div 2 \tfrac{3}{4} = \tfrac{11}{2} \div \tfrac{11}{4} = \tfrac{11}{2} \times \tfrac{4}{11} = 2$
- f. $4 \tfrac{4}{5} \div 2 \tfrac{2}{3} = \tfrac{24}{5} \div \tfrac{8}{3} = \tfrac{24}{5} \times \tfrac{3}{8} = \tfrac{72}{40} = \tfrac{9}{5} = 1 \tfrac{4}{5}$
- g. $1 \tfrac{1}{4} \div \tfrac{10}{11} = \tfrac{5}{4} \div \tfrac{10}{11} = \tfrac{5}{4} \times \tfrac{11}{10} = \tfrac{55}{40} = \tfrac{11}{8} = 1 \tfrac{3}{8}$
- h. $\tfrac{3}{5} \div 2 \tfrac{1}{10} = \tfrac{3}{5} \div \tfrac{21}{10} = \tfrac{3}{5} \times \tfrac{10}{21} = \tfrac{30}{105} = \tfrac{2}{7}$
🧠 Think like a Mathematician
Question 15:
a) Read what Zara says:
“If I divide any positive number by a proper fraction, the answer will always be greater than the original number.”
b) Use specialising to complete these general statements:
- When you divide any positive number by an improper fraction, the answer will always be ……. than the original number.
- When you divide any positive number by a mixed number, the answer will always be ……. than the original number.
🔎 Reasoning Tip
Specialising with examples: You can test cases by trying examples such as \( 3 \div \tfrac{1}{2} \), \( 1 \tfrac{1}{2} \div \tfrac{2}{3} \), \( \tfrac{5}{8} \div \tfrac{1}{6} \), etc.
👀 show answer
a) Zara is correct because dividing by a proper fraction (e.g., $\tfrac{1}{2}$ or $\tfrac{2}{3}$) is the same as multiplying by its reciprocal (e.g., 2 or $\tfrac{3}{2}$), which makes the result larger.
b) i) When dividing by an improper fraction (greater than 1), the answer will always be smaller than the original number.
ii) When dividing by a mixed number (greater than 1), the answer will also be smaller than the original number.
✅ Example: $12 \div \tfrac{1}{3} = 36$ (greater), but $12 \div \tfrac{3}{2} = 8$ (smaller).
❓ EXERCISES
16. Work out these calculations. Before you do each calculation, write down if the answer to the division should be bigger or smaller than the first number in the calculation.
a. $7 \div \tfrac{3}{4}$
b. $4 \tfrac{2}{5} \div 1 \tfrac{1}{10}$
c. $3 \tfrac{2}{3} \div \tfrac{7}{4}$
👀 Show answer
Answers:
- a. Prediction: bigger than $7$. $7 \div \tfrac{3}{4} = 7 \times \tfrac{4}{3} = \tfrac{28}{3} = 9 \tfrac{1}{3}$
- b. Prediction: bigger than $4 \tfrac{2}{5}$. $4 \tfrac{2}{5} = \tfrac{22}{5}, \quad 1 \tfrac{1}{10} = \tfrac{11}{10}$ $\tfrac{22}{5} \div \tfrac{11}{10} = \tfrac{22}{5} \times \tfrac{10}{11} = \tfrac{220}{55} = 4$
- c. Prediction: smaller than $3 \tfrac{2}{3}$. $3 \tfrac{2}{3} = \tfrac{11}{3}$ $\tfrac{11}{3} \div \tfrac{7}{4} = \tfrac{11}{3} \times \tfrac{4}{7} = \tfrac{44}{21} = 2 \tfrac{2}{21}$
🧠 Think like a Mathematician
Question 17:
Arun makes this conjecture:
“If I divide a mixed number by a different mixed number, my answer will always be a mixed number.”
Task: Do you think Arun’s conjecture is true? Show working to support your decision.
👀 show answer
Arun’s conjecture is false.
Example 1: $2 \tfrac{1}{2} \div 1 \tfrac{1}{4}$ $= \dfrac{5}{2} \div \dfrac{5}{4} = \dfrac{5}{2} \times \dfrac{4}{5} = 2$ The answer is a whole number, not a mixed number.
Example 2: $3 \tfrac{1}{2} \div 7$ $= \dfrac{7}{2} \div 7 = \dfrac{7}{2} \times \dfrac{1}{7} = \dfrac{1}{2}$ The answer is a proper fraction, not a mixed number.
✅ So dividing two mixed numbers can result in a whole number, a proper fraction, or a mixed number — not always a mixed number.
❓ EXERCISES
18. This is part of Helen’s homework. She uses inverse operations to check her answer is correct. Work out the answers to these divisions. Use Helen’s method to check your answers are correct.
a. $\tfrac{2}{5} \div \tfrac{3}{7}$
b. $\tfrac{4}{7} \div \tfrac{1}{5}$
c. $\tfrac{6}{7} \div \tfrac{3}{4}$
d. $\tfrac{8}{9} \div \tfrac{4}{5}$
e. $\tfrac{2}{9} \div \tfrac{6}{11}$
f. $\tfrac{10}{11} \div \tfrac{5}{6}$
👀 Show answer
Answers:
- a. $\tfrac{2}{5} \div \tfrac{3}{7} = \tfrac{2}{5} \times \tfrac{7}{3} = \tfrac{14}{15}$
- b. $\tfrac{4}{7} \div \tfrac{1}{5} = \tfrac{4}{7} \times 5 = \tfrac{20}{7} = 2 \tfrac{6}{7}$
- c. $\tfrac{6}{7} \div \tfrac{3}{4} = \tfrac{6}{7} \times \tfrac{4}{3} = \tfrac{24}{21} = \tfrac{8}{7} = 1 \tfrac{1}{7}$
- d. $\tfrac{8}{9} \div \tfrac{4}{5} = \tfrac{8}{9} \times \tfrac{5}{4} = \tfrac{40}{36} = \tfrac{10}{9} = 1 \tfrac{1}{9}$
- e. $\tfrac{2}{9} \div \tfrac{6}{11} = \tfrac{2}{9} \times \tfrac{11}{6} = \tfrac{22}{54} = \tfrac{11}{27}$
- f. $\tfrac{10}{11} \div \tfrac{5}{6} = \tfrac{10}{11} \times \tfrac{6}{5} = \tfrac{60}{55} = \tfrac{12}{11} = 1 \tfrac{1}{11}$
19. The circumference of a circle is $14 \tfrac{1}{7}$ cm. Sofia makes this conjecture:
💡 Sofia says:
“Without actually calculating the answer, I estimate the diameter of the circle to be just under 5 cm.”
a. Explain how Sofia estimated this value for the diameter.
b. Show that Sofia’s estimate is a good estimate of the accurate answer. Use $\pi = \tfrac{22}{7}$.
👀 Show answer
Answer:
- a. Sofia estimated using the formula $C = \pi d$. Since circumference $\approx 15$ and $\pi \approx 3$, she reasoned $d \approx 15 \div 3 = 5$. So the diameter should be just under 5 cm.
- b. Accurate calculation: $C = 14 \tfrac{1}{7} = \tfrac{99}{7}$ cm. Using $C = \pi d$: $d = \dfrac{C}{\pi} = \dfrac{\tfrac{99}{7}}{\tfrac{22}{7}} = \dfrac{99}{22} = 4.5$ cm. So the true diameter is $4.5$ cm, which is indeed just under 5 cm.
20. Work out:
a. $(1 - \tfrac{1}{3}) \div (1 - \tfrac{3}{5})$
b. $(\tfrac{2}{5} \tfrac{3}{10}) \div \tfrac{7}{15}$
c. $5 \tfrac{1}{3} - 1 \tfrac{3}{7} \div \tfrac{5}{14}$
👀 Show answer
Answers:
- a. $(1 - \tfrac{1}{3}) \div (1 - \tfrac{3}{5}) = \tfrac{2}{3} \div \tfrac{2}{5} = \tfrac{2}{3} \times \tfrac{5}{2} = \tfrac{5}{3} = 1 \tfrac{2}{3}$
- b. $(\tfrac{2}{5} \tfrac{3}{10}) \div \tfrac{7}{15} = \tfrac{7}{10} \div \tfrac{7}{15} = \tfrac{7}{10} \times \tfrac{15}{7} = \tfrac{15}{10} = \tfrac{3}{2} = 1 \tfrac{1}{2}$
- c. $5 \tfrac{1}{3} - (1 \tfrac{3}{7} \div \tfrac{5}{14})$ $= \tfrac{16}{3} - \Big(\tfrac{10}{7} \div \tfrac{5}{14}\Big)$ $= \tfrac{16}{3} - \Big(\tfrac{10}{7} \times \tfrac{14}{5}\Big)$ $= \tfrac{16}{3} - \tfrac{140}{35}$ $= \tfrac{16}{3} - 4 = \tfrac{4}{3}$
21. Sebastian uses this formula to work out the average speed of his car, in kilometres per hour, when he knows the distance, in kilometres, and the time in hours.
average speed = $\dfrac{\text{distance}}{\text{time}}$
Sebastian travels $155 \tfrac{5}{8}$ km in $1 \tfrac{1}{4}$ hours. What is Sebastian’s average speed?
👀 Show answer
Answer:
Distance $= 155 \tfrac{5}{8} = \tfrac{1245}{8}$ km
Time $= 1 \tfrac{1}{4} = \tfrac{5}{4}$ h
Average speed $= \tfrac{1245}{8} \div \tfrac{5}{4} = \tfrac{1245}{8} \times \tfrac{4}{5} = \tfrac{4980}{40} = 124.5$ km/h
22. Which is greater: $(2 \tfrac{1}{2} - \tfrac{4}{5}) \div \tfrac{34}{15}$ or $(\tfrac{2}{3})^2 \tfrac{15}{6} \div 5 \tfrac{1}{2}$? Show your working.
👀 Show answer
Answer:
First expression: $(2 \tfrac{1}{2} - \tfrac{4}{5}) \div \tfrac{34}{15}$
$= (\tfrac{5}{2} - \tfrac{4}{5}) \div \tfrac{34}{15}$
$= \tfrac{25}{10} - \tfrac{8}{10} = \tfrac{17}{10}$
$= \tfrac{17}{10} \div \tfrac{34}{15} = \tfrac{17}{10} \times \tfrac{15}{34} = \tfrac{255}{340} = \tfrac{3}{4}$
Second expression: $(\tfrac{2}{3})^2 \tfrac{15}{6} \div 5 \tfrac{1}{2}$
$(\tfrac{2}{3})^2 = \tfrac{4}{9}$
$\tfrac{15}{6} \div 5 \tfrac{1}{2} = \tfrac{15}{6} \div \tfrac{11}{2} = \tfrac{15}{6} \times \tfrac{2}{11} = \tfrac{30}{66} = \tfrac{5}{11}$
Total $= \tfrac{4}{9} \tfrac{5}{11} = \tfrac{44}{99} \tfrac{45}{99} = \tfrac{89}{99} \approx 0.9$
Comparison: $\tfrac{3}{4} = 0.75$, $\tfrac{89}{99} \approx 0.9$
✅ The second expression is greater.
⚠️ Be careful!
When dividing fractions, only the second fraction (the divisor) is turned upside down. The first fraction stays the same.
For example, to work out: $\tfrac{8}{9} \div \tfrac{4}{5}$
The correct method is: $\tfrac{8}{9} \times \tfrac{5}{4} = \tfrac{40}{36} = \tfrac{10}{9} = 1\tfrac{1}{9}$
Incorrect method (mistake shown in the lesson): flipping the first fraction instead, e.g. $\tfrac{9}{8} \times \tfrac{4}{5} = \tfrac{36}{40} = \tfrac{9}{10}$ This gives the wrong result.