Dividing an integer by a fraction
🎯 In this topic you will
- Divide an integer by a proper fraction
🧠 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.
Look at this diagram.
It shows three rectangles, each divided in half.
When you work out $3 \div \tfrac{1}{2}$, the question is asking you ‘How many halves are in three?’
You can see that there are six, so $3 \div \tfrac{1}{2} = 6$.
Another method you can use is to turn the fraction upside down, then multiply by the integer.
This is called multiplying by the reciprocal of the fraction.
So, $3 \div \tfrac{1}{2} = 3 \times \tfrac{2}{1} = \tfrac{6}{1} = 6$.
🔎 Reasoning Tip
Seedlings: Seedlings are seeds that are just starting to grow into plants.
❓ EXERCISES
1. Work out the answers to these calculations. Use the diagrams to help you.
a. $2 \div \dfrac{1}{3}$
b. $4 \div \dfrac{1}{2}$
c. $3 \div \dfrac{1}{4}$
d. $5 \div \dfrac{1}{5}$
👀 Show answer
a. $2 \div \dfrac{1}{3} = 2 \times 3 = 6$
b. $4 \div \dfrac{1}{2} = 4 \times 2 = 8$
c. $3 \div \dfrac{1}{4} = 3 \times 4 = 12$
d. $5 \div \dfrac{1}{5} = 5 \times 5 = 25$
2. Read what Sofia says about dividing an integer by a unit fraction.
The quick way to divide an integer by a unit fraction is to multiply the integer by the denominator of the fraction.
a. Can you explain why Sofia’s method works?
b. Check your answers to Question 1 using Sofia’s method.
c. Use Sofia’s method to work out
i. $12 \div \dfrac{1}{3}$
ii. $25 \div \dfrac{1}{4}$
iii. $8 \div \dfrac{1}{9}$
👀 Show answer
a. Sofia’s method works because dividing by $\dfrac{1}{n}$ is the same as multiplying by $n$. This uses the reciprocal relationship between multiplication and division.
b. Answers to Question 1 match using this method.
c.
i. $12 \div \dfrac{1}{3} = 12 \times 3 = 36$
ii. $25 \div \dfrac{1}{4} = 25 \times 4 = 100$
iii. $8 \div \dfrac{1}{9} = 8 \times 9 = 72$
3. The area of a rectangle is $16\,m^2$.
The width of the rectangle is $\dfrac{1}{5}\,m$.
What is the length of the rectangle?
🔎 Reasoning Tip
Rectangle formula: Area of a rectangle = length × width, so length = area ÷ width.
👀 Show answer
Area $= \text{length} \times \text{width}$
$16 = \text{length} \times \dfrac{1}{5}$
$\text{length} = 16 \div \dfrac{1}{5} = 16 \times 5 = 80\,m$
4. Kai uses this formula to work out the average speed of a car in kilometres per hour (km/h), when he knows the distance it has travelled and the time it has taken.
$\text{speed} = \dfrac{\text{distance}}{\text{time}}$
Work out the average speed of these cars. The first one has been done for you.
a. distance $=30$ km, time $=\dfrac{1}{4}$ hour
So, $\text{speed}=30 \div \dfrac{1}{4} = 30 \times 4 = 120$ km/h
b. distance $=45$ km, time $=\dfrac{1}{2}$ hour
c. distance $=16$ km, time $=\dfrac{1}{6}$ hour
👀 Show answer
b. $\text{speed} = 45 \div \dfrac{1}{2} = 45 \times 2 = 90$ km/h
c. $\text{speed} = 16 \div \dfrac{1}{6} = 16 \times 6 = 96$ km/h
🧠 Think like a Mathematician
Task: Work through the following division problems using the diagrams provided. Then reflect on which method you find most effective.
Questions:
👀 show answer
- a) Think of the question as “How many $\dfrac{2}{3}$ are in 2?”. There are 3 groups of $\dfrac{2}{3}$ in 2, so $2 \div \dfrac{2}{3} = 3$.
- b) Think of the question as “How many $\dfrac{3}{4}$ are in 3?”. There are 4 groups of $\dfrac{3}{4}$ in 3, so $3 \div \dfrac{3}{4} = 4$.
- c) Either method works, but many learners prefer the visual model because it shows how repeated groups of a fraction fit into a whole number. Both lead to the same conclusion: dividing by a fraction is equivalent to multiplying by its reciprocal.
❓ EXERCISES
6. Work out the answers to these calculations. Use the diagrams to help you.
a. $4 \div \dfrac{2}{3}$
b. $6 \div \dfrac{3}{4}$
c. $4 \div \dfrac{2}{5}$
d. $8 \div \dfrac{4}{7}$
👀 Show answer
a. $4 \div \dfrac{2}{3} = 4 \times \dfrac{3}{2} = \dfrac{12}{2} = 6$
b. $6 \div \dfrac{3}{4} = 6 \times \dfrac{4}{3} = \dfrac{24}{3} = 8$
c. $4 \div \dfrac{2}{5} = 4 \times \dfrac{5}{2} = \dfrac{20}{2} = 10$
d. $8 \div \dfrac{4}{7} = 8 \times \dfrac{7}{4} = \dfrac{56}{4} = 14$
🧠 Think like a Mathematician
Task: Work through the following division questions, using both diagram and reciprocal methods. Reflect on when each method is more effective.
Questions:
👀 show answer
- a) The diagram shows that there are 4 full groups of $\dfrac{2}{3}$ in 3, plus half of another group. Therefore, $3 \div \dfrac{2}{3} = 4\dfrac{1}{2}$.
- b) The diagram shows that there are 5 groups of $\dfrac{3}{4}$ in 4. So $4 \div \dfrac{3}{4} = 5\dfrac{1}{3}$ (or $\dfrac{16}{3}$).
- c) Reciprocal method: $4 \div \dfrac{3}{4} = 4 \times \dfrac{4}{3} = \dfrac{16}{3} = 5\dfrac{1}{3}$, which matches part b.
- d) If answers differ, the likely mistake is in counting groups in the diagram or misapplying the reciprocal method. Both should agree.
- e) The diagram method is helpful for building understanding and visualising fractional groups. The reciprocal method is faster and more efficient for larger numbers or abstract calculations.
❓ EXERCISES
8. Work out the answers to these calculations. Use the reciprocal method.
The first two have been started for you.
a. $11 \div \dfrac{3}{4} = 11 \times \dfrac{4}{3}$
b. $9 \div \dfrac{5}{6} = 9 \times \dfrac{6}{5}$
c. $7 \div \dfrac{4}{5}$
d. $12 \div \dfrac{7}{10}$
e. $10 \div \dfrac{4}{11}$
👀 Show answer
a. $11 \div \dfrac{3}{4} = 11 \times \dfrac{4}{3} = \dfrac{44}{3} = 14\dfrac{2}{3}$
b. $9 \div \dfrac{5}{6} = 9 \times \dfrac{6}{5} = \dfrac{54}{5} = 10\dfrac{4}{5}$
c. $7 \div \dfrac{4}{5} = 7 \times \dfrac{5}{4} = \dfrac{35}{4} = 8\dfrac{3}{4}$
d. $12 \div \dfrac{7}{10} = 12 \times \dfrac{10}{7} = \dfrac{120}{7} = 17\dfrac{1}{7}$
e. $10 \div \dfrac{4}{11} = 10 \times \dfrac{11}{4} = \dfrac{110}{4} = 27\dfrac{1}{2}$
9. This is part of Anil’s homework.
You can see that he simplified the improper fraction to its lowest terms before he changed it into a mixed number.
Read what Marcus says.
I use a different method. I change the improper fraction to a mixed number, and then simplify the fraction to its lowest terms like this.
$\dfrac{50}{4} = 12\dfrac{2}{4} = 12\dfrac{1}{2}$
a. Whose method do you prefer, Anil’s or Marcus’s? Explain why.
b. Work out these calculations. Give each answer as a mixed number in its lowest terms.
i. $6 \div \dfrac{4}{7}$
ii. $4 \div \dfrac{6}{11}$
iii. $12 \div \dfrac{9}{10}$
iv. $9 \div \dfrac{12}{13}$
👀 Show answer
a. Marcus’s method is often clearer because you see the mixed number form directly, but Anil’s method is efficient because he simplifies earlier. Preference may depend on which is easier for you to follow.
b.
i. $6 \div \dfrac{4}{7} = 6 \times \dfrac{7}{4} = \dfrac{42}{4} = \dfrac{21}{2} = 10\dfrac{1}{2}$
ii. $4 \div \dfrac{6}{11} = 4 \times \dfrac{11}{6} = \dfrac{44}{6} = \dfrac{22}{3} = 7\dfrac{1}{3}$
iii. $12 \div \dfrac{9}{10} = 12 \times \dfrac{10}{9} = \dfrac{120}{9} = \dfrac{40}{3} = 13\dfrac{1}{3}$
iv. $9 \div \dfrac{12}{13} = 9 \times \dfrac{13}{12} = \dfrac{117}{12} = \dfrac{39}{4} = 9\dfrac{3}{4}$
10. Sofia is looking for patterns in the division questions.
She has come up with two ideas.
Are Sofia’s ideas correct? Explain your answers.
Look back at the questions you have done in this exercise to help you explain.
1. When you divide an integer by a proper fraction, the answer is always bigger than the integer you started with.
2. When you divide an integer by two different proper fractions, the larger fraction will give you the larger answer.
👀 Show answer
Answer:
1. Yes, dividing by a proper fraction (less than $1$) is the same as multiplying by a number greater than $1$, so the result is always bigger than the starting integer.
2. No, dividing by a larger proper fraction gives a smaller result. For example, $12 \div \dfrac{3}{4} = 16$, but $12 \div \dfrac{2}{3} = 18$. Since $\dfrac{3}{4} > \dfrac{2}{3}$, the larger fraction actually gives the smaller answer.
11a. Here is a sequence of calculations.
$1 \div \dfrac{1}{6},\; 2 \div \dfrac{1}{6},\; 3 \div \dfrac{1}{6},\; 4 \div \dfrac{1}{6},\;\dots$
i. Work out the sequence of answers.
ii. Write the next two terms of the sequence.
iii. Describe the sequence of answers in words.
👀 Show answer
i. $1 \div \dfrac{1}{6} = 6,\; 2 \div \dfrac{1}{6} = 12,\; 3 \div \dfrac{1}{6} = 18,\; 4 \div \dfrac{1}{6} = 24$
ii. Next two terms: $30,\; 36$
iii. The answers form the sequence of multiples of $6$.
11b. Here is a different sequence of calculations.
$1 \div \dfrac{2}{6},\; 2 \div \dfrac{2}{6},\; 3 \div \dfrac{2}{6},\; 4 \div \dfrac{2}{6},\;\dots$
i. Work out the sequence of answers.
ii. Write the next two terms of the sequence.
iii. Describe the sequence of answers in words.
👀 Show answer
i. $1 \div \dfrac{2}{6} = 3,\; 2 \div \dfrac{2}{6} = 6,\; 3 \div \dfrac{2}{6} = 9,\; 4 \div \dfrac{2}{6} = 12$
ii. Next two terms: $15,\; 18$
iii. The answers form the sequence of multiples of $3$.
11c. Compare your sequences of answers in parts a and b. What do you notice?
Explain why this happens.
👀 Show answer
The sequence in part (a) increases in steps of $6$, while in part (b) it increases in steps of $3$. This happens because dividing by $\dfrac{1}{6}$ is the same as multiplying by $6$, whereas dividing by $\dfrac{2}{6}=\dfrac{1}{3}$ is the same as multiplying by $3$.
11d. Look at your answers to part bi and, without actually completing the calculations, write down the sequence of answers for this sequence of calculations:
$1 \div \dfrac{3}{6},\; 2 \div \dfrac{3}{6},\; 3 \div \dfrac{3}{6},\; 4 \div \dfrac{3}{6},\;\dots$
Explain how you worked out your answer.
👀 Show answer
$\dfrac{3}{6} = \dfrac{1}{2}$, so dividing by $\dfrac{3}{6}$ is the same as multiplying by $2$.
The sequence is $2,\; 4,\; 6,\; 8,\;\dots$
11e. Here is another sequence of answers for a sequence of calculations.
Calculations: $1 \div \dfrac{1}{15},\; 2 \div \dfrac{1}{15},\; 3 \div \dfrac{1}{15},\; 4 \div \dfrac{1}{15},\;\dots$
Answers: $15,\; 30,\; 45,\; 60,\;\dots$
Use this information to write down the sequence of answers for this sequence of calculations:
$1 \div \dfrac{5}{15},\; 2 \div \dfrac{5}{15},\; 3 \div \dfrac{5}{15},\; 4 \div \dfrac{5}{15},\;\dots$
Explain how you worked out your answer.
👀 Show answer
$\dfrac{5}{15} = \dfrac{1}{3}$, so dividing by $\dfrac{5}{15}$ is the same as multiplying by $3$.
The sequence is $3,\; 6,\; 9,\; 12,\;\dots$
⚠️ Be careful! Dividing by a fraction
- Only flip the fraction you are dividing by (the divisor). Do not flip the integer.
- The answer will always be larger than the integer when dividing by a proper fraction (e.g. $3 \div \tfrac{1}{2} = 6$).
- Use the denominator directly when dividing by a unit fraction: $n \div \tfrac{1}{d} = n \times d$.
- Simplify at the end and write as a mixed number if needed. Example: $10 \div \tfrac{3}{4} = \tfrac{40}{3} = 13\tfrac{1}{3}$.
- Check with estimation: $10 \div \tfrac{3}{4}$ should be a bit more than $10 \div 1 = 10$.