Sec 2 Maths · Algebraic fractions and formulae

Multiplying and dividing algebraic fractions questions with answers

Practice questions on multiplying and dividing algebraic fractions, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

Simplify $\dfrac{11mn}{2} \times \dfrac{6mn}{11}$ , giving your answer in its simplest form.

Step 1 of 3

Multiply the tops together and the bottoms together (write the letters out in full)

\dfrac{11mn}{2} \times \dfrac{6mn}{11} = \dfrac{66 \times m \times m \times n \times n}{22}

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Questions

Question 1Multiply: cancel the numbers
Simplify $\dfrac{9q}{7p} \times \dfrac{q}{3p}$ , giving your answer in its simplest form.
Show the worked solution
1. Step 1: Multiply the tops together and the bottoms together (write the letters out in full) $\dfrac{9q}{7p} \times \dfrac{q}{3p} = \dfrac{9 \times q \times q}{21 \times p \times p}$
2. Step 2: Cancel the numbers: divide the top and the bottom by their HCFThe HCF of $9$ and $21$ is $3$ . $\dfrac{9}{21} = \dfrac{3}{7}$
3. Step 3: Cancel each letter that is on both the top and the bottom, and write the answer $= \dfrac{3q^2}{7p^2}$
Question 2Multiply: cancel the numbers
Simplify $\dfrac{4q}{3p^2} \times \dfrac{9q}{5}$ , giving your answer in its simplest form.
Show the worked solution
1. Step 1: Multiply the tops together and the bottoms together (write the letters out in full) $\dfrac{4q}{3p^2} \times \dfrac{9q}{5} = \dfrac{36 \times q \times q}{15 \times p \times p}$
2. Step 2: Cancel the numbers: divide the top and the bottom by their HCFThe HCF of $36$ and $15$ is $3$ . $\dfrac{36}{15} = \dfrac{12}{5}$
3. Step 3: Cancel each letter that is on both the top and the bottom, and write the answer $= \dfrac{12q^2}{5p^2}$
Question 3Multiply: cancel numbers and letters
Simplify $\dfrac{5}{12m^2} \times \dfrac{11m}{5}$ , giving your answer in its simplest form.
Show the worked solution
1. Step 1: Multiply the tops together and the bottoms together (write the letters out in full) $\dfrac{5}{12m^2} \times \dfrac{11m}{5} = \dfrac{55 \times m}{60 \times m \times m}$
2. Step 2: Cancel the numbers: divide the top and the bottom by their HCFThe HCF of $55$ and $60$ is $5$ . $\dfrac{55}{60} = \dfrac{11}{12}$
3. Step 3: Cancel each letter that is on both the top and the bottom, and write the answer $= \dfrac{11}{12m}$
Question 4Multiply: cancel numbers and letters
Simplify $\dfrac{11}{6x} \times \dfrac{4xy}{9}$ , giving your answer in its simplest form.
Show the worked solution
1. Step 1: Multiply the tops together and the bottoms together (write the letters out in full) $\dfrac{11}{6x} \times \dfrac{4xy}{9} = \dfrac{44 \times x \times y}{54 \times x}$
2. Step 2: Cancel the numbers: divide the top and the bottom by their HCFThe HCF of $44$ and $54$ is $2$ . $\dfrac{44}{54} = \dfrac{22}{27}$
3. Step 3: Cancel each letter that is on both the top and the bottom, and write the answer $= \dfrac{22y}{27}$
Question 5Divide: flip the second fraction and multiply
Simplify $\dfrac{3p}{4q} \div \dfrac{7}{8pq^2}$ , giving your answer in its simplest form.
Show the worked solution
1. Step 1: Change the division into a multiplication $\dfrac{3p}{4q} \div \dfrac{7}{8pq^2} = \dfrac{3p}{4q} \times \dfrac{8pq^2}{7}$
2. Step 2: Multiply the tops together and the bottoms together (write the letters out in full) $= \dfrac{24 \times p \times p \times q \times q}{28 \times q}$
3. Step 3: Cancel the numbers: divide the top and the bottom by their HCFThe HCF of $24$ and $28$ is $4$ . $\dfrac{24}{28} = \dfrac{6}{7}$
4. Step 4: Cancel each letter that is on both the top and the bottom, and write the answer $= \dfrac{6p^2q}{7}$
Question 6Divide: flip the second fraction and multiply
Simplify $\dfrac{6b}{11a} \div \dfrac{10b}{7a^2}$ , giving your answer in its simplest form.
Show the worked solution
1. Step 1: Change the division into a multiplication $\dfrac{6b}{11a} \div \dfrac{10b}{7a^2} = \dfrac{6b}{11a} \times \dfrac{7a^2}{10b}$
2. Step 2: Multiply the tops together and the bottoms together (write the letters out in full) $= \dfrac{42 \times a \times a \times b}{110 \times a \times b}$
3. Step 3: Cancel the numbers: divide the top and the bottom by their HCFThe HCF of $42$ and $110$ is $2$ . $\dfrac{42}{110} = \dfrac{21}{55}$
4. Step 4: Cancel each letter that is on both the top and the bottom, and write the answer $= \dfrac{21a}{55}$

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Worked example

Simplify

\dfrac{11mn}{2} \times \dfrac{6mn}{11}

, giving your answer in its simplest form.

Step 1 of 3

Multiply the tops together and the bottoms together (write the letters out in full)

\dfrac{11mn}{2} \times \dfrac{6mn}{11} = \dfrac{66 \times m \times m \times n \times n}{22}

Questions

Question 1Multiply: cancel the numbers

Simplify $\dfrac{9q}{7p} \times \dfrac{q}{3p}$ , giving your answer in its simplest form.

Show the worked solution

Step 1: Multiply the tops together and the bottoms together (write the letters out in full) $\dfrac{9q}{7p} \times \dfrac{q}{3p} = \dfrac{9 \times q \times q}{21 \times p \times p}$
Step 2: Cancel the numbers: divide the top and the bottom by their HCFThe HCF of $9$ and $21$ is $3$ . $\dfrac{9}{21} = \dfrac{3}{7}$
Step 3: Cancel each letter that is on both the top and the bottom, and write the answer $= \dfrac{3q^2}{7p^2}$

Question 2Multiply: cancel the numbers

Simplify $\dfrac{4q}{3p^2} \times \dfrac{9q}{5}$ , giving your answer in its simplest form.

Show the worked solution

Step 1: Multiply the tops together and the bottoms together (write the letters out in full) $\dfrac{4q}{3p^2} \times \dfrac{9q}{5} = \dfrac{36 \times q \times q}{15 \times p \times p}$
Step 2: Cancel the numbers: divide the top and the bottom by their HCFThe HCF of $36$ and $15$ is $3$ . $\dfrac{36}{15} = \dfrac{12}{5}$
Step 3: Cancel each letter that is on both the top and the bottom, and write the answer $= \dfrac{12q^2}{5p^2}$

Question 3Multiply: cancel numbers and letters

Simplify $\dfrac{5}{12m^2} \times \dfrac{11m}{5}$ , giving your answer in its simplest form.

Show the worked solution

Step 1: Multiply the tops together and the bottoms together (write the letters out in full) $\dfrac{5}{12m^2} \times \dfrac{11m}{5} = \dfrac{55 \times m}{60 \times m \times m}$
Step 2: Cancel the numbers: divide the top and the bottom by their HCFThe HCF of $55$ and $60$ is $5$ . $\dfrac{55}{60} = \dfrac{11}{12}$
Step 3: Cancel each letter that is on both the top and the bottom, and write the answer $= \dfrac{11}{12m}$

Question 4Multiply: cancel numbers and letters

Simplify $\dfrac{11}{6x} \times \dfrac{4xy}{9}$ , giving your answer in its simplest form.

Show the worked solution

Step 1: Multiply the tops together and the bottoms together (write the letters out in full) $\dfrac{11}{6x} \times \dfrac{4xy}{9} = \dfrac{44 \times x \times y}{54 \times x}$
Step 2: Cancel the numbers: divide the top and the bottom by their HCFThe HCF of $44$ and $54$ is $2$ . $\dfrac{44}{54} = \dfrac{22}{27}$
Step 3: Cancel each letter that is on both the top and the bottom, and write the answer $= \dfrac{22y}{27}$

Question 5Divide: flip the second fraction and multiply

Simplify $\dfrac{3p}{4q} \div \dfrac{7}{8pq^2}$ , giving your answer in its simplest form.

Show the worked solution

Step 1: Change the division into a multiplication $\dfrac{3p}{4q} \div \dfrac{7}{8pq^2} = \dfrac{3p}{4q} \times \dfrac{8pq^2}{7}$
Step 2: Multiply the tops together and the bottoms together (write the letters out in full) $= \dfrac{24 \times p \times p \times q \times q}{28 \times q}$
Step 3: Cancel the numbers: divide the top and the bottom by their HCFThe HCF of $24$ and $28$ is $4$ . $\dfrac{24}{28} = \dfrac{6}{7}$
Step 4: Cancel each letter that is on both the top and the bottom, and write the answer $= \dfrac{6p^2q}{7}$

Question 6Divide: flip the second fraction and multiply

Simplify $\dfrac{6b}{11a} \div \dfrac{10b}{7a^2}$ , giving your answer in its simplest form.

Show the worked solution

Step 1: Change the division into a multiplication $\dfrac{6b}{11a} \div \dfrac{10b}{7a^2} = \dfrac{6b}{11a} \times \dfrac{7a^2}{10b}$
Step 2: Multiply the tops together and the bottoms together (write the letters out in full) $= \dfrac{42 \times a \times a \times b}{110 \times a \times b}$
Step 3: Cancel the numbers: divide the top and the bottom by their HCFThe HCF of $42$ and $110$ is $2$ . $\dfrac{42}{110} = \dfrac{21}{55}$
Step 4: Cancel each letter that is on both the top and the bottom, and write the answer $= \dfrac{21a}{55}$