Dividing Fractions worksheet with exercises on reciprocals, integer division by fractions, and general fraction division, designed for GCSE Foundation level.

Here are the solutions to the problems on the worksheet, broken down by section.

Section A: Reciprocals

1) Show that $\frac{3}{4} \times \frac{4}{3} = 1$
To multiply fractions, you multiply the top numbers (numerators) together and the bottom numbers (denominators) together.
* Top: $3 \times 4 = 12$
* Bottom: $4 \times 3 = 12$
* Result: $\frac{12}{12}$
Any number divided by itself is equal to 1. So, $\frac{12}{12} = 1$.

2) Fill in the blanks:
To get an answer of 1 when multiplying, you need to use the reciprocal (flip the fraction).
* a) The reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$. ($\frac{2}{3} \times \frac{3}{2} = \frac{6}{6} = 1$)
* b) The reciprocal of $\frac{5}{7}$ is $\frac{7}{5}$. ($\frac{7}{5} \times \frac{5}{7} = \frac{35}{35} = 1$)
* c) The reciprocal of $\frac{1}{2}$ is $\frac{2}{1}$ or just $2$. ($1 = \frac{1}{2} \times 2$)
* d) Think of 8 as $\frac{8}{1}$. Its reciprocal is $\frac{1}{8}$. ($\frac{1}{8} \times 8 = 1$)

Statement: Any number multiplied by its reciprocal is equal to 1.

3) Find the reciprocal of each of the following numbers:
* a) Flip $\frac{6}{11}$ $\rightarrow$ $\frac{11}{6}$
* b) Flip $-\frac{2}{3}$ $\rightarrow$ $-\frac{3}{2}$ (Keep the negative sign)
* c) Think of 5 as $\frac{5}{1}$. Flip it $\rightarrow$ $\frac{1}{5}$
* d) Flip $\frac{1}{2}$ $\rightarrow$ $\frac{2}{1}$ (or just 2)
* e) Flip $\frac{8}{19}$ $\rightarrow$ $\frac{19}{8}$
* f) First, change the mixed number $4\frac{2}{3}$ into an improper fraction: $(4 \times 3) + 2 = 14$, so it is $\frac{14}{3}$. Now flip it $\rightarrow$ $\frac{3}{14}$

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Section B: Dividing integers by fractions

1) Explain how this diagram shows that $1 \div \frac{1}{3} = 3$.
The big yellow rectangle represents the whole number 1. It is cut into three equal pieces. Each piece is labeled $\frac{1}{3}$. Because there are exactly 3 of these $\frac{1}{3}$ pieces inside the whole, it shows that 1 divided by $\frac{1}{3}$ equals 3.

2) Calculate the following:
*Rule: To divide by a fraction, keep the first number, change division to multiplication, and flip the second fraction (Keep-Change-Flip).*

* a) $2 \div \frac{1}{3} = 2 \times \frac{3}{1} = 2 \times 3 =$ 6
* b) $2 \div \frac{2}{3} = 2 \times \frac{3}{2} = \frac{6}{2} =$ 3
* c) $10 \div \frac{2}{3} = 10 \times \frac{3}{2} = \frac{30}{2} =$ 15
* d) $10 \div \frac{2}{5} = 10 \times \frac{5}{2} = \frac{50}{2} =$ 25
* e) $10 \div \frac{3}{5} = 10 \times \frac{5}{3} = \frac{50}{3} =$ $16\frac{2}{3}$
* f) $21 \div 2\frac{1}{3}$
* Change $2\frac{1}{3}$ to improper fraction: $\frac{7}{3}$.
* Equation: $21 \div \frac{7}{3} = 21 \times \frac{3}{7}$.
* $21 \div 7 = 3$, then $3 \times 3 = 9$. Answer: 9

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Section C: Dividing any pair of fractions

1) Calculate:

* a) $\frac{1}{3} \div \frac{1}{3} = \frac{1}{3} \times \frac{3}{1} = \frac{3}{3} =$ 1
* b) $\frac{2}{3} \div \frac{1}{2} = \frac{2}{3} \times \frac{2}{1} = \frac{4}{3} =$ $1\frac{1}{3}$
* c) $4\frac{2}{3} \div \frac{1}{2}$
* Convert $4\frac{2}{3}$ to $\frac{14}{3}$.
* $\frac{14}{3} \times \frac{2}{1} = \frac{28}{3} =$ $9\frac{1}{3}$
* d) $\frac{5}{7} \div \frac{5}{12} = \frac{5}{7} \times \frac{12}{5}$. The 5s cancel out. Answer: $\frac{12}{7}$ (or $1\frac{5}{7}$)
* e) $-\frac{5}{12} \div \frac{4}{9} = -\frac{5}{12} \times \frac{9}{4}$.
* Simplify 12 and 9 by dividing both by 3 (becomes 4 and 3).
* $-\frac{5}{4} \times \frac{3}{4} = -\frac{15}{16}$. Answer: $-\frac{15}{16}$
* f) $2\frac{1}{8} \div \frac{9}{10}$
* Convert $2\frac{1}{8}$ to $\frac{17}{8}$.
* $\frac{17}{8} \times \frac{10}{9} = \frac{170}{72}$.
* Simplify by dividing top and bottom by 2: $\frac{85}{36}$. Answer: $2\frac{13}{36}$
* g) $\frac{9}{11} \div \frac{9}{11}$. Any number divided by itself is 1.
* h) $\frac{7}{12} \div \frac{3}{4} \div \frac{1}{2}$
* Do the first part: $\frac{7}{12} \times \frac{4}{3}$. Simplify 12 and 4 (becomes 3 and 1). Result: $\frac{7}{3} \times \frac{1}{3} = \frac{7}{9}$.
* Now do the second part: $\frac{7}{9} \div \frac{1}{2} = \frac{7}{9} \times \frac{2}{1} = \frac{14}{9}$. Answer: $1\frac{5}{9}$
* i) $3\frac{1}{7} \div 5\frac{1}{2}$
* Convert to improper fractions: $\frac{22}{7} \div \frac{11}{2}$.
* Flip and multiply: $\frac{22}{7} \times \frac{2}{11}$.
* Simplify 22 and 11 (becomes 2 and 1).
* $\frac{2}{7} \times \frac{2}{1} = \frac{4}{7}$. Answer: $\frac{4}{7}$

Final Answer:
Section A
1) Shown above ($\frac{12}{12}=1$)
2) a) $\frac{3}{2}$, b) $\frac{7}{5}$, c) 2, d) $\frac{1}{8}$. Statement: reciprocal
3) a) $\frac{11}{6}$, b) $-\frac{3}{2}$, c) $\frac{1}{5}$, d) 2, e) $\frac{19}{8}$, f) $\frac{3}{14}$

Section B
1) The diagram shows one whole split into three parts of size $\frac{1}{3}$, meaning there are 3 thirds in 1.
2) a) 6, b) 3, c) 15, d) 25, e) $16\frac{2}{3}$, f) 9

Section C
1) a) 1, b) $1\frac{1}{3}$, c) $9\frac{1}{3}$, d) $1\frac{5}{7}$, e) $-\frac{15}{16}$, f) $2\frac{13}{36}$, g) 1, h) $1\frac{5}{9}$, i) $\frac{4}{7}$