The Multiplicative Inverse - Maths with Mum - Free Printable

Problem Analysis:

The task involves finding the multiplicative inverse of given numbers. The multiplicative inverse of a number \( x \) is defined as a number \( y \) such that:

\[
x \cdot y = 1
\]

For whole numbers and fractions, the multiplicative inverse can be determined as follows:
1. For a whole number \( n \), its multiplicative inverse is \( \frac{1}{n} \).
2. For a fraction \( \frac{a}{b} \), its multiplicative inverse is \( \frac{b}{a} \).

Solution Explanation:

#### Part 1: Multiplicative Inverse of Whole Numbers
The given whole numbers are:
- \( 5 \)
- \( 9 \)
- \( 3 \)
- \( 12 \)
- \( 25 \)
- \( 19 \)
- \( 1 \)
- \( 8 \)

The multiplicative inverse of each whole number \( n \) is \( \frac{1}{n} \). Let's compute them step by step:

1. \( 5 \):
\[
\text{Multiplicative inverse} = \frac{1}{5}
\]

2. \( 9 \):
\[
\text{Multiplicative inverse} = \frac{1}{9}
\]

3. \( 3 \):
\[
\text{Multiplicative inverse} = \frac{1}{3}
\]

4. \( 12 \):
\[
\text{Multiplicative inverse} = \frac{1}{12}
\]

5. \( 25 \):
\[
\text{Multiplicative inverse} = \frac{1}{25}
\]

6. \( 19 \):
\[
\text{Multiplicative inverse} = \frac{1}{19}
\]

7. \( 1 \):
\[
\text{Multiplicative inverse} = \frac{1}{1} = 1
\]

8. \( 8 \):
\[
\text{Multiplicative inverse} = \frac{1}{8}
\]

#### Part 2: Multiplicative Inverse of Fractions
The given fractions are:
- \( \frac{2}{3} \)
- \( \frac{4}{9} \)
- \( \frac{1}{6} \)
- \( \frac{12}{7} \)
- \( \frac{3}{11} \)
- \( \frac{4}{3} \)
- \( \frac{8}{9} \)
- \( \frac{2}{17} \)

The multiplicative inverse of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \). Let's compute them step by step:

1. \( \frac{2}{3} \):
\[
\text{Multiplicative inverse} = \frac{3}{2}
\]

2. \( \frac{4}{9} \):
\[
\text{Multiplicative inverse} = \frac{9}{4}
\]

3. \( \frac{1}{6} \):
\[
\text{Multiplicative inverse} = \frac{6}{1} = 6
\]

4. \( \frac{12}{7} \):
\[
\text{Multiplicative inverse} = \frac{7}{12}
\]

5. \( \frac{3}{11} \):
\[
\text{Multiplicative inverse} = \frac{11}{3}
\]

6. \( \frac{4}{3} \):
\[
\text{Multiplicative inverse} = \frac{3}{4}
\]

7. \( \frac{8}{9} \):
\[
\text{Multiplicative inverse} = \frac{9}{8}
\]

8. \( \frac{2}{17} \):
\[
\text{Multiplicative inverse} = \frac{17}{2}
\]

Final Answer:

The multiplicative inverses are as follows:

#### Whole Numbers:
\[
\begin{aligned}
&\text{(a)} \ 5 \rightarrow \frac{1}{5} \\
&\text{(b)} \ 9 \rightarrow \frac{1}{9} \\
&\text{(c)} \ 3 \rightarrow \frac{1}{3} \\
&\text{(d)} \ 12 \rightarrow \frac{1}{12} \\
&\text{(e)} \ 25 \rightarrow \frac{1}{25} \\
&\text{(f)} \ 19 \rightarrow \frac{1}{19} \\
&\text{(g)} \ 1 \rightarrow 1 \\
&\text{(h)} \ 8 \rightarrow \frac{1}{8}
\end{aligned}
\]

#### Fractions:
\[
\begin{aligned}
&\text{(i)} \ \frac{2}{3} \rightarrow \frac{3}{2} \\
&\text{(j)} \ \frac{4}{9} \rightarrow \frac{9}{4} \\
&\text{(k)} \ \frac{1}{6} \rightarrow 6 \\
&\text{(l)} \ \frac{12}{7} \rightarrow \frac{7}{12} \\
&\text{(m)} \ \frac{3}{11} \rightarrow \frac{11}{3} \\
&\text{(n)} \ \frac{4}{3} \rightarrow \frac{3}{4} \\
&\text{(o)} \ \frac{8}{9} \rightarrow \frac{9}{8} \\
&\text{(p)} \ \frac{2}{17} \rightarrow \frac{17}{2}
\end{aligned}
\]

\boxed{
\begin{aligned}
&\text{Whole Numbers:} \\
&\text{(a)} \ 5 \rightarrow \frac{1}{5}, \ \text{(b)} \ 9 \rightarrow \frac{1}{9}, \ \text{(c)} \ 3 \rightarrow \frac{1}{3}, \ \text{(d)} \ 12 \rightarrow \frac{1}{12}, \\
&\text{(e)} \ 25 \rightarrow \frac{1}{25}, \ \text{(f)} \ 19 \rightarrow \frac{1}{19}, \ \text{(g)} \ 1 \rightarrow 1, \ \text{(h)} \ 8 \rightarrow \frac{1}{8} \\
&\text{Fractions:} \\
&\text{(i)} \ \frac{2}{3} \rightarrow \frac{3}{2}, \ \text{(j)} \ \frac{4}{9} \rightarrow \frac{9}{4}, \ \text{(k)} \ \frac{1}{6} \rightarrow 6, \ \text{(l)} \ \frac{12}{7} \rightarrow \frac{7}{12}, \\
&\text{(m)} \ \frac{3}{11} \rightarrow \frac{11}{3}, \ \text{(n)} \ \frac{4}{3} \rightarrow \frac{3}{4}, \ \text{(o)} \ \frac{8}{9} \rightarrow \frac{9}{8}, \ \text{(p)} \ \frac{2}{17} \rightarrow \frac{17}{2}
\end{aligned}
}