Find the multiplicative inverse (or reciprocal) of each of the following rational numbers :
(i) `6/25`, (ii) `15/11`, (iii) `-8/9`, (iv) `-23/16`, (v) 12, (vi) `1/10`, (vii) `-6`, (viii) `-1`, (ix) `-1/5`, (x) `-7/(-9)`

Answer

To find the multiplicative inverse (or reciprocal) of each of the given rational numbers, we will follow the rule that states the reciprocal of a fraction \( \frac{a}{b} \) is \( \frac{b}{a} \). If the number is a whole number, we can express it as a fraction by putting it over 1.

Let's solve each part step by step:

### Step-by-Step Solution:

**(i) Find the reciprocal of \( \frac{6}{25} \)**
- The reciprocal is obtained by flipping the fraction:
\[ \text{Reciprocal} = \frac{25}{6} \]

**(ii) Find the reciprocal of \( \frac{15}{11} \)**
- Flip the fraction:
\[ \text{Reciprocal} = \frac{11}{15} \]

**(iii) Find the reciprocal of \( -\frac{8}{9} \)**
- Flip the fraction, keeping the negative sign:
\[ \text{Reciprocal} = -\frac{9}{8} \]

**(iv) Find the reciprocal of \( -\frac{23}{16} \)**
- Flip the fraction, keeping the negative sign:
\[ \text{Reciprocal} = -\frac{16}{23} \]

**(v) Find the reciprocal of \( 12 \)**
- Express 12 as a fraction: \( \frac{12}{1} \)
- Flip the fraction:
\[ \text{Reciprocal} = \frac{1}{12} \]

**(vi) Find the reciprocal of \( \frac{1}{10} \)**
- Flip the fraction:
\[ \text{Reciprocal} = \frac{10}{1} = 10 \]

**(vii) Find the reciprocal of \( -6 \)**
- Express -6 as a fraction: \( -\frac{6}{1} \)
- Flip the fraction:
\[ \text{Reciprocal} = -\frac{1}{6} \]

**(viii) Find the reciprocal of \( -1 \)**
- Express -1 as a fraction: \( -\frac{1}{1} \)
- The reciprocal is:
\[ \text{Reciprocal} = -1 \]

**(ix) Find the reciprocal of \( -\frac{1}{5} \)**
- Flip the fraction, keeping the negative sign:
\[ \text{Reciprocal} = -\frac{5}{1} = -5 \]

**(x) Find the reciprocal of \( -\frac{7}{-9} \)**
- Two negatives make a positive, so it simplifies to \( \frac{7}{9} \)
- Flip the fraction:
\[ \text{Reciprocal} = \frac{9}{7} \]

### Final Answers:
1. \( \frac{25}{6} \)
2. \( \frac{11}{15} \)
3. \( -\frac{9}{8} \)
4. \( -\frac{16}{23} \)
5. \( \frac{1}{12} \)
6. \( 10 \)
7. \( -\frac{1}{6} \)
8. \( -1 \)
9. \( -5 \)
10. \( \frac{9}{7} \)