Add each pair of rational numbers, given below, and show that their addition (sum) is also a rational number :
(i) `(7)/(15) "and" (3)/(5)`
(ii) `(2)/(7) "and" 2`
(iii) `(3)/(8) "and" (-5)/(12)`
(iv) `(7)/(-15) "and" (2)/(-3)`
(v) `(5)/(-13) "and" (11)/(26)`

Answer

Let's solve the problem step by step for each pair of rational numbers.

### (i) Add \( \frac{7}{15} \) and \( \frac{3}{5} \)

1. **Find the LCM of the denominators (15 and 5)**:
- The LCM of 15 and 5 is 15.

2. **Convert \( \frac{3}{5} \) to have a denominator of 15**:
- \( \frac{3}{5} = \frac{3 \times 3}{5 \times 3} = \frac{9}{15} \)

3. **Add the two fractions**:
- \( \frac{7}{15} + \frac{9}{15} = \frac{7 + 9}{15} = \frac{16}{15} \)

4. **Conclusion**:
- The sum \( \frac{16}{15} \) is a rational number.

### (ii) Add \( \frac{2}{7} \) and \( 2 \)

1. **Convert 2 to a fraction**:
- \( 2 = \frac{2}{1} \)

2. **Find the LCM of the denominators (7 and 1)**:
- The LCM of 7 and 1 is 7.

3. **Convert \( \frac{2}{1} \) to have a denominator of 7**:
- \( \frac{2}{1} = \frac{2 \times 7}{1 \times 7} = \frac{14}{7} \)

4. **Add the two fractions**:
- \( \frac{2}{7} + \frac{14}{7} = \frac{2 + 14}{7} = \frac{16}{7} \)

5. **Conclusion**:
- The sum \( \frac{16}{7} \) is a rational number.

### (iii) Add \( \frac{3}{8} \) and \( -\frac{5}{12} \)

1. **Find the LCM of the denominators (8 and 12)**:
- The LCM of 8 and 12 is 24.

2. **Convert both fractions to have a denominator of 24**:
- \( \frac{3}{8} = \frac{3 \times 3}{8 \times 3} = \frac{9}{24} \)
- \( -\frac{5}{12} = -\frac{5 \times 2}{12 \times 2} = -\frac{10}{24} \)

3. **Add the two fractions**:
- \( \frac{9}{24} + \left(-\frac{10}{24}\right) = \frac{9 - 10}{24} = -\frac{1}{24} \)

4. **Conclusion**:
- The sum \( -\frac{1}{24} \) is a rational number.

### (iv) Add \( -\frac{7}{15} \) and \( -\frac{2}{3} \)

1. **Find the LCM of the denominators (15 and 3)**:
- The LCM of 15 and 3 is 15.

2. **Convert \( -\frac{2}{3} \) to have a denominator of 15**:
- \( -\frac{2}{3} = -\frac{2 \times 5}{3 \times 5} = -\frac{10}{15} \)

3. **Add the two fractions**:
- \( -\frac{7}{15} + \left(-\frac{10}{15}\right) = -\frac{7 + 10}{15} = -\frac{17}{15} \)

4. **Conclusion**:
- The sum \( -\frac{17}{15} \) is a rational number.

### (v) Add \( -\frac{5}{13} \) and \( \frac{11}{26} \)

1. **Find the LCM of the denominators (13 and 26)**:
- The LCM of 13 and 26 is 26.

2. **Convert \( -\frac{5}{13} \) to have a denominator of 26**:
- \( -\frac{5}{13} = -\frac{5 \times 2}{13 \times 2} = -\frac{10}{26} \)

3. **Add the two fractions**:
- \( -\frac{10}{26} + \frac{11}{26} = \frac{-10 + 11}{26} = \frac{1}{26} \)

4. **Conclusion**:
- The sum \( \frac{1}{26} \) is a rational number.

### Final Summary
In all cases, the sums obtained are rational numbers.

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