(v) Assertion (A): [tex]\left(\frac{1}{5}\right)+\left(\frac{5}{6}\right)\neq\left(\frac{1}{5}\right) \times \left(\frac{6}{5}\right)[/tex]
Reason (R): While dividing one fraction by another fraction, we multiply the first fraction by the reciprocal of the other fraction.

(a) Both A and R are true and R is the correct explanation of A.
(b) Both A and R are true but R is not the correct explanation of A.
(c) A is true but R is false.
(d) A is false but R is true.



Answer :

Let's evaluate the assertion (A) and the reason (R) step-by-step.

### Assertion (A)

We are given:
[tex]\[ \left(\frac{1}{5}\right) + \left(\frac{5}{6}\right) = \left(\frac{1}{5}\right) \times \left(\frac{6}{5}\right) \][/tex]

Let's compute both sides.

#### Left Side:
[tex]\[ \frac{1}{5} + \frac{5}{6} \][/tex]

To add these fractions, we need a common denominator. The lowest common multiple of 5 and 6 is 30.

[tex]\[ \frac{1}{5} = \frac{6}{30} \][/tex]
[tex]\[ \frac{5}{6} = \frac{25}{30} \][/tex]

Adding these fractions:
[tex]\[ \frac{6}{30} + \frac{25}{30} = \frac{31}{30} \][/tex]

#### Right Side:
[tex]\[ \frac{1}{5} \times \frac{6}{5} \][/tex]

Multiplying these fractions:
[tex]\[ \frac{1 \times 6}{5 \times 5} = \frac{6}{25} \][/tex]

Thus, the left side is [tex]\(\frac{31}{30}\)[/tex] and the right side is [tex]\(\frac{6}{25}\)[/tex]. Clearly, these two values are not equal:
[tex]\[ \frac{31}{30} \neq \frac{6}{25} \][/tex]

Therefore, the assertion [tex]\(A\)[/tex] is false.

### Reason (R)

The reason states that when dividing one fraction by another, we multiply the first fraction by the reciprocal of the other fraction. This is indeed a true statement as it is a fundamental rule of dividing fractions.

### Conclusion

- [tex]\(A\)[/tex] is false.
- [tex]\(R\)[/tex] is true.

Therefore, the correct answer is:
[tex]\[ \boxed{(d) \text{A is false but R is true.}} \][/tex]