The HCF and LCM of two numbers a and b are respectively 6 and 210. If a + b = 72, then $\frac{1}{a} + \frac{1}{b}$ = ?

Question

Accepted Answer

To solve the problem and justify why Option 4 is the correct answer, we need to use the given information about the HCF, LCM, and the sum of the two numbers.

Given:
- HCF(a, b) = 6
- LCM(a, b) = 210
- a + b = 72

We know the relationship between two numbers, their HCF, and LCM:
$$ a \times b = \text{HCF}(a, b) \times \text{LCM}(a, b) $$

Substituting the given values:
$$ a \times b = 6 \times 210 = 1260 $$

We also have:
$$ a + b = 72 $$

We need to find:
$$ \frac{1}{a} + \frac{1}{b} = \frac{a + b}{a \times b} $$

Substituting the known values:
$$ \frac{1}{a} + \frac{1}{b} = \frac{72}{1260} $$

Simplifying the fraction:
$$ \frac{72}{1260} = \frac{6}{105} = \frac{2}{35} $$

Thus, the correct answer is $\frac{2}{35}$, which corresponds to Option 4.

### Explanation for Other Options:

1. **Option 1: $\frac{1}{35}$**
   - This option is incorrect because when we simplify $\frac{72}{1260}$, it does not reduce to $\frac{1}{35}$.

2. **Option 2: $\frac{3}{35}$**
   - This option is incorrect because the simplified form of $\frac{72}{1260}$ is $\frac{2}{35}$, not $\frac{3}{35}$.

3. **Option 3: $\frac{5}{37}$**
   - This option is incorrect because $\frac{72}{1260}$ simplifies to $\frac{2}{35}$, and $\frac{5}{37}$ is not equivalent to this value.

By following the logical steps and calculations, we confirm that Option 4 is indeed the correct answer.