The ratio of two numbers is 3 : 7 and their LCM is 630. Then, sum of the LCM and HCF of these numbers is :

Question

Accepted Answer

To solve the problem, we need to find the sum of the LCM and HCF of two numbers given that their ratio is 3:7 and their LCM is 630.

1. **Understanding the Ratio and LCM:**
   - Let the two numbers be $3x$ and $7x$ based on the given ratio 3:7.
   - The LCM of two numbers $a$ and $b$ is given by the formula: $\text{LCM}(a, b) = \frac{a \times b}{\text{HCF}(a, b)}$.

2. **Using the LCM:**
   - We know the LCM of $3x$ and $7x$ is 630.
   - Therefore, $\text{LCM}(3x, 7x) = \frac{3x \times 7x}{\text{HCF}(3x, 7x)} = 630$.

3. **Finding the HCF:**
   - The HCF of $3x$ and $7x$ is $x$ because 3 and 7 are coprime (they have no common factors other than 1).
   - So, $\frac{21x^2}{x} = 630$.
   - Simplifying gives $21x = 630$.
   - Solving for $x$, we get $x = 30$.

4. **Calculating the Numbers:**
   - The numbers are $3x = 90$ and $7x = 210$.

5. **Finding the HCF:**
   - The HCF of 90 and 210 is 30 (since 30 is the greatest number that divides both 90 and 210).

6. **Sum of LCM and HCF:**
   - LCM = 630
   - HCF = 30
   - Sum = 630 + 30 = 660

Therefore, the sum of the LCM and HCF of the numbers is 660, which corresponds to Option 4.

**Explanation of Incorrect Options:**

- **Option 1 (30):** This option is incorrect because it only represents the HCF, not the sum of the LCM and HCF.
- **Option 2 (300):** This option is incorrect because it does not match the calculated sum of the LCM and HCF.
- **Option 3 (600):** This option is incorrect because it is less than the actual sum of the LCM and HCF.

Thus, the correct answer is Option 4, which is 660.