Two numbers are in the ratio 3 : 4 . The product of their H.C.F. and L.C.M. Is 2028 . The sum of the numbers is

Options

A. 68
B. 72
C. 86
D. 91

Answer

To solve the problem, we need to find the sum of two numbers that are in the ratio of 3:4, given that the product of their H.C.F. (Highest Common Factor) and L.C.M. (Lowest Common Multiple) is 2028.

### Step-by-Step Solution:

1. **Define the Numbers**:
Let the two numbers be \(3k\) and \(4k\), where \(k\) is a common multiplier.

2. **Calculate H.C.F. and L.C.M.**:
- The H.C.F. of \(3k\) and \(4k\) is \(k\) (since 3 and 4 are co-prime).
- The L.C.M. of \(3k\) and \(4k\) is \(12k\) (since \(L.C.M(a, b) = \frac{a \times b}{H.C.F(a, b)}\)).

3. **Set Up the Equation**:
According to the problem, the product of H.C.F. and L.C.M. is given by:
\[ H.C.F. \times L.C.M. = k \times 12k = 12k^2 \]
We know this equals 2028:
\[ 12k^2 = 2028 \]

4. **Solve for \(k^2\)**:
Divide both sides by 12:
\[ k^2 = \frac{2028}{12} \]
Calculate \(2028 \div 12\):
\[ k^2 = 169 \]

5. **Find \(k\)**:
Taking the square root of both sides:
\[ k = \sqrt{169} = 13 \]

6. **Calculate the Numbers**:
Now substitute \(k\) back into the expressions for the numbers:
- First number: \(3k = 3 \times 13 = 39\)
- Second number: \(4k = 4 \times 13 = 52\)

7. **Find the Sum of the Numbers**:
Now, we can find the sum:
\[ \text{Sum} = 39 + 52 = 91 \]

### Final Answer:
The sum of the two numbers is **91**.

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