The HCF and LCM of two numbers are 48 and 720. Ratio of two numbers is 3 : 5. Then find the larger number.

Options

A. 280
B. 360
C. 240
D. 300

Answer

To find the larger number given the HCF, LCM, and the ratio of two numbers, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the given values**:
- HCF (Highest Common Factor) = 48
- LCM (Lowest Common Multiple) = 720
- Ratio of the two numbers = 3:5

2. **Let the two numbers be represented in terms of HCF**:
- Since the HCF is 48, we can express the two numbers as:
- First number = 48a
- Second number = 48b
- Here, a and b are the parts of the ratio.

3. **Using the ratio**:
- From the ratio 3:5, we can set:
- a = 3k
- b = 5k
- Where k is a common multiplier.

4. **Substituting a and b into the expressions for the numbers**:
- First number = 48 * 3k = 144k
- Second number = 48 * 5k = 240k

5. **Using the relationship between HCF, LCM, and the product of the numbers**:
- The formula states that:
\[ \text{HCF} \times \text{LCM} = \text{First number} \times \text{Second number} \]
- Substituting the known values:
\[ 48 \times 720 = (144k) \times (240k) \]

6. **Calculating the left side**:
- Calculate \(48 \times 720\):
\[ 48 \times 720 = 34560 \]

7. **Calculating the right side**:
- Calculate \((144k) \times (240k)\):
\[ 144 \times 240 \times k^2 = 34560k^2 \]

8. **Setting the equations equal**:
- Now we have:
\[ 34560 = 34560k^2 \]
- Dividing both sides by 34560:
\[ 1 = k^2 \]
- Therefore, \(k = 1\).

9. **Finding the actual numbers**:
- Substitute \(k = 1\) back into the expressions for the numbers:
- First number = \(144 \times 1 = 144\)
- Second number = \(240 \times 1 = 240\)

10. **Identifying the larger number**:
- The larger number is 240.

### Final Answer:
The larger number is **240**.