Three numbers are in the ratio 1 : 2 : 3 and their HCF is 12 . The numbers are

Options

A. `12 , 24, 36`
B. `5, 10, 15`
C. `4, 8, 12`
D. `10, 20, 30`

Answer

To find the three numbers that are in the ratio 1:2:3 and have a Highest Common Factor (HCF) of 12, we can follow these steps:

### Step 1: Understand the Ratio
The numbers are in the ratio 1:2:3. This means we can express the numbers in terms of a common variable. Let the three numbers be:
- First number = 1x
- Second number = 2x
- Third number = 3x

### Step 2: Use the HCF
We know that the HCF of these three numbers is 12. The HCF can also be expressed in terms of the common variable. Since the HCF is the greatest number that divides all three numbers, we can set:
- HCF = 12

### Step 3: Relate HCF to the Variable
Since the HCF of the numbers (1x, 2x, 3x) is equal to x multiplied by the HCF of the coefficients (1, 2, 3), we can express this as:
- HCF(1, 2, 3) = 1 (since 1 is the highest number that divides 1, 2, and 3)
- Therefore, HCF(1x, 2x, 3x) = x * HCF(1, 2, 3) = x * 1 = x

Since we know the HCF is 12, we can equate:
- x = 12

### Step 4: Calculate the Numbers
Now that we have the value of x, we can find the three numbers:
- First number = 1x = 1 * 12 = 12
- Second number = 2x = 2 * 12 = 24
- Third number = 3x = 3 * 12 = 36

### Step 5: Final Answer
Thus, the three numbers are:
- 12, 24, and 36.

### Summary
The three numbers in the ratio 1:2:3 with an HCF of 12 are 12, 24, and 36.

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