Three numbers are in the ratio `1/2: 2/3:3/4`
The difference between the greatest and the smallest number is 36. The numbers are

Options

A. 72, 84, 108
B. 60, 72, 96
C. 72, 84, 96
D. 72. 96, 108

Answer

To solve the problem, we need to find three numbers that are in the ratio \( \frac{1}{2} : \frac{2}{3} : \frac{3}{4} \) and have a difference of 36 between the greatest and smallest numbers.

### Step-by-Step Solution:

1. **Identify the Ratios**:
The given ratios are \( \frac{1}{2}, \frac{2}{3}, \frac{3}{4} \).

2. **Find the LCM of the Denominators**:
The denominators are 2, 3, and 4. The least common multiple (LCM) of these numbers is 12.

3. **Convert the Ratios to a Common Scale**:
To express the ratios in whole numbers, we multiply each ratio by the LCM (12):
- For \( \frac{1}{2} \): \( \frac{1}{2} \times 12 = 6 \)
- For \( \frac{2}{3} \): \( \frac{2}{3} \times 12 = 8 \)
- For \( \frac{3}{4} \): \( \frac{3}{4} \times 12 = 9 \)

So, the three numbers are 6, 8, and 9.

4. **Identify the Greatest and Smallest Numbers**:
From the numbers 6, 8, and 9:
- The smallest number is 6.
- The greatest number is 9.

5. **Calculate the Difference**:
The difference between the greatest and smallest numbers is:
\[ 9 - 6 = 3 \]
However, we need to find the actual numbers that satisfy the condition of the difference being 36.

6. **Set Up the Equation**:
Let the common multiplier be \( x \). Then the numbers can be expressed as:
- First number: \( 6x \)
- Second number: \( 8x \)
- Third number: \( 9x \)

The difference between the greatest and smallest numbers is:
\[ 9x - 6x = 3x \]

7. **Equate the Difference to 36**:
Set the equation:
\[ 3x = 36 \]

8. **Solve for \( x \)**:
Dividing both sides by 3:
\[ x = \frac{36}{3} = 12 \]

9. **Find the Actual Numbers**:
Substitute \( x \) back into the expressions for the numbers:
- First number: \( 6x = 6 \times 12 = 72 \)
- Second number: \( 8x = 8 \times 12 = 96 \)
- Third number: \( 9x = 9 \times 12 = 108 \)

### Final Answer:
The three numbers are **72, 96, and 108**.