The three numbers are in the ratio of $\frac{1}{2} : \frac{2}{3} : \frac{3}{4}$. The difference between the greatest and smallest numbers is 36. Three numbers are equal to :

Question

Accepted Answer

To solve the problem, we need to determine the 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 Explanation:

1. **Convert the Ratios to a Common Base:**
   - The given ratios are $\frac{1}{2}$, $\frac{2}{3}$, and $\frac{3}{4}$.
   - To simplify these ratios, find a common denominator. The least common multiple of 2, 3, and 4 is 12.
   - Convert each fraction to have a denominator of 12:
     - $\frac{1}{2} = \frac{6}{12}$
     - $\frac{2}{3} = \frac{8}{12}$
     - $\frac{3}{4} = \frac{9}{12}$

2. **Express the Numbers in Terms of a Variable:**
   - Let the numbers be $6x$, $8x$, and $9x$ based on the simplified ratios.

3. **Calculate the Difference:**
   - The difference between the greatest and smallest numbers is given as 36.
   - Therefore, $9x - 6x = 36$.

4. **Solve for $x$:**
   - $3x = 36$
   - $x = 12$

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

Thus, the three numbers are 72, 96, and 108, which corresponds to **Option 4**.

### Explanation of Why Other Options are Incorrect:

- **Option 1: 72, 84, 108**
  - The difference between the greatest (108) and smallest (72) is 36, which is correct. However, the middle number (84) does not fit the ratio $\frac{8}{12}$ or $\frac{2}{3}$.

- **Option 2: 60, 72, 96**
  - The difference between the greatest (96) and smallest (60) is 36, which is correct. However, the numbers do not fit the ratio $\frac{6}{12} : \frac{8}{12} : \frac{9}{12}$.

- **Option 3: 72, 84, 96**
  - The difference between the greatest (96) and smallest (72) is 24, not 36. Therefore, this option does not satisfy the condition of the problem.

In conclusion, **Option 4** is the only set of numbers that satisfies both the given ratio and the specified difference, making it the correct answer.