Three number are in the ratio `(1)/(4),(1)/(5),(1)/(2)` . The difference between the greater and the smallest number is 1800 . The sum of the three numbers is

Options

A. 4500
B. 5800
C. 5700
D. 4700

Answer

To solve the problem step by step, we will follow these instructions:

1. **Identify the Ratios**: The three numbers are in the ratio \( \frac{1}{4}, \frac{1}{5}, \frac{1}{2} \). To make calculations easier, we will convert these ratios into a common form.

2. **Find the LCM of the Denominators**: The denominators are 4, 5, and 2. The least common multiple (LCM) of these numbers is 20.

3. **Express the Ratios with a Common Denominator**: We will multiply each ratio by 20 to eliminate the fractions:
- For \( \frac{1}{4} \): \( 20 \times \frac{1}{4} = 5 \)
- For \( \frac{1}{5} \): \( 20 \times \frac{1}{5} = 4 \)
- For \( \frac{1}{2} \): \( 20 \times \frac{1}{2} = 10 \)

So, the three numbers can be expressed as \( 5x, 4x, \) and \( 10x \).

4. **Identify the Greatest and Smallest Numbers**: In this case, the greatest number is \( 10x \) and the smallest number is \( 4x \).

5. **Set Up the Equation for the Difference**: According to the problem, the difference between the greatest and smallest number is 1800:
\[ 10x - 4x = 1800 \]
Simplifying this gives:
\[ 6x = 1800 \]

6. **Solve for x**: To find the value of \( x \), divide both sides by 6:
\[ x = \frac{1800}{6} = 300 \]

7. **Calculate the Values of the Numbers**: Now that we have \( x \), we can find the actual values of the three numbers:
- First number: \( 5x = 5 \times 300 = 1500 \)
- Second number: \( 4x = 4 \times 300 = 1200 \)
- Third number: \( 10x = 10 \times 300 = 3000 \)

8. **Calculate the Sum of the Three Numbers**: Finally, we can find the sum of the three numbers:
\[ 1500 + 1200 + 3000 = 5700 \]

Thus, the sum of the three numbers is **5700**.