Three numbers are in the ratio $$\frac{4}{10} : \frac{20}{18} : \frac{20}{11}$$​. The difference between the greatest and the smallest numbers is 39. Find the largest number.

Correct option is D

Given
Ratio of three numbers = $$\frac{4}{10} : \frac{20}{18} : \frac{20}{11}$$​​
Difference between the greatest and smallest numbers = 39
Formula Used
Difference = Greatest value - Smallest value

Solution
Simplify the given ratio fractions:
​$$\frac{2}{5} : \frac{10}{9} : \frac{20}{11}$$​​
To remove fractions, find the LCM of denominators 5, 9, and 11, which is $$5 \times 9 \times 11 = 495$$​.
Multiply each term by 495:
​$$\frac{2}{5} \times 495 = 198 \\\ \\\frac{10}{9} \times 495 = 550 \\\ \\\frac{20}{11} \times 495 = 900$$​​
The simplified ratio is 198 : 550 : 900.
Dividing by their common factor 2:
99 : 275 : 450.
Let the numbers be 99x, 275x, and 450x.
The greatest number is 450x and the smallest is 99x.
According to the given condition:
450x - 99x = 39
351x = 39
​$$x = \frac{39}{351} = \frac{1}{9}$$​​
The largest number is 450x:
Largest number = $$450 \times \frac{1}{9} = 50$$​​
Final Answer
So the correct answer is (d)