Question : If the three numbers in the ratio 3 : 2 : 5 such that the sum of the squares is equal to 1862, then which number is the middle one?
Option 1: 16
Option 2: 14
Option 3: 13
Option 4: 15
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Correct Answer: 14
Solution :
Given: The ratio of three numbers is 3 : 2 : 5.
The sum of the squares of the numbers = 1862
Let the numbers be $3x,2x,$ and $5x$.
According to the given condition,
$(3x)^{2}+(2x)^{2}+(5x)^{2}=1862$
⇒ $9x^{2}+4x^{2}+25x^{2}=1862$
⇒ $38x^{2}=1862$
⇒ $x^{2}=\frac{1862}{38}$
⇒ $x=\sqrt{49}$
⇒ $x=7$
So, the middle number is 7 × 2 = 14
Hence, the correct answer is 14.
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