Three numbers are in the ratio 3 : 2 : 5. If the sum of the squares of these three numbers is 1862, what is the middle number?

Introduction / Context:
This problem combines ratio with the sum of squares of numbers. We know the ratio of three numbers and the sum of their squares. The task is to identify the middle number. Such questions test algebraic manipulation skills and the ability to represent ratios algebraically using a common factor, then apply the given condition to solve for that factor.

Given Data / Assumptions:

• Three numbers are in the ratio 3 : 2 : 5.
• Let the numbers be 3k, 2k and 5k for some positive k.
• The sum of the squares of these numbers is 1862.
• We need to find the value of the middle number, which is 2k.

Concept / Approach:
From the ratio, we represent the three numbers as 3k, 2k and 5k. The sum of their squares is then (3k)^2 + (2k)^2 + (5k)^2. We set this equal to 1862, and solve for k. Once k is known, the individual numbers can be found by multiplying the ratio terms with k. The middle number is simply 2k.

Step-by-Step Solution:
Let the three numbers be 3k, 2k and 5k.Compute their squares: (3k)^2 = 9k^2, (2k)^2 = 4k^2 and (5k)^2 = 25k^2.Sum of squares = 9k^2 + 4k^2 + 25k^2 = 38k^2.We are given that 38k^2 = 1862.So k^2 = 1862 / 38.Compute 1862 / 38 = 49.Thus k^2 = 49, so k = 7 (taking the positive value as we are dealing with magnitudes).The three numbers are then 3k = 21, 2k = 14 and 5k = 35.The middle number is 14.

Verification / Alternative check:
Check the sum of squares using the found numbers.21^2 = 441, 14^2 = 196 and 35^2 = 1225.Sum = 441 + 196 + 1225 = 1862, which matches the given sum.Thus the values 21, 14 and 35 satisfy both the ratio and the sum of squares condition.

Why Other Options Are Wrong:
If the middle number were 16, 13, 15 or 18, then the implied k and the corresponding numbers 3k, 2k and 5k would not give a sum of squares equal to 1862. Only 14 produces the exact required sum. Therefore, those options are incorrect even if they seem close in magnitude.

Common Pitfalls:
Some students mistakenly square the ratio terms and equate the sum of squared ratios directly to 1862, without introducing the variable k. Others may make arithmetic errors when computing the sum of squares. The correct process is to express the numbers as multiples of k, compute the squared sum, equate it to the given value, solve for k, and then determine the actual numbers.

Final Answer:
The middle number is 14.