Question

If three numbers in the ratio 3 : 2 : 5 be such that the sum of their squares is 1862, the middle number will be

• A. 10
• B. 12
• C. 14
• D. 16

Answer 1

Answer:

C. 14

Understanding Ratio & Proportion Problems

• When numbers are given in a specific ratio, like A : B : C, we represent them as Ax, Bx, and Cx, where 'x' is a common multiplier or proportionality constant. This method helps convert a ratio problem into an algebraic equation.

• In this problem, the three numbers are in the ratio 3 : 2 : 5. Therefore, we can denote them as 3x, 2x, and 5x.

Setting up the Equation

• The problem states that the sum of the squares of these numbers is 1862.

• Squaring each number:

  • Square of the first number: (3x)² = 9x²

  • Square of the second number: (2x)² = 4x²

  • Square of the third number: (5x)² = 25x²

• Now, sum these squares and equate to the given total: 9x² + 4x² + 25x² = 1862.

Solving for 'x'

• Combine the like terms on the left side of the equation: (9 + 4 + 25)x² = 1862.

• This simplifies to 38x² = 1862.

• To find x², divide both sides by 38: x² = 1862 / 38.

• Performing the division: x² = 49.

• To find 'x', take the square root of 49: x = √49. Since we are dealing with numbers and their squares, we consider the positive root. So, x = 7.

Finding the Middle Number

• The three numbers were represented as 3x, 2x, and 5x.

• The middle number in the ratio 3 : 2 : 5 is the one corresponding to '2', which is 2x.

• Substitute the value of x = 7 into 2x: Middle Number = 2 × 7 = 14.