Question

The mean proportional of the two positive numbers obtained by subtracting 77 from a positive number and adding 11 to the same number is 33. What is the number?

• A. 154
• B. 143
• C. 121
• D. 88

Hint:

Use the mean proportional definition to form a quadratic equation and solve for the number.

Answer 1

The Correct Option is D

Step 1: Set up the two numbers. Let the positive number be x. The two numbers formed are \(x - 77\) and \(x + 11\).
Step 2: Apply the mean proportional definition. The mean proportional of two numbers a and b is \(\sqrt{ab}\). Here this equals 33, so \(\sqrt{(x-77)(x+11)} = 33\).
Step 3: Square both sides. \((x-77)(x+11) = 33^2 = 1089\).
Step 4: Expand. \(x^2 + 11x - 77x - 847 = 1089\), which simplifies to \(x^2 - 66x - 847 = 1089\), so \(x^2 - 66x - 1936 = 0\).
Step 5: Solve the quadratic. Using the quadratic formula, \(x = \dfrac{66 \pm \sqrt{66^2 + 4 \times 1936}}{2} = \dfrac{66 \pm \sqrt{4356+7744}}{2} = \dfrac{66 \pm \sqrt{12100}}{2} = \dfrac{66 \pm 110}{2}\).
Step 6: Pick the valid root. This gives \(x = 88\) or \(x = -22\). Since x must be a positive number and \(x-77\) must also be positive, \(x = 88\) is the only valid solution.
Step 7: Verify. \(x - 77 = 11\) and \(x + 11 = 99\), both positive, and \(\sqrt{11 \times 99} = \sqrt{1089} = 33\), which matches.\[\boxed{88}\]