Question:medium

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?

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Use the mean proportional definition to form a quadratic equation and solve for the number.
Updated On: Jul 21, 2026
  • 154
  • 143
  • 121
  • 88
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The Correct Option is D

Solution and Explanation

This can also be solved using the sum and difference of the two numbers instead of expanding and solving a quadratic directly.
Step 1: Name the two numbers. Let \(a = x - 77\) and \(b = x + 11\).
Step 2: Find their difference directly. \(b - a = (x+11)-(x-77) = 88\), a fixed value that does not depend on x at all.
Step 3: Find their product from the mean proportional condition. \(\sqrt{ab} = 33\) means \(ab = 1089\).
Step 4: Use the identity connecting sum, difference and product. \((b-a)^2 + 4ab = (a+b)^2\). Substituting the known values, \(88^2 + 4 \times 1089 = (a+b)^2\), that is \(7744 + 4356 = 12100 = (a+b)^2\), so \(a+b = 110\).
Step 5: Solve for a and b. With \(b - a = 88\) and \(b + a = 110\), adding gives \(2b = 198\), so \(b = 99\), and then \(a = 110 - 99 = 11\).
Step 6: Recover x. Since \(a = x - 77 = 11\), \(x = 88\), matching the earlier answer.\[\boxed{88}\]
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