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. 99
• E. 88

Hint:

Write the mean proportional condition as \(\sqrt{(x-77)(x+11)}=33\), square both sides, and solve the resulting quadratic, keeping only the positive root that keeps both parts positive.

Answer 1

Answer: (E)

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 condition.
The mean proportional of two numbers \(m\) and \(n\) is \(\sqrt{mn}\). So:
$$\sqrt{(x-77)(x+11)}=33$$
Squaring both sides:
$$(x-77)(x+11)=33^2=1089$$

Step 3: Expand and form a quadratic equation.
$$x^2+11x-77x-847=1089$$
$$x^2-66x-847-1089=0$$
$$x^2-66x-1936=0$$

Step 4: Solve the quadratic equation.
Using the quadratic formula with \(a=1,\ b=-66,\ c=-1936\):
$$x=\frac{66\pm\sqrt{66^2+4(1936)}}{2}=\frac{66\pm\sqrt{4356+7744}}{2}=\frac{66\pm\sqrt{12100}}{2}=\frac{66\pm110}{2}$$
This gives \(x=\dfrac{176}{2}=88\) or \(x=\dfrac{-44}{2}=-22\).

Step 5: Reject the invalid root and verify.
Since the number must be positive and \(x-77\) must also be a positive number (as stated in the question), \(x=-22\) is rejected. Check \(x=88\): \(x-77=11\) and \(x+11=99\), both positive. Mean proportional \(=\sqrt{11\times99}=\sqrt{1089}=33\), which matches. So the number is 88. The correct option is (e) 88.