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Ratio and Proportion

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Question : 5 of 112

Marks: +1, -0

If

(\sqrt{13+x} + \sqrt{13-x}) : (\sqrt{13+x} - \sqrt{13-x}) = 5

x

is

4
5
6
3

Solution:

Concept:

Use cross-multiplication and algebraic simplification to solve the given ratio equation.

Explanation:

Given:

\frac{\sqrt{13+x} + \sqrt{13-x}}{\sqrt{13+x} - \sqrt{13-x}} = \frac{5}{1}

.

Cross-multiply:

\sqrt{13+x} + \sqrt{13-x} = 5(\sqrt{13+x} - \sqrt{13-x})

.

Simplify:

\sqrt{13+x} + \sqrt{13-x} = 5\sqrt{13+x} - 5\sqrt{13-x}

.

Bring like terms together:

\sqrt{13-x} + 5\sqrt{13-x} = 5\sqrt{13+x} - \sqrt{13+x}

.

Thus

6\sqrt{13-x} = 4\sqrt{13+x}

.

Divide by 2:

3\sqrt{13-x} = 2\sqrt{13+x}

.

Square both sides:

9(13-x) = 4(13+x)

.

Expand:

117 - 9x = 52 + 4x

.

Rearrange:

117 - 52 = 4x + 9x \implies 65 = 13x

.

Hence

x = 5

.

Answer:

x = 5

, which corresponds to option B.

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