If \(\sqrt{2}:(1+\sqrt{5}):: \sqrt{10}: x\), then find the value of x.
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → \(5+\sqrt{5}\)
In a proportion \(a:b::c:x\), the product of the extremes equals the product of the means: \(a\times x=b\times c\).
Extremes \(\sqrt2\) and \(x\), means \(1+\sqrt5\) and \(\sqrt{10}\): \(\sqrt2\times x=(1+\sqrt5)\times\sqrt{10}\)
\(\displaystyle x=\frac{(1+\sqrt5)\sqrt{10}}{\sqrt2}=(1+\sqrt5)\sqrt5\), since \(\frac{\sqrt{10}}{\sqrt2}=\sqrt5\)
\(x=\sqrt5+5=5+\sqrt5\)