If \(P:Q=2:\sqrt{3}\), then what is the value of \((3P+Q):(3P-2Q)\)?
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle \frac{1}{4}(7+3\sqrt{3})\)
A ratio does not change when both terms are scaled, so we may take \(P=2\) and \(Q=\sqrt{3}\). The surd in the denominator is then removed by multiplying by its conjugate.
\(3P+Q=6+\sqrt{3}\)
\(3P-2Q=6-2\sqrt{3}\)
\(\displaystyle \frac{6+\sqrt{3}}{6-2\sqrt{3}}\times\frac{6+2\sqrt{3}}{6+2\sqrt{3}}\)
Numerator \(=36+12\sqrt{3}+6\sqrt{3}+6=42+18\sqrt{3}\)
Denominator \(=36-12=24\)
\(\displaystyle \frac{42+18\sqrt{3}}{24}=\frac{7+3\sqrt{3}}{4}\)
The value is \(\displaystyle \frac{1}{4}(7+3\sqrt{3})\).