If \(\frac{a}{b}=\frac{1}{2}\) and \(\frac{b}{c}=\frac{3}{4}\), then find the value of \(\frac{a^2-c^2}{ac-bc}\).
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\frac{55}{24}\)
Every term of the expression has the same degree (two letters multiplied), so its value depends only on the ratio \(a:b:c\); any numbers in that ratio give the same answer. The two given ratios are joined through the common letter \(b\).
Assuming \(b=3\):
The second ratio \(\displaystyle \frac{b}{c}=\frac{3}{4}\) then gives \(c=4\)
The first ratio \(\displaystyle \frac{a}{b}=\frac{1}{2}\) then gives \(\displaystyle a=\frac{3}{2}\)
The numerator becomes \(\displaystyle a^2-c^2=\frac{9}{4}-16=-\frac{55}{4}\)
Denominator, taking \(c\) out as a common factor: \(\displaystyle ac-bc=c(a-b)=4\left(\frac{3}{2}-3\right)=-6\)
The value of the expression is \(\displaystyle \frac{-55/4}{-6}=\frac{55}{24}\)