The difference between two numbers is 15, and their HCF and LCM are 3 and 108 respectively. What is the difference between their reciprocals?
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \(\frac{5}{108}\)
For any two numbers, HCF \(\times\) LCM \(=\) product of the numbers. The difference of the reciprocals of \(x\) and \(y\) is \(\displaystyle\frac{1}{y}-\frac{1}{x}=\frac{x-y}{xy}\), so only the difference and the product are needed.
Product \(xy=3\times108=324\)
Difference \(x-y=15\)
\(\displaystyle\frac{1}{y}-\frac{1}{x}=\frac{x-y}{xy}=\frac{15}{324}=\frac{5}{108}\)
Check: the numbers are 27 and 12 (HCF 3, LCM 108, difference 15), and \(\displaystyle\frac{1}{12}-\frac{1}{27}=\frac{9-4}{108}=\frac{5}{108}\)
The difference between the reciprocals is \(\frac{5}{108}\).