Question : If the sum of two numbers is 60 and their HCF and LCM are 5 and 60, respectively, then the sum of the reciprocals of the numbers will be:
Option 1: $\frac{1}{11}$
Option 2: $\frac{1}{6}$
Option 3: $\frac{1}{4}$
Option 4: $\frac{1}{5}$
Correct Answer: $\frac{1}{5}$
Solution :
Let the two numbers be $x$ and $y$.
The sum of two numbers = 60
⇒ $x+y=60$
HCF of the given numbers = 5
LCM of the given numbers = 60
We know that,
Product of two numbers = LCM × HCF
⇒ $xy$ = LCM × HCF
⇒ $xy$ = 5 × 60
The required sum = $\frac{1}{x}+\frac{1}{y}=\frac{x+y}{xy}$
Putting the values, we get:
= $\frac{60}{5 × 60}$
= $\frac{1}{5}$
Hence, the correct answer is $\frac{1}{5}$.
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