Question : The ratio of the LCM of two numbers to the sum of the same two numbers is 12 : 7. If their HCF is 4, what is the product of these two numbers?
Option 1: 192
Option 2: 172
Option 3: 196
Option 4: 169
Correct Answer: 192
Solution :
HCF = 4
Let the numbers be $4a$ and $4b$.
So, LCM = $4ab$
The ratio of LCM to the sum of numbers = 12 : 7
⇒ $\frac{4ab}{4a+4b} = \frac{12}{7}$
⇒ $\frac{ab}{a+b} = \frac{12}{7}$
⇒ $ab = 12$ and $a+b = 7$
Now, $a(7-a) = 12$
⇒ $a^2 - 7a + 12 = 0$
⇒ $(a-4)(a-3) = 0$
⇒ $a = 3$ or $4$
$\therefore b = 4$ or $3$
First number = $4a$ = 12
Second number = $4b$ = 16
Product of two numbers = 12 × 16 = 192
Hence, the correct answer is 192.
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