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
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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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