Ratio and Proportion

Concept:If two numbers are in the ratio $a : b$, they can be written as $a k$ and $b k$.When $a$ and $b$ are coprime, the LCM of $a k$ and $b k$ is $a b k$.Explanation:Let the two numbers be $3k$ and $4k$, where $k$ is a common factor.Since $3$ and $4$ are coprime (they share no common factor), their LCM is $3 \times 4 \times k = 12k$.The problem states that the LCM is $180$. Therefore, we have the equation:$12k = 180$Solving for $k$: $k = \frac{180}{12} = 15$.Now find the numbers:First number = $3k = 3 \times 15 = 45$Second number = $4k = 4 \times 15 = 60$Thus, the numbers are $45$ and $60$.Comparing with the options, this matches option B.Answer:$45$ and $60$