Three numbers are in the ratio of $3: 4: 5$ and their L.C.M. is $2400$. Their H.C.F. is

### Concept & Formula When a set of numbers is given in a ratio $a:b:c$, they can be represented as $ax$, $bx$, and $cx$, where the variable $x$ represents the Highest Common Factor (H.C.F.) of the numbers. If the base ratio integers ($a, b, c$) are mutually co-prime (meaning no common factor exists across all of them), the Least Common Multiple (L.C.M.) is calculated by multiplying the ratio integers together with the H.C.F. $$\text{L.C.M.} = a \times b \times c \times \text{H.C.F.}$$ ### Step-by-Step Solution * **Given:** Ratio of numbers = $3: 4: 5$ L.C.M. = $2400$ * **Step 1:** Assign algebraic variables. Let the three numbers be $3x$, $4x$, and $5x$. Here, $x$ represents the H.C.F. we need to find. * **Step 2:** Formulate the L.C.M. expression. The integers $3, 4$, and $5$ are pairwise co-prime (they share no common factors). The L.C.M. of $3x, 4x$, and $5x$ is simply the L.C.M. of $(3, 4, 5)$ multiplied by the common variable $x$. $$\text{L.C.M.} = (3 \times 4 \times 5) \times x$$ $$\text{L.C.M.} = 60x$$ * **Step 3:** Solve for $x$ using the given L.C.M. value. $$60x = 2400$$ $$x = \frac{2400}{60}$$ $$x = 40$$ * **Conclusion:** Since $x$ represents the common factor, the H.C.F. is exactly $40$. ### Exam Strategy & Shortcut **Direct Formula Application:** If you recognize that $3, 4$, and $5$ are co-prime, you can skip writing equations entirely. You just multiply the ratio numbers and divide the L.C.M. by that product. $$3 \times 4 \times 5 = 60$$ $$\text{H.C.F.} = \frac{2400}{60} = 40$$ This is a $5$-second mental math problem. ### Common Pitfall The main danger occurs if the ratio integers are *not* co-prime (for example, if the ratio was $2:4:5$). In that case, you cannot simply multiply them together ($2 \times 4 \times 5 \times x$). You would have to find the actual L.C.M. of the ratio integers first (L.C.M. of $2, 4, 5$ is $20$, so it would be $20x$). Always verify that the ratio components share no common factors before applying the direct multiplication shortcut. Here, $3, 4, 5$ are clear, so the shortcut works flawlessly. ### Final Answer **Therefore, the correct answer is 40.**