Two numbers are in the ratio of $3 : 4$. Their L.C.M. is $84$. Find the numbers.

### Concept & Logic When two numbers are in a ratio $a : b$, they can be algebraically represented as $ax$ and $bx$, where $x$ represents their Highest Common Factor (H.C.F.). If the ratio components $a$ and $b$ are co-prime (which they should be in a simplified ratio), the formula for the Least Common Multiple (L.C.M.) of these two numbers is simply the product of the ratio terms and their H.C.F.: $$ \text{L.C.M.} = a \times b \times x $$ ### Step-by-Step Solution **Step 1: Define the numbers.** * The given ratio is $3 : 4$. * Let the required numbers be $3x$ and $4x$, where $x$ is their H.C.F. **Step 2: Formulate an equation using the given L.C.M.** * The L.C.M. of $3x$ and $4x$ is $3 \times 4 \times x = 12x$. * We are given that the actual L.C.M. is $84$. * Therefore, equate the two: $12x = 84$. **Step 3: Solve for $x$ and find the numbers.** * Divide both sides by 12: $x = \frac{84}{12}$ $x = 7$ * Now substitute $x$ back into the original expressions: First number = $3 \times 7 = 21$ Second number = $4 \times 7 = 28$ ### Exam Strategy & Shortcut Don't use algebra if you don't have to! Check the options against the primary constraints. First, which option provides numbers in a $3 : 4$ ratio? $21 : 28$ simplifies to $3 : 4$. Next, quickly check their L.C.M. The L.C.M. of $21$ and $28$ is indeed $84$. This confirms the answer instantly by working backward from the choices. ### Common Pitfall A common mistake is multiplying the ratio numbers ($3 \times 4 = 12$) and assuming that is the answer, or confusing the L.C.M. formula and mistakenly assuming $x = 84$, which results in extremely large, incorrect numbers ($252$ and $336$). Always remember that the ratio terms must be multiplied by the H.C.F. ### Final Answer **Therefore, the correct answer is $21$ and $28$.**