The H.C.F. and L.C.M. of two numbers are 21 and 84 respectively. If the ratio of the two numbers is 1 : 4, then the larger of the two numbers is

### Concept & Formula When two numbers are given in a ratio $a : b$, the actual numbers can be represented as $ax$ and $bx$, where the common multiplier $x$ is the Highest Common Factor (H.C.F.) of the two numbers. $$ \text{Numbers} = (\text{Ratio Part } 1 \times \text{H.C.F.}) \text{ and } (\text{Ratio Part } 2 \times \text{H.C.F.}) $$ ### Step-by-Step Solution * **Given:** * H.C.F. = 21 * L.C.M. = 84 * Ratio of numbers = 1 : 4 * Let the two numbers be $1x$ and $4x$. * By definition, the common factor $x$ is the H.C.F. Therefore, $x = 21$. * The two numbers are: * First number = $1 \times 21 = 21$ * Second number = $4 \times 21 = 84$ * The question asks for the larger of the two numbers. * Comparing 21 and 84, the larger number is 84. ### Exam Strategy & Shortcut You don't even need the L.C.M. to solve this specific question. The problem explicitly gives the ratio (1:4) and the H.C.F. (21). The larger number corresponds to the larger ratio part (4). Simply multiply the larger ratio part by the H.C.F.: $4 \times 21 = 84$. You can solve this in 3 seconds mentally. ### Common Pitfall A common trap is assuming you need to use all the provided numbers (like the L.C.M. = 84) in a complex formula. While you *could* use the formula $\text{Product} = \text{H.C.F.} \times \text{L.C.M.}$, it unnecessarily complicates a straightforward ratio multiplication. Don't over-engineer the solution. ### Final Answer **Therefore, the correct answer is 84.**