Two numbers are in the ratio of $15 : 11$. If their H.C.F. is $13$, find the numbers.

### Concept & Formula When two numbers are given in a simplified ratio $a : b$, the numbers themselves can be expressed as $ax$ and $bx$, where $x$ is their Highest Common Factor (H.C.F.). Because a ratio is in its simplest form, the terms $a$ and $b$ are co-prime (they share no common factors other than $1$). Therefore, multiplying each term of the ratio by the H.C.F. directly yields the original numbers. ### Step-by-Step Solution **Step 1: Define the numbers using the given ratio.** * Given ratio = $15 : 11$. * Let the two numbers be $15x$ and $11x$. **Step 2: Identify the H.C.F.** * Since $15$ and $11$ are co-prime, the highest common factor of $15x$ and $11x$ is exactly $x$. * The problem states the H.C.F. is $13$. * Therefore, $x = 13$. **Step 3: Calculate the actual numbers.** * First number = $15 \times 13 = 195$. * Second number = $11 \times 13 = 143$. ### Exam Strategy & Shortcut You don't need to write out algebraic steps. As a strict rule: **Original Numbers = Ratio Terms $\times$ H.C.F.** If you see this exact question type on an exam, instantly multiply $15 \times 13$ and $11 \times 13$. Even faster, you can use unit digit checking on the options: $5 \times 3$ ends in $5$. $1 \times 3$ ends in $3$. Look for an option where the numbers end in $5$ and $3$ respectively. Option A ($195$ and $143$) perfectly fits without even doing the full multiplication. ### Common Pitfall A common beginner mistake is assuming the ratio terms ($15$ and $11$) are factors to be multiplied together to find the L.C.M., and confusing this with finding the actual numbers. Always remember that the ratio terms scaled by the H.C.F. gives the numbers directly. ### Final Answer **Therefore, the correct answer is $195$ and $143$.**