Three numbers are in the ratio of $3 : 4 : 6$ and their product is $1944$. The largest of these numbers is

### Concept & Formula When multiplying numbers based on ratios, remember that the common multiplier variable (e.g., $x$) will be raised to a power corresponding to the number of terms. Let the common multiplier be $x$. ### Step-by-Step Solution * **Given:** The ratio of the three numbers is $3 : 4 : 6$. Let the three numbers be $3x$, $4x$, and $6x$. Their product is $1944$. * **Calculation:** Set up the product equation: $$ 3x \times 4x \times 6x = 1944 $$ Multiply the coefficients and the variables: $$ (3 \times 4 \times 6) \times (x \cdot x \cdot x) = 1944 $$ $$ 72x^3 = 1944 $$ Isolate $x^3$ by dividing by $72$: $$ x^3 = \frac{1944}{72} $$ To simplify, you can divide step-by-step. Let's divide by $12$ first: $$ 1944 \div 12 = 162 $$ $$ 72 \div 12 = 6 $$ $$ x^3 = \frac{162}{6} $$ $$ x^3 = 27 $$ Take the cube root: $$ x = \sqrt[3]{27} = 3 $$ The question asks for the largest number. The largest ratio part is $6x$. Largest number $= 6 \times 3 = 18$. ### Exam Strategy & Shortcut **Digital Sum / Divisibility Rules** can speed up finding $x^3$. $1944$ sum of digits is $1+9+4+4 = 18$, meaning it's divisible by $9$. $72$ is also divisible by $9$. If you suspect $x=3$, test the final answer using the options. If the largest number is $18$ (Option C), then $6x = 18 \Rightarrow x=3$. Check the product: $3(3) \times 4(3) \times 6(3) = 9 \times 12 \times 18$. $9 \times 12 = 108$. $108 \times 18 = 1944$. Matches perfectly. ### Common Pitfall The most dangerous pitfall is forgetting the $x^3$ and writing $72x = 1944 \Rightarrow x = 27$. This happens when students treat a product of ratios like a sum of ratios. Always mind your exponents when multiplying variables. ### Final Answer **Therefore, the correct answer is 18.**