The L.C.M. of $\frac{3}{4}, \frac{6}{7}, \frac{9}{8}$ is

### Concept & Formula To evaluate the Least Common Multiple (LCM) of given fractions, the corresponding rule is: $$\text{L.C.M. of fractions} = \frac{\text{L.C.M. of Numerators}}{\text{H.C.F. of Denominators}}$$ ### Step-by-Step Solution * **Given Fractions:** $\frac{3}{4}$, $\frac{6}{7}$, and $\frac{9}{8}$. * **Step 1: Determine the L.C.M. of numerators ($3, 6, 9$).** The numbers are $3$, $2 \times 3$, and $3^2$. The L.C.M. is $2 \times 3^2 = 18$. * **Step 2: Determine the H.C.F. of denominators ($4, 7, 8$).** The factors are $2^2$, $7$, and $2^3$. Since $7$ is a prime number and doesn't divide $4$ or $8$, the only common factor is $1$. * **Step 3: Final Calculation.** $$\text{Result} = \frac{18}{1} = 18$$ ### Exam Strategy & Shortcut Recognize immediately that the denominator $7$ is prime, while $4$ and $8$ are powers of $2$. Their H.C.F. must be $1$. Thus, the answer is just the L.C.M. of $3, 6$, and $9$. Since $18$ is the smallest number divisible by all three, the answer is $18$. Takes 5 seconds visually. ### Common Pitfall Students might mistakenly calculate the L.C.M. of both numerator and denominator, or just focus on the largest numbers. Another mistake is choosing $9$ (option c) by just looking at the largest numerator, forgetting that $9$ is not divisible by $6$. ### Final Answer **Therefore, the correct answer is 18.**