The L.C.M. of $\frac{1}{3}, \frac{5}{6}, \frac{2}{9}, \frac{4}{27}$ is

### Concept & Formula To find the Least Common Multiple (LCM) of a set of fractions, we use the standard formula: $$\text{L.C.M. of fractions} = \frac{\text{L.C.M. of Numerators}}{\text{H.C.F. of Denominators}}$$ ### Step-by-Step Solution * **Given Fractions:** $\frac{1}{3}$, $\frac{5}{6}$, $\frac{2}{9}$, and $\frac{4}{27}$. * **Step 1: Calculate the L.C.M. of the numerators ($1, 5, 2, 4$).** The multiples of the highest number ($5$) that are also divisible by $4$ and $2$ is $20$. $$\text{L.C.M. (1, 5, 2, 4)} = 20$$ * **Step 2: Calculate the H.C.F. of the denominators ($3, 6, 9, 27$).** The factors are $3$, $2 \times 3$, $3^2$, and $3^3$. The highest common factor shared by all is clearly $3$. $$\text{H.C.F. (3, 6, 9, 27)} = 3$$ * **Step 3: Assemble the fraction.** $$\text{Result} = \frac{20}{3}$$ ### Exam Strategy & Shortcut First, scan the numerators: $1, 5, 2, 4$. Their L.C.M. is obviously $20$. Look at the options—only option (c) has $20$ in the numerator. You can confidently select (c) without even calculating the H.C.F. for the denominators. ### Common Pitfall Students often mix up the formula, calculating the L.C.M. of the denominators and the H.C.F. of the numerators (which would give $\frac{1}{54}$, option a). Remember: "Whatever they ask for goes on top." If they ask for L.C.M., put L.C.M. in the numerator. ### Final Answer **Therefore, the correct answer is $\frac{20}{3}$.**