The H.C.F. of $\frac{2}{3}, \frac{8}{9}, \frac{64}{81}$ and $\frac{10}{27}$ is :

### Concept & Formula To find the Highest Common Factor (HCF) of a set of fractions, use the standard fraction formula: $$\text{H.C.F.} = \frac{\text{H.C.F. of Numerators}}{\text{L.C.M. of Denominators}}$$ ### Step-by-Step Solution * **Given Fractions:** $\frac{2}{3}$, $\frac{8}{9}$, $\frac{64}{81}$, and $\frac{10}{27}$. * **Step 1: Find the HCF of the numerators ($2, 8, 64, 10$).** The factors are $2$, $2^3$, $2^6$, and $2 \times 5$. The highest common factor shared by all four numbers is clearly $2$. * **Step 2: Find the LCM of the denominators ($3, 9, 81, 27$).** The denominators are all powers of $3$: $3 = 3^1$ $9 = 3^2$ $27 = 3^3$ $81 = 3^4$ The LCM of numbers with the same base is simply the one with the highest power, which is $81$. * **Step 3: Combine them into the final fraction.** $$\text{Final H.C.F.} = \frac{2}{81}$$ ### Exam Strategy & Shortcut You can solve this visually in three seconds. Look at the numerators ($2, 8, 64, 10$); the smallest number is $2$, and $2$ divides all the others perfectly, so the top must be $2$. Eliminate options (c) and (d). Look at the denominators; they are all multiples of $3$, with $81$ being the largest and a multiple of the others. So the bottom must be $81$. Option (b) is the instant winner. ### Common Pitfall Students who forget the formula often calculate the HCF for both numerators and denominators (resulting in $\frac{2}{3}$) or calculate the LCM for both (resulting in $\frac{160}{81}$). Always remember: "HCF on top for HCF questions." ### Final Answer **Therefore, the correct answer is $\frac{2}{81}$.**