Find the H.C.F. and L.C.M. of $\frac{2}{3}$, $\frac{8}{9}$, $\frac{16}{81}$ and $\frac{10}{27}$.

### Concept & Formula To find the H.C.F. and L.C.M. of a set of fractions, we use the following standard formulas: $$ \text{H.C.F. of fractions} = \frac{\text{H.C.F. of Numerators}}{\text{L.C.M. of Denominators}} $$ $$ \text{L.C.M. of fractions} = \frac{\text{L.C.M. of Numerators}}{\text{H.C.F. of Denominators}} $$ ### Step-by-Step Solution **Given fractions:** $\frac{2}{3}$, $\frac{8}{9}$, $\frac{16}{81}$, $\frac{10}{27}$ * **Numerators:** $2, 8, 16, 10$ * **Denominators:** $3, 9, 81, 27$ **Step 1: Calculate the H.C.F. of the fractions.** * Find H.C.F. of Numerators ($2, 8, 16, 10$): The highest common factor dividing all these numbers is clearly $2$. * Find L.C.M. of Denominators ($3, 9, 81, 27$): Since $3, 9$, and $27$ are all factors of $81$, the least common multiple is $81$. * Apply the formula: $\text{H.C.F.} = \frac{2}{81}$ **Step 2: Calculate the L.C.M. of the fractions.** * Find L.C.M. of Numerators ($2, 8, 16, 10$): $16$ covers the factors of $2$ and $8$. We just need the L.C.M. of $16$ and $10$. $16 = 2^4$ $10 = 2 \times 5$ L.C.M. = $2^4 \times 5 = 16 \times 5 = 80$. * Find H.C.F. of Denominators ($3, 9, 81, 27$): The highest common factor dividing all these numbers is $3$. * Apply the formula: $\text{L.C.M.} = \frac{80}{3}$ ### Exam Strategy & Shortcut In multiple-choice exams, you rarely need to calculate both values entirely. Calculate the easiest component first. The H.C.F. of the numerators ($2, 8, 16, 10$) is instantly visible as $2$. The L.C.M. of denominators ($3, 9, 81, 27$) is instantly visible as $81$ because $81$ is a multiple of the others. Thus, H.C.F. is $\frac{2}{81}$. You can often eliminate 2-3 incorrect options just by finding this first half of the answer. ### Common Pitfall The most common mistake is swapping the formulas under pressure—calculating the L.C.M. of numerators when asked for the H.C.F. of the fractions. To avoid this, remember that the operation requested (H.C.F. or L.C.M.) always applies to the **numerators**, while the opposite operation applies to the denominators. ### Final Answer **Therefore, the correct answer is H.C.F. = $\frac{2}{81}$, L.C.M. = $\frac{80}{3}$.**