The H.C.F. of $\frac{9}{10}, \frac{12}{25}, \frac{18}{35}$ and $\frac{21}{40}$ is

### Concept & Formula When calculating the Highest Common Factor (HCF) of several fractions, we apply the foundational fraction rule: $$\text{H.C.F.} = \frac{\text{H.C.F. of Numerators}}{\text{L.C.M. of Denominators}}$$ ### Step-by-Step Solution * **Given Fractions:** $\frac{9}{10}$, $\frac{12}{25}$, $\frac{18}{35}$, and $\frac{21}{40}$. * **Step 1: Calculate the HCF of the numerators ($9, 12, 18, 21$).** All of these numbers are clearly multiples of $3$. $9 = 3 \times 3$ $12 = 3 \times 4$ $18 = 3 \times 6$ $21 = 3 \times 7$ The highest number that divides all of them is $3$. So, the numerator of our answer is $3$. * **Step 2: Calculate the LCM of the denominators ($10, 25, 35, 40$).** Prime factorize each: $10 = 2 \times 5$ $25 = 5^2$ $35 = 5 \times 7$ $40 = 2^3 \times 5$ Take the highest power of each prime factor ($2, 5, 7$): $$LCM = 2^3 \times 5^2 \times 7$$ $$LCM = 8 \times 25 \times 7$$ $$LCM = 200 \times 7 = 1400$$ * **Step 3: Assemble the fraction.** $$\text{Result} = \frac{3}{1400}$$ ### Exam Strategy & Shortcut Calculate the HCF of the numerators first, which is $3$. Instantly eliminate option (b) and (d). You are left with $\frac{3}{5}$ and $\frac{3}{1400}$. Now look at the denominators. For the denominator of the answer to be $5$, the LCM of $10, 25, 35, 40$ would have to be $5$, which is logically impossible (the LCM must be greater than or equal to the largest number, $40$). Thus, without doing the heavy math, option (c) is the only mathematically viable choice. ### Common Pitfall Calculating the full LCM for the denominators using the ladder method wastes a lot of time. In multiple-choice exams, determining just the numerator and doing a quick logical check on the remaining denominator options is the key to finishing on time. ### Final Answer **Therefore, the correct answer is $\frac{3}{1400}$.**