HCF Calculation for Fractions

To determine the Highest Common Factor (HCF) of several fractions, apply the following formula:

$ \text{HCF}\left(\frac{a}{b}, \frac{c}{d}, \frac{e}{f}, ...\right) = \frac{\text{HCF of Numerators }(a, c, e, ...)}{\text{LCM of Denominators }(b, d, f, ...)} $

Applying the HCF Formula

We need to find the HCF of the fractions $\frac{1}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}$.

Step 1: HCF of Numerators

Identify the numerators: 1, 3, 4, 5.

Calculate their HCF:

$ \text{HCF}(1, 3, 4, 5) = 1 $

The HCF is 1 because 1 is present in the set of numerators.

Step 2: LCM of Denominators

Identify the denominators: 3, 4, 5, 6.

Calculate their Least Common Multiple (LCM):

• Prime factorizations: $3 = 3$, $4 = 2^2$, $5 = 5$, $6 = 2 \times 3$.
• The LCM requires the highest power of each prime factor present: $2^2, 3^1, 5^1$.
• $ \text{LCM}(3, 4, 5, 6) = 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60 $

Step 3: Final HCF Calculation

Substitute the calculated HCF of numerators and LCM of denominators into the formula:

$ \text{HCF} = \frac{\text{HCF}(1, 3, 4, 5)}{\text{LCM}(3, 4, 5, 6)} = \frac{1}{60} $

Thus, the HCF of the given fractions is $\frac{1}{60}$.