HCF of Fractions Calculation

The question asks for the Highest Common Factor (HCF) of the fractions $\frac{2}{3}$, $\frac{3}{4}$, and $\frac{5}{6}$.

Finding HCF of Fractions

To find the HCF of multiple fractions, we use the formula:

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

Here, $a, c, e$ are the numerators and $b, d, f$ are the denominators.

Step 1: Identify Numerators and Denominators

• Numerators: 2, 3, 5
• Denominators: 3, 4, 6

Step 2: Calculate HCF of Numerators

We need to find the HCF of 2, 3, and 5.

• Factors of 2: 1, 2
• Factors of 3: 1, 3
• Factors of 5: 1, 5
• The highest common factor is 1.

Therefore, HCF(2, 3, 5) = 1.

Step 3: Calculate LCM of Denominators

We need to find the Least Common Multiple (LCM) of 3, 4, and 6.

• Prime factorization of 3 = 3
• Prime factorization of 4 = $2^2$
• Prime factorization of 6 = 2 $\times$ 3
• LCM(3, 4, 6) = $2^2 \times 3 = 4 \times 3 = 12$.

Therefore, LCM(3, 4, 6) = 12.

Step 4: Calculate HCF of the Fractions

Using the formula from Step 1:

$ \text{HCF}\left(\frac{2}{3}, \frac{3}{4}, \frac{5}{6}\right) = \frac{\text{HCF}(2, 3, 5)}{\text{LCM}(3, 4, 6)} = \frac{1}{12} $

The HCF of the given fractions is $\frac{1}{12}$.