H.C.F. of $\frac{4}{5}$ and $\frac{7}{15}$ is

Question

Accepted Answer

The correct option is 1:
To find the H.C.F. (Highest Common Factor) of fractions, we use the following formula:
$$H.C.F. \text{ of fractions} = \frac{H.C.F. \text{ of numerators}}{L.C.M. \text{ of denominators}}$$

Given fractions are $\frac{4}{5}$ and $\frac{7}{15}$.

Step 1: Find the H.C.F. of the numerators.
The numerators are 4 and 7.
Prime factorization of 4: $2 \times 2$
Prime factorization of 7: $7$
The only common factor is 1.
So, H.C.F. (4, 7) = 1.

Step 2: Find the L.C.M. of the denominators.
The denominators are 5 and 15.
Prime factorization of 5: $5$
Prime factorization of 15: $3 \times 5$
To find the L.C.M., we take the highest power of all prime factors involved: $3 \times 5 = 15$.
So, L.C.M. (5, 15) = 15.

Step 3: Apply the H.C.F. of fractions formula.
$$H.C.F. \left(\frac{4}{5}, \frac{7}{15}\right) = \frac{H.C.F. (4, 7)}{L.C.M. (5, 15)} = \frac{1}{15}$$
Therefore, the H.C.F. of $\frac{4}{5}$ and $\frac{7}{15}$ is $\frac{1}{15}$.

The incorrect option is 2: $\frac{28}{75}$ is incorrect because it might result from multiplying the numerators and multiplying the denominators, which is not the method for finding the H.C.F. of fractions.
The incorrect option is 3: $\frac{1}{25}$ is incorrect because it appears to have taken the H.C.F. of numerators correctly, but used an incorrect L.C.M. of the denominators or made a mistake in calculation.
The incorrect option is 4: $\frac{1}{5}$ is incorrect because it has the correct denominator as a result of L.C.M. but an incorrect numerator, or a mistake in applying the formula.