Finding the L.C.M. of Fractions Explained

To find the Least Common Multiple (L.C.M.) of fractions, we use a specific formula that relates the L.C.M. of the numerators and the Highest Common Factor (H.C.F.) of the denominators.

Formula for L.C.M. of Fractions

The formula is:

$$ \text{L.C.M. of fractions} = \frac{\text{L.C.M. of numerators}}{\text{H.C.F. of denominators}} $$

Applying the Formula to the Given Fractions

The given fractions are $\frac{4}{5}$, $\frac{2}{3}$, and $\frac{5}{7}$.

• The numerators are 4, 2, and 5.
• The denominators are 5, 3, and 7.

Step 1: Find the L.C.M. of the Numerators

We need to find the L.C.M. of 4, 2, and 5.

Let's list the prime factors:

• 4 = $2 \times 2 = 2^2$
• 2 = $2^1$
• 5 = $5^1$

To find the L.C.M., we take the highest power of all prime factors that appear in any of the numbers.

• Highest power of 2 is $2^2$.
• Highest power of 5 is $5^1$.

L.C.M.(4, 2, 5) = $2^2 \times 5^1 = 4 \times 5 = 20$.

Step 2: Find the H.C.F. of the Denominators

We need to find the H.C.F. of 5, 3, and 7.

Let's find the factors of each number:

• Factors of 5: 1, 5
• Factors of 3: 1, 3
• Factors of 7: 1, 7

The H.C.F. is the largest factor that is common to all the numbers. In this case, the only common factor is 1.

H.C.F.(5, 3, 7) = 1.

Step 3: Calculate the L.C.M. of the Fractions

Now, we use the formula:

$$ \text{L.C.M. of } \frac{4}{5}, \frac{2}{3}, \frac{5}{7} = \frac{\text{L.C.M. of (4, 2, 5)}}{\text{H.C.F. of (5, 3, 7)}} $$

Substituting the values we found:

$$ \text{L.C.M. of fractions} = \frac{20}{1} = 20 $$

So, the L.C.M. of $\frac{4}{5}$, $\frac{2}{3}$, and $\frac{5}{7}$ is 20.

Revision Table: L.C.M. and H.C.F. Concepts

Let's quickly review the definitions of L.C.M. and H.C.F.

Additional Information: H.C.F. of Fractions

While this question is about the L.C.M. of fractions, it's useful to know how to find the H.C.F. of fractions as well. The formula is similar but inverted:

$$ \text{H.C.F. of fractions} = \frac{\text{H.C.F. of numerators}}{\text{L.C.M. of denominators}} $$

For the fractions $\frac{4}{5}$, $\frac{2}{3}$, $\frac{5}{7}$:

• H.C.F. of numerators (4, 2, 5): H.C.F.(4, 2, 5) = 1 (as found earlier).
• L.C.M. of denominators (5, 3, 7): Since 5, 3, and 7 are prime numbers, their L.C.M. is their product: L.C.M.(5, 3, 7) = $5 \times 3 \times 7 = 105$.

So, the H.C.F. of $\frac{4}{5}$, $\frac{2}{3}$, and $\frac{5}{7}$ would be $\frac{1}{105}$.

Understanding both formulas helps in solving problems involving either L.C.M. or H.C.F. of fractions.