To find the LCM and HCF of the fractions \( \frac{25}{12} \) and \( \frac{35}{18} \), we will follow the steps outlined in the video transcript.
### Step 1: Find the LCM of the numerators
We need to find the LCM of the numerators \( 25 \) and \( 35 \).
- **Prime factorization of 25**: \( 25 = 5^2 \)
- **Prime factorization of 35**: \( 35 = 5^1 \times 7^1 \)
To find the LCM, we take the highest power of each prime factor:
- For \( 5 \): Highest power is \( 5^2 \)
- For \( 7 \): Highest power is \( 7^1 \)
Thus, the LCM of \( 25 \) and \( 35 \) is:
\[
LCM(25, 35) = 5^2 \times 7^1 = 25 \times 7 = 175
\]
### Step 2: Find the HCF of the denominators
Now, we need to find the HCF of the denominators \( 12 \) and \( 18 \).
- **Prime factorization of 12**: \( 12 = 2^2 \times 3^1 \)
- **Prime factorization of 18**: \( 18 = 2^1 \times 3^2 \)
To find the HCF, we take the lowest power of each common prime factor:
- For \( 2 \): Lowest power is \( 2^1 \)
- For \( 3 \): Lowest power is \( 3^1 \)
Thus, the HCF of \( 12 \) and \( 18 \) is:
\[
HCF(12, 18) = 2^1 \times 3^1 = 2 \times 3 = 6
\]
### Step 3: Calculate the LCM of the fractions
Using the results from Steps 1 and 2, we can find the LCM of the fractions:
\[
LCM\left(\frac{25}{12}, \frac{35}{18}\right) = \frac{LCM(25, 35)}{HCF(12, 18)} = \frac{175}{6}
\]
### Step 4: Find the HCF of the fractions
Now, we will find the HCF of the fractions:
\[
HCF\left(\frac{25}{12}, \frac{35}{18}\right) = \frac{HCF(25, 35)}{LCM(12, 18)}
\]
First, we find the HCF of the numerators \( 25 \) and \( 35 \):
- The common factor is \( 5 \).
Now, we find the LCM of the denominators \( 12 \) and \( 18 \):
- The LCM is \( 36 \) (calculated by taking the highest powers of all prime factors).
Thus, the HCF of the fractions is:
\[
HCF\left(\frac{25}{12}, \frac{35}{18}\right) = \frac{5}{36}
\]
### Final Answers
- LCM of \( \frac{25}{12} \) and \( \frac{35}{18} \) is \( \frac{175}{6} \).
- HCF of \( \frac{25}{12} \) and \( \frac{35}{18} \) is \( \frac{5}{36} \).
