To find the Least Common Multiple (LCM) and Highest Common Factor (HCF) of the fractions [tex]\(\frac{3}{5}\)[/tex], [tex]\(\frac{1}{3}\)[/tex], and [tex]\(\frac{5}{6}\)[/tex], we need to follow a set of steps involving numerators and denominators. Here's the detailed step-by-step solution:
### Step 1: Finding the Least Common Multiple (LCM)
#### 1. Identify the denominators of each fraction:
- [tex]\(\frac{3}{5}\)[/tex]: denominator = 5
- [tex]\(\frac{1}{3}\)[/tex]: denominator = 3
- [tex]\(\frac{5}{6}\)[/tex]: denominator = 6
#### 2. Calculate the LCM of the denominators:
- Find the LCM of 5, 3, and 6.
To find the LCM, identify the prime factors:
- 5 is prime, so its prime factor is [tex]\(5\)[/tex].
- 3 is prime, so its prime factor is [tex]\(3\)[/tex].
- 6 is [tex]\(2 \times 3\)[/tex].
The LCM is found by taking the highest power of each prime factor appearing:
- LCM [tex]\(= 2^1 \times 3^1 \times 5^1 = 30\)[/tex].
Thus, the LCM of the denominators 5, 3, and 6 is 30.
### Step 2: Adjust Numerators to the Common Denominator
We need to adjust the numerators so that each fraction has the LCM as the common denominator:
- [tex]\(\frac{3}{5}\)[/tex] converts to [tex]\(\frac{3 \times 6}{5 \times 6} = \frac{18}{30}\)[/tex]
- [tex]\(\frac{1}{3}\)[/tex] converts to [tex]\(\frac{1 \times 10}{3 \times 10} = \frac{10}{30}\)[/tex]
- [tex]\(\frac{5}{6}\)[/tex] converts to [tex]\(\frac{5 \times 5}{6 \times 5} = \frac{25}{30}\)[/tex]
### Step 3: Finding the Highest Common Factor (HCF)
#### 1. Identify the adjusted numerators:
- The adjusted numerators are 18, 10, and 25.
#### 2. Calculate the HCF of these adjusted numerators:
- To find the HCF of 18, 10, and 25, consider the prime factorizations:
- 18 = [tex]\(2 \times 3^2\)[/tex]
- 10 = [tex]\(2 \times 5\)[/tex]
- 25 = [tex]\(5^2\)[/tex]
The only common factor among the numerators 18, 10, and 25 is 1, as they share no other common factors.
#### 3. Form the HCF fraction with the LCM as the denominator:
- The HCF of the adjusted numerators is 1.
- Therefore, the HCF of the fractions is [tex]\(\frac{1}{30}\)[/tex].
### Results
- The LCM of the fractions [tex]\(\frac{3}{5}\)[/tex], [tex]\(\frac{1}{3}\)[/tex], and [tex]\(\frac{5}{6}\)[/tex] is represented by the common denominator, which is 30.
- The HCF of these fractions is [tex]\(\frac{1}{30}\)[/tex], as the numerators share a common factor of 1 when adjusted to the common denominator.
Therefore:
- LCM (denominator): 30
- HCF (fraction): [tex]\(\frac{1}{30}\)[/tex]
