Difference between the greatest common factor of `(2)/(5) and (3)/(4)` and smallest common multiple of `(5)/(2) and (4)/(3)` is

Options

A. `(398)/(20)`
B. `-(399)/(20)`
C. `(399)/(21)`
D. `(396)/(20)`

Answer

To solve the problem, we need to find the difference between the greatest common factor (GCF) of \( \frac{2}{5} \) and \( \frac{3}{4} \), and the least common multiple (LCM) of \( \frac{5}{2} \) and \( \frac{4}{3} \).

### Step-by-Step Solution:

**Step 1: Find the GCF of \( \frac{2}{5} \) and \( \frac{3}{4} \)**

- The GCF of two fractions is calculated using the formula:
\[ \text{GCF} \left( \frac{a}{b}, \frac{c}{d} \right) = \frac{\text{GCF}(a, c)}{\text{LCM}(b, d)} \]

- Here, \( a = 2 \), \( b = 5 \), \( c = 3 \), and \( d = 4 \).

- **Find GCF of the numerators (2 and 3)**:
- The factors of 2 are: 1, 2
- The factors of 3 are: 1, 3
- The GCF of 2 and 3 is 1.

- **Find LCM of the denominators (5 and 4)**:
- The multiples of 5 are: 5, 10, 15, 20, ...
- The multiples of 4 are: 4, 8, 12, 16, 20, ...
- The LCM of 5 and 4 is 20.

- Therefore, the GCF of \( \frac{2}{5} \) and \( \frac{3}{4} \) is:
\[ \text{GCF} \left( \frac{2}{5}, \frac{3}{4} \right) = \frac{1}{20} \]

**Step 2: Find the LCM of \( \frac{5}{2} \) and \( \frac{4}{3} \)**

- The LCM of two fractions is calculated using the formula:
\[ \text{LCM} \left( \frac{a}{b}, \frac{c}{d} \right) = \frac{\text{LCM}(a, c)}{\text{GCF}(b, d)} \]

- Here, \( a = 5 \), \( b = 2 \), \( c = 4 \), and \( d = 3 \).

- **Find LCM of the numerators (5 and 4)**:
- The multiples of 5 are: 5, 10, 15, 20, ...
- The multiples of 4 are: 4, 8, 12, 16, 20, ...
- The LCM of 5 and 4 is 20.

- **Find GCF of the denominators (2 and 3)**:
- The factors of 2 are: 1, 2
- The factors of 3 are: 1, 3
- The GCF of 2 and 3 is 1.

- Therefore, the LCM of \( \frac{5}{2} \) and \( \frac{4}{3} \) is:
\[ \text{LCM} \left( \frac{5}{2}, \frac{4}{3} \right) = \frac{20}{1} = 20 \]

**Step 3: Find the difference between the GCF and the LCM**

- Now we need to calculate:
\[ \text{Difference} = \text{GCF} - \text{LCM} = \frac{1}{20} - 20 \]

- To perform the subtraction, we convert 20 into a fraction with a denominator of 20:
\[ 20 = \frac{400}{20} \]

- Thus, the difference becomes:
\[ \frac{1}{20} - \frac{400}{20} = \frac{1 - 400}{20} = \frac{-399}{20} \]

### Final Answer:
The difference between the greatest common factor of \( \frac{2}{5} \) and \( \frac{3}{4} \), and the least common multiple of \( \frac{5}{2} \) and \( \frac{4}{3} \) is:
\[ \frac{-399}{20} \]