Simplify : `4(3)/(5)-2(7)/(9)-1(2)/(15)-(2)/(5)`

Options

A. `(14)/(45)`
B. `(13)/(45)`
C. `(16)/(45)`
D. `(12)/(45)`

Answer

To simplify the expression \( \frac{4 \cdot 3}{5} - \frac{2 \cdot 7}{9} - \frac{1 \cdot 2}{15} - \frac{2}{5} \), we can follow these steps:

### Step 1: Convert the mixed fractions to improper fractions
1. The first fraction \( \frac{4 \cdot 3}{5} \) is already in improper form: \( \frac{12}{5} \).
2. The second fraction \( \frac{2 \cdot 7}{9} \) is also in improper form: \( \frac{14}{9} \).
3. The third fraction \( \frac{1 \cdot 2}{15} \) is in improper form: \( \frac{2}{15} \).
4. The last fraction \( \frac{2}{5} \) is already in proper form.

So we rewrite the expression as:
\[ \frac{12}{5} - \frac{14}{9} - \frac{2}{15} - \frac{2}{5} \]

### Step 2: Find a common denominator
The denominators are 5, 9, 15, and 5. The least common multiple (LCM) of these numbers is 45.

### Step 3: Convert each fraction to have the common denominator
1. For \( \frac{12}{5} \):
\[ \frac{12}{5} = \frac{12 \cdot 9}{5 \cdot 9} = \frac{108}{45} \]
2. For \( \frac{14}{9} \):
\[ \frac{14}{9} = \frac{14 \cdot 5}{9 \cdot 5} = \frac{70}{45} \]
3. For \( \frac{2}{15} \):
\[ \frac{2}{15} = \frac{2 \cdot 3}{15 \cdot 3} = \frac{6}{45} \]
4. For \( \frac{2}{5} \):
\[ \frac{2}{5} = \frac{2 \cdot 9}{5 \cdot 9} = \frac{18}{45} \]

Now we can rewrite the expression as:
\[ \frac{108}{45} - \frac{70}{45} - \frac{6}{45} - \frac{18}{45} \]

### Step 4: Combine the fractions
Now that all fractions have the same denominator, we can combine them:
\[ \frac{108 - 70 - 6 - 18}{45} \]

### Step 5: Perform the arithmetic in the numerator
1. First, subtract \( 70 \) from \( 108 \):
\[ 108 - 70 = 38 \]
2. Next, subtract \( 6 \):
\[ 38 - 6 = 32 \]
3. Finally, subtract \( 18 \):
\[ 32 - 18 = 14 \]

So we have:
\[ \frac{14}{45} \]

### Final Result
The simplified form of the expression is:
\[ \frac{14}{45} \]