Simplify : `4 1/3 - 2 4/9+1 3/5 -2/5`

Options

A. `3 4/45`
B. `1 4/45`
C. `3 7/45`
D. `1 2/45`

Answer

To simplify the expression \( 4 \frac{1}{3} - 2 \frac{4}{9} + 1 \frac{3}{5} - \frac{2}{5} \), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we convert the mixed numbers into improper fractions:

- \( 4 \frac{1}{3} = \frac{4 \times 3 + 1}{3} = \frac{12 + 1}{3} = \frac{13}{3} \)
- \( 2 \frac{4}{9} = \frac{2 \times 9 + 4}{9} = \frac{18 + 4}{9} = \frac{22}{9} \)
- \( 1 \frac{3}{5} = \frac{1 \times 5 + 3}{5} = \frac{5 + 3}{5} = \frac{8}{5} \)

Now, we rewrite the expression:
\[ \frac{13}{3} - \frac{22}{9} + \frac{8}{5} - \frac{2}{5} \]

### Step 2: Simplify the Fractions
Next, we can combine the fractions with the same denominator:
\[ \frac{8}{5} - \frac{2}{5} = \frac{8 - 2}{5} = \frac{6}{5} \]

Now our expression looks like this:
\[ \frac{13}{3} - \frac{22}{9} + \frac{6}{5} \]

### Step 3: Find a Common Denominator
To combine all fractions, we need a common denominator. The denominators are 3, 9, and 5. The least common multiple (LCM) of these numbers is 45.

Now, we convert each fraction:
- For \( \frac{13}{3} \):
\[ \frac{13}{3} = \frac{13 \times 15}{3 \times 15} = \frac{195}{45} \]
- For \( \frac{22}{9} \):
\[ \frac{22}{9} = \frac{22 \times 5}{9 \times 5} = \frac{110}{45} \]
- For \( \frac{6}{5} \):
\[ \frac{6}{5} = \frac{6 \times 9}{5 \times 9} = \frac{54}{45} \]

### Step 4: Combine the Fractions
Now we can combine the fractions:
\[ \frac{195}{45} - \frac{110}{45} + \frac{54}{45} = \frac{195 - 110 + 54}{45} = \frac{139}{45} \]

### Step 5: Final Result
Thus, the simplified form of the expression is:
\[ \frac{139}{45} \]

### Step 6: Convert to Mixed Number (if needed)
To convert \( \frac{139}{45} \) to a mixed number:
- Divide 139 by 45, which gives 3 with a remainder of 4.
- Therefore, \( \frac{139}{45} = 3 \frac{4}{45} \).

So, the final answer can be expressed as:
\[ 3 \frac{4}{45} \]