`1(3)/( 5) + 1( 8)/( 9) + 2 ( 4)/( 5) = ?`

Options

A. `6(19)/( 45)`
B. `6(16)/(45)`
C. `6(17)/( 45)`
D. `6(13)/( 45)`

Answer

To solve the question \( 1 \frac{3}{5} + 1 \frac{8}{9} + 2 \frac{4}{5} \), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
1. Convert \( 1 \frac{3}{5} \) to an improper fraction:
\[ 1 \frac{3}{5} = \frac{1 \times 5 + 3}{5} = \frac{5 + 3}{5} = \frac{8}{5} \]

2. Convert \( 1 \frac{8}{9} \) to an improper fraction:
\[ 1 \frac{8}{9} = \frac{1 \times 9 + 8}{9} = \frac{9 + 8}{9} = \frac{17}{9} \]

3. Convert \( 2 \frac{4}{5} \) to an improper fraction:
\[ 2 \frac{4}{5} = \frac{2 \times 5 + 4}{5} = \frac{10 + 4}{5} = \frac{14}{5} \]

### Step 2: Add the Improper Fractions
Now we need to add \( \frac{8}{5} + \frac{17}{9} + \frac{14}{5} \).

1. Combine the fractions with the same denominator:
\[ \frac{8}{5} + \frac{14}{5} = \frac{8 + 14}{5} = \frac{22}{5} \]

2. Now we have:
\[ \frac{22}{5} + \frac{17}{9} \]

### Step 3: Find the Least Common Multiple (LCM)
To add \( \frac{22}{5} \) and \( \frac{17}{9} \), we need a common denominator. The LCM of 5 and 9 is 45.

### Step 4: Convert Each Fraction to Have the Common Denominator
1. Convert \( \frac{22}{5} \) to a fraction with a denominator of 45:
\[ \frac{22}{5} = \frac{22 \times 9}{5 \times 9} = \frac{198}{45} \]

2. Convert \( \frac{17}{9} \) to a fraction with a denominator of 45:
\[ \frac{17}{9} = \frac{17 \times 5}{9 \times 5} = \frac{85}{45} \]

### Step 5: Add the Converted Fractions
Now we can add the two fractions:
\[ \frac{198}{45} + \frac{85}{45} = \frac{198 + 85}{45} = \frac{283}{45} \]

### Step 6: Convert to Mixed Number
To convert \( \frac{283}{45} \) to a mixed number:
1. Divide 283 by 45:
- \( 283 \div 45 = 6 \) (since \( 6 \times 45 = 270 \))
- Remainder: \( 283 - 270 = 13 \)

Thus, we can write:
\[ \frac{283}{45} = 6 \frac{13}{45} \]

### Final Answer
The final answer is:
\[ 6 \frac{13}{45} \]