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

Options

A. `(8)/(45)`
B. `(7)/(47)`
C. `(7)/(45)`
D. `(8)/(51)`

Explanation

To solve the equation \( 3\frac{2}{5} + 1\frac{2}{9} = 4\frac{4}{5} - ? \), we will follow these steps:

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

1. For \( 3\frac{2}{5} \):
\[ 3\frac{2}{5} = \frac{3 \times 5 + 2}{5} = \frac{15 + 2}{5} = \frac{17}{5} \]

2. For \( 1\frac{2}{9} \):
\[ 1\frac{2}{9} = \frac{1 \times 9 + 2}{9} = \frac{9 + 2}{9} = \frac{11}{9} \]

3. For \( 4\frac{4}{5} \):
\[ 4\frac{4}{5} = \frac{4 \times 5 + 4}{5} = \frac{20 + 4}{5} = \frac{24}{5} \]

### Step 2: Substitute the Improper Fractions into the Equation
Now we substitute the improper fractions back into the equation:
\[ \frac{17}{5} + \frac{11}{9} = \frac{24}{5} - x \]

### Step 3: Isolate the Variable \( x \)
To isolate \( x \), we rearrange the equation:
\[ x = \frac{24}{5} - \left( \frac{17}{5} + \frac{11}{9} \right) \]

### Step 4: Combine the Terms on the Right Side
First, we need to find a common denominator to combine \( \frac{17}{5} \) and \( \frac{11}{9} \). The least common multiple (LCM) of 5 and 9 is 45.

1. Convert \( \frac{17}{5} \) to have a denominator of 45:
\[ \frac{17}{5} = \frac{17 \times 9}{5 \times 9} = \frac{153}{45} \]

2. Convert \( \frac{11}{9} \) to have a denominator of 45:
\[ \frac{11}{9} = \frac{11 \times 5}{9 \times 5} = \frac{55}{45} \]

3. Now, add these two fractions:
\[ \frac{153}{45} + \frac{55}{45} = \frac{208}{45} \]

### Step 5: Substitute Back and Solve for \( x \)
Now we substitute back into the equation for \( x \):
\[ x = \frac{24}{5} - \frac{208}{45} \]

Convert \( \frac{24}{5} \) to have a denominator of 45:
\[ \frac{24}{5} = \frac{24 \times 9}{5 \times 9} = \frac{216}{45} \]

Now we can subtract:
\[ x = \frac{216}{45} - \frac{208}{45} = \frac{8}{45} \]

### Step 6: Final Answer
Thus, the value of \( ? \) is:
\[ \boxed{\frac{8}{45}} \]