In each of the following questions, find the exact value of (?)
`2 (3)/(5) + 3 (2)/(3) - 1 (1)/(2) = ? + 1 (23)/(30)`

Options

A. 2
B. 4
C. 3
D. 1

Answer

To solve the equation \( 2 \frac{3}{5} + 3 \frac{2}{3} - 1 \frac{1}{2} = ? + 1 \frac{23}{30} \), we will first convert the mixed fractions into improper fractions, then simplify and solve for the unknown.

### Step-by-Step Solution:

1. **Convert Mixed Fractions to Improper Fractions:**
- For \( 2 \frac{3}{5} \):
\[ 2 \frac{3}{5} = \frac{2 \times 5 + 3}{5} = \frac{10 + 3}{5} = \frac{13}{5} \]
- For \( 3 \frac{2}{3} \):
\[ 3 \frac{2}{3} = \frac{3 \times 3 + 2}{3} = \frac{9 + 2}{3} = \frac{11}{3} \]
- For \( 1 \frac{1}{2} \):
\[ 1 \frac{1}{2} = \frac{1 \times 2 + 1}{2} = \frac{2 + 1}{2} = \frac{3}{2} \]
- For \( 1 \frac{23}{30} \):
\[ 1 \frac{23}{30} = \frac{1 \times 30 + 23}{30} = \frac{30 + 23}{30} = \frac{53}{30} \]

2. **Substituting Back into the Equation:**
Now, substituting the improper fractions back into the equation, we have:
\[ \frac{13}{5} + \frac{11}{3} - \frac{3}{2} = ? + \frac{53}{30} \]

3. **Finding a Common Denominator:**
The least common multiple (LCM) of the denominators \(5\), \(3\), and \(2\) is \(30\). We will convert each fraction to have a denominator of \(30\):
- For \( \frac{13}{5} \):
\[ \frac{13}{5} = \frac{13 \times 6}{5 \times 6} = \frac{78}{30} \]
- For \( \frac{11}{3} \):
\[ \frac{11}{3} = \frac{11 \times 10}{3 \times 10} = \frac{110}{30} \]
- For \( \frac{3}{2} \):
\[ \frac{3}{2} = \frac{3 \times 15}{2 \times 15} = \frac{45}{30} \]

4. **Substituting the Converted Fractions:**
Now, substituting these into the equation:
\[ \frac{78}{30} + \frac{110}{30} - \frac{45}{30} = ? + \frac{53}{30} \]

5. **Combining the Left Side:**
Combine the fractions on the left:
\[ \frac{78 + 110 - 45}{30} = \frac{143}{30} \]
So, we have:
\[ \frac{143}{30} = ? + \frac{53}{30} \]

6. **Isolating the Unknown:**
To isolate \( ? \), subtract \( \frac{53}{30} \) from both sides:
\[ ? = \frac{143}{30} - \frac{53}{30} = \frac{143 - 53}{30} = \frac{90}{30} \]

7. **Simplifying the Result:**
Simplifying \( \frac{90}{30} \):
\[ ? = 3 \]

### Final Answer:
The exact value of \( ? \) is \( 3 \).