Find the value of x when `3 (1)/(2) : x : : 2 (1)/(2) : 4`

Answer

To solve the problem, we need to find the value of \( x \) in the proportion \( 3 \frac{1}{2} : x :: 2 \frac{1}{2} : 4 \).

### Step-by-Step Solution:

1. **Convert Mixed Numbers to Improper Fractions:**
- Convert \( 3 \frac{1}{2} \) to an improper fraction:
\[ 3 \frac{1}{2} = \frac{3 \times 2 + 1}{2} = \frac{6 + 1}{2} = \frac{7}{2} \]
- Convert \( 2 \frac{1}{2} \) to an improper fraction:
\[ 2 \frac{1}{2} = \frac{2 \times 2 + 1}{2} = \frac{4 + 1}{2} = \frac{5}{2} \]

Now we can rewrite the proportion:
\[ \frac{7}{2} : x :: \frac{5}{2} : 4 \]

2. **Set Up the Proportion:**
- We can express the proportion as:
\[ \frac{7}{2} : x = \frac{5}{2} : 4 \]
- This can be written as:
\[ \frac{7/2}{x} = \frac{5/2}{4} \]

3. **Cross Multiply:**
- Using the property of proportions, we can cross multiply:
\[ 7 \cdot 4 = x \cdot 5 \]
- This simplifies to:
\[ 28 = 5x \]

4. **Solve for \( x \):**
- Now, divide both sides by 5:
\[ x = \frac{28}{5} \]

5. **Convert to Mixed Number (if needed):**
- To convert \( \frac{28}{5} \) to a mixed number:
\[ 28 \div 5 = 5 \quad \text{(remainder 3)} \]
- So, \( \frac{28}{5} = 5 \frac{3}{5} \).

### Final Answer:
Thus, the value of \( x \) is \( 5 \frac{3}{5} \).

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