Find the fourth proportional to :
`2 (1)/(2), 2 (6)/(7), 3(1)/(2)`

Answer

To find the fourth proportional to the numbers \( 2 \frac{1}{2}, 2 \frac{6}{7}, 3 \frac{1}{2} \), we can follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
1. **Convert \( 2 \frac{1}{2} \)**:
\[ 2 \frac{1}{2} = 2 \times 2 + 1 = 4 + 1 = \frac{5}{2} \]

2. **Convert \( 2 \frac{6}{7} \)**:
\[ 2 \frac{6}{7} = 2 \times 7 + 6 = 14 + 6 = \frac{20}{7} \]

3. **Convert \( 3 \frac{1}{2} \)**:
\[ 3 \frac{1}{2} = 3 \times 2 + 1 = 6 + 1 = \frac{7}{2} \]

### Step 2: Set Up the Proportion
Now that we have converted the mixed numbers, we can set up the proportion:
\[ \frac{5}{2} : \frac{20}{7} :: \frac{7}{2} : x \]

### Step 3: Use the Property of Proportions
According to the property of proportions, the product of the means equals the product of the extremes:
\[ \frac{5}{2} \times x = \frac{20}{7} \times \frac{7}{2} \]

### Step 4: Simplify the Right Side
Calculating the right side:
\[ \frac{20}{7} \times \frac{7}{2} = \frac{20 \times 7}{7 \times 2} = \frac{20}{2} = 10 \]

### Step 5: Solve for \( x \)
Now we have:
\[ \frac{5}{2} \times x = 10 \]
To isolate \( x \), multiply both sides by \( \frac{2}{5} \):
\[ x = 10 \times \frac{2}{5} \]
\[ x = \frac{20}{5} = 4 \]

### Conclusion
The fourth proportional is \( 4 \).

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