Check whether the given two ratios form a proportion or not:
`3 (1)/(2) : 2 (1)/(2) and 21: 15`

Answer

To determine whether the two ratios \(3 \frac{1}{2} : 2 \frac{1}{2}\) and \(21 : 15\) form a proportion, we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
First, we need to convert the mixed numbers \(3 \frac{1}{2}\) and \(2 \frac{1}{2}\) into improper fractions.

- For \(3 \frac{1}{2}\):
\[ 3 \frac{1}{2} = 3 \times 2 + 1 = 6 + 1 = 7 \quad \text{(improper fraction: } \frac{7}{2}\text{)} \]

- For \(2 \frac{1}{2}\):
\[ 2 \frac{1}{2} = 2 \times 2 + 1 = 4 + 1 = 5 \quad \text{(improper fraction: } \frac{5}{2}\text{)} \]

Thus, we have:
\[ 3 \frac{1}{2} : 2 \frac{1}{2} = \frac{7}{2} : \frac{5}{2} \]

### Step 2: Write the Ratios as Fractions
Now we can express the ratios as fractions:
\[ \frac{7}{2} : \frac{5}{2} = \frac{7}{2} \div \frac{5}{2} = \frac{7}{2} \times \frac{2}{5} = \frac{7 \times 2}{2 \times 5} = \frac{7}{5} \]

### Step 3: Simplify the Second Ratio
Next, we simplify the second ratio \(21 : 15\):
\[ 21 : 15 = \frac{21}{15} \]
To simplify, we find the greatest common divisor (GCD) of 21 and 15, which is 3:
\[ \frac{21 \div 3}{15 \div 3} = \frac{7}{5} \]

### Step 4: Compare the Two Ratios
Now we compare the two simplified ratios:
\[ \frac{7}{5} \quad \text{and} \quad \frac{7}{5} \]

Since both ratios are equal, we conclude that:
\[ 3 \frac{1}{2} : 2 \frac{1}{2} \text{ is equivalent to } 21 : 15 \]

### Conclusion
Thus, the two ratios \(3 \frac{1}{2} : 2 \frac{1}{2}\) and \(21 : 15\) are in proportion.

---