Simplify the ratio:
` 2 (1)/(3) : 3 (1)/(2)`

Answer

To simplify the ratio \( 2 \frac{1}{3} : 3 \frac{1}{2} \), we can follow these steps:

### Step 1: Convert the mixed fractions to improper fractions
- For \( 2 \frac{1}{3} \):
\[ 2 \frac{1}{3} = 2 \times 3 + 1 = 6 + 1 = \frac{7}{3} \]

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

Now, we can rewrite the ratio:
\[ \frac{7}{3} : \frac{7}{2} \]

### Step 2: Find the LCM of the denominators
The denominators are 3 and 2. The least common multiple (LCM) of 3 and 2 is 6.

### Step 3: Multiply both fractions by the LCM
Now we will multiply each part of the ratio by the LCM (6):
\[ \left(\frac{7}{3} \times 6\right) : \left(\frac{7}{2} \times 6\right) \]

Calculating each part:
- For \( \frac{7}{3} \times 6 \):
\[ \frac{7 \times 6}{3} = \frac{42}{3} = 14 \]

- For \( \frac{7}{2} \times 6 \):
\[ \frac{7 \times 6}{2} = \frac{42}{2} = 21 \]

Now the ratio is:
\[ 14 : 21 \]

### Step 4: Simplify the ratio
To simplify \( 14 : 21 \), we can divide both terms by their greatest common divisor (GCD), which is 7:
\[ \frac{14}{7} : \frac{21}{7} = 2 : 3 \]

### Final Answer
The simplified ratio is:
\[ 2 : 3 \]

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