The average of the fractions `1 1/2,2 1/3,3 1/3,4 5/6` is -

Options

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

Answer

To find the average of the fractions \(1 \frac{1}{2}, 2 \frac{1}{3}, 3 \frac{1}{3}, 4 \frac{5}{6}\), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
1. Convert \(1 \frac{1}{2}\) to an improper fraction:
\[ 1 \frac{1}{2} = \frac{2 \times 1 + 1}{2} = \frac{3}{2} \]

2. Convert \(2 \frac{1}{3}\) to an improper fraction:
\[ 2 \frac{1}{3} = \frac{3 \times 2 + 1}{3} = \frac{7}{3} \]

3. Convert \(3 \frac{1}{3}\) to an improper fraction:
\[ 3 \frac{1}{3} = \frac{3 \times 3 + 1}{3} = \frac{10}{3} \]

4. Convert \(4 \frac{5}{6}\) to an improper fraction:
\[ 4 \frac{5}{6} = \frac{6 \times 4 + 5}{6} = \frac{29}{6} \]

### Step 2: Find the Sum of the Improper Fractions
Now, we will add the improper fractions:
\[ \text{Sum} = \frac{3}{2} + \frac{7}{3} + \frac{10}{3} + \frac{29}{6} \]

### Step 3: Find a Common Denominator
The least common multiple (LCM) of the denominators \(2, 3, 3, 6\) is \(6\). We will convert each fraction to have a denominator of \(6\):
1. Convert \(\frac{3}{2}\):
\[ \frac{3}{2} = \frac{3 \times 3}{2 \times 3} = \frac{9}{6} \]

2. Convert \(\frac{7}{3}\):
\[ \frac{7}{3} = \frac{7 \times 2}{3 \times 2} = \frac{14}{6} \]

3. Convert \(\frac{10}{3}\):
\[ \frac{10}{3} = \frac{10 \times 2}{3 \times 2} = \frac{20}{6} \]

4. \(\frac{29}{6}\) remains the same.

### Step 4: Add the Converted Fractions
Now we can add them:
\[ \text{Sum} = \frac{9}{6} + \frac{14}{6} + \frac{20}{6} + \frac{29}{6} = \frac{9 + 14 + 20 + 29}{6} = \frac{72}{6} \]

### Step 5: Simplify the Sum
Now simplify:
\[ \frac{72}{6} = 12 \]

### Step 6: Calculate the Average
To find the average, divide the sum by the number of observations (which is \(4\)):
\[ \text{Average} = \frac{12}{4} = 3 \]

### Final Answer
The average of the fractions \(1 \frac{1}{2}, 2 \frac{1}{3}, 3 \frac{1}{3}, 4 \frac{5}{6}\) is \(3\).

---