The average of `4(3/5)`,`2(2/3)`,`6(8/9)`,`7(7/15)`,`3(5/9)` is:

Options

A. `5(3/225)`
B. `5(8/225)`
C. `6(3/45)`
D. `25(8/45)`

Answer

To find the average of the numbers \(4 \frac{3}{5}\), \(2 \frac{2}{3}\), \(6 \frac{8}{9}\), \(7 \frac{7}{15}\), and \(3 \frac{5}{9}\), we will follow these steps:

### Step 1: Convert Mixed Numbers to Improper Fractions
1. Convert each mixed number to an improper fraction:
- \(4 \frac{3}{5} = \frac{4 \times 5 + 3}{5} = \frac{20 + 3}{5} = \frac{23}{5}\)
- \(2 \frac{2}{3} = \frac{2 \times 3 + 2}{3} = \frac{6 + 2}{3} = \frac{8}{3}\)
- \(6 \frac{8}{9} = \frac{6 \times 9 + 8}{9} = \frac{54 + 8}{9} = \frac{62}{9}\)
- \(7 \frac{7}{15} = \frac{7 \times 15 + 7}{15} = \frac{105 + 7}{15} = \frac{112}{15}\)
- \(3 \frac{5}{9} = \frac{3 \times 9 + 5}{9} = \frac{27 + 5}{9} = \frac{32}{9}\)

### Step 2: Find a Common Denominator
2. The denominators are \(5\), \(3\), \(9\), and \(15\). The least common multiple (LCM) of these numbers is \(45\).

### Step 3: Convert Each Fraction to the Common Denominator
3. Convert each fraction to have a denominator of \(45\):
- \(\frac{23}{5} = \frac{23 \times 9}{5 \times 9} = \frac{207}{45}\)
- \(\frac{8}{3} = \frac{8 \times 15}{3 \times 15} = \frac{120}{45}\)
- \(\frac{62}{9} = \frac{62 \times 5}{9 \times 5} = \frac{310}{45}\)
- \(\frac{112}{15} = \frac{112 \times 3}{15 \times 3} = \frac{336}{45}\)
- \(\frac{32}{9} = \frac{32 \times 5}{9 \times 5} = \frac{160}{45}\)

### Step 4: Add All the Fractions
4. Now, add all the fractions:
\[ \frac{207}{45} + \frac{120}{45} + \frac{310}{45} + \frac{336}{45} + \frac{160}{45} = \frac{207 + 120 + 310 + 336 + 160}{45} = \frac{1133}{45} \]

### Step 5: Calculate the Average
5. To find the average, divide the total sum by the number of values (which is \(5\)):
\[ \text{Average} = \frac{1133}{45 \times 5} = \frac{1133}{225} \]

### Step 6: Simplify the Result
6. The average in mixed number form:
- Divide \(1133\) by \(225\) to find the whole number and the remainder:
- \(1133 \div 225 = 5\) remainder \(8\), so:
\[ \text{Average} = 5 \frac{8}{225} \]

### Final Answer
The average of the numbers \(4 \frac{3}{5}\), \(2 \frac{2}{3}\), \(6 \frac{8}{9}\), \(7 \frac{7}{15}\), and \(3 \frac{5}{9}\) is:
\[ \boxed{5 \frac{8}{225}} \]