Find the mean of the following distribution:
`{:(x,10,30,50,70,90),(f,7,5,10,3,5):}`

Answer

To find the mean of the given distribution, we will follow these steps:

### Step 1: Write down the data
We have the following values:
- \( x = (10, 30, 50, 70, 90) \)
- \( f = (7, 5, 10, 3, 5) \)

### Step 2: Calculate \( f \cdot x \)
We need to multiply each value of \( x \) by its corresponding frequency \( f \):
- For \( x = 10 \) and \( f = 7 \):
\[ f \cdot x = 10 \cdot 7 = 70 \]
- For \( x = 30 \) and \( f = 5 \):
\[ f \cdot x = 30 \cdot 5 = 150 \]
- For \( x = 50 \) and \( f = 10 \):
\[ f \cdot x = 50 \cdot 10 = 500 \]
- For \( x = 70 \) and \( f = 3 \):
\[ f \cdot x = 70 \cdot 3 = 210 \]
- For \( x = 90 \) and \( f = 5 \):
\[ f \cdot x = 90 \cdot 5 = 450 \]

Now we can summarize these calculations:
\[ f \cdot x = (70, 150, 500, 210, 450) \]

### Step 3: Calculate \( \Sigma f \cdot x \)
Now we will sum up all the values of \( f \cdot x \):
\[ \Sigma f \cdot x = 70 + 150 + 500 + 210 + 450 \]
Calculating this step-by-step:
- \( 70 + 150 = 220 \)
- \( 220 + 500 = 720 \)
- \( 720 + 210 = 930 \)
- \( 930 + 450 = 1380 \)

So, \( \Sigma f \cdot x = 1380 \).

### Step 4: Calculate \( \Sigma f \)
Now we will sum up the frequencies \( f \):
\[ \Sigma f = 7 + 5 + 10 + 3 + 5 \]
Calculating this:
- \( 7 + 5 = 12 \)
- \( 12 + 10 = 22 \)
- \( 22 + 3 = 25 \)
- \( 25 + 5 = 30 \)

So, \( \Sigma f = 30 \).

### Step 5: Calculate the mean
Now we can calculate the mean using the formula:
\[ \bar{x} = \frac{\Sigma f \cdot x}{\Sigma f} = \frac{1380}{30} \]
Calculating this:
\[ \bar{x} = 46 \]

### Final Answer
The mean of the distribution is \( 46 \).

---