The average of eight number is 20 . If the sum of first two numbers is 31 , the average of the next three numbers is `21 "" (1)/(3)` and the seventh and eighth numbers exceed the sixth number by 4 and 7 respectively , then the eighth number is

Options

A. 20
B. 25
C. 21.6
D. 25.3

Answer

To solve the problem step by step, we need to break down the information given and use it to find the eighth number.

### Step 1: Calculate the total sum of the eight numbers
The average of eight numbers is given as 20. Therefore, the total sum of the eight numbers can be calculated as:
\[ \text{Total Sum} = \text{Average} \times \text{Number of items} = 20 \times 8 = 160 \]

**Hint:** Remember that the total sum can be found by multiplying the average by the total number of items.

### Step 2: Find the sum of the first two numbers
We are given that the sum of the first two numbers is 31.

**Hint:** This is a direct piece of information provided in the question.

### Step 3: Calculate the sum of the next three numbers
The average of the next three numbers is given as \(21 \frac{1}{3}\). First, convert this mixed number into an improper fraction:
\[ 21 \frac{1}{3} = \frac{64}{3} \]
Now, to find the sum of these three numbers:
\[ \text{Sum of next three numbers} = \text{Average} \times \text{Number of items} = \frac{64}{3} \times 3 = 64 \]

**Hint:** When dealing with averages, always remember to multiply by the number of items to find the total sum.

### Step 4: Set up the equation for the sixth number
Let the sixth number be \(x\). According to the problem, the seventh number exceeds the sixth number by 4, and the eighth number exceeds the sixth number by 7. Therefore:
- Seventh number = \(x + 4\)
- Eighth number = \(x + 7\)

**Hint:** Use variables to represent unknown numbers to simplify calculations.

### Step 5: Write the equation for the total sum
Now we can write the equation for the total sum of all eight numbers:
\[ 31 + 64 + x + (x + 4) + (x + 7) = 160 \]
This simplifies to:
\[ 31 + 64 + x + x + 4 + x + 7 = 160 \]
Combining like terms gives:
\[ 3x + 106 = 160 \]

**Hint:** Combine all known quantities on one side to isolate the variable.

### Step 6: Solve for \(x\)
Now, isolate \(x\):
\[ 3x = 160 - 106 \]
\[ 3x = 54 \]
\[ x = \frac{54}{3} = 18 \]

**Hint:** Always perform arithmetic operations carefully to avoid mistakes.

### Step 7: Find the eighth number
Now that we have \(x\), we can find the eighth number:
\[ \text{Eighth number} = x + 7 = 18 + 7 = 25 \]

**Hint:** Substitute the value of \(x\) back into the expression for the eighth number.

### Final Answer
The eighth number is **25**.