Five numbers a,b,c,d and e are in the ratio of 2:3:5:8:9 and their sum is 162. Find the average of all these numbers:

Options

A. 27
B. 30
C. 32.4
D. can't be determined

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Ratio
The numbers a, b, c, d, and e are in the ratio of 2:3:5:8:9. This means that we can express these numbers in terms of a common variable, say x.

### Step 2: Express the Numbers in Terms of x
Let:
- \( a = 2x \)
- \( b = 3x \)
- \( c = 5x \)
- \( d = 8x \)
- \( e = 9x \)

### Step 3: Set Up the Equation for the Sum
According to the problem, the sum of these numbers is 162. Therefore, we can write the equation:
\[ 2x + 3x + 5x + 8x + 9x = 162 \]

### Step 4: Combine Like Terms
Now, we combine the terms on the left side:
\[ (2 + 3 + 5 + 8 + 9)x = 162 \]
Calculating the sum:
\[ 27x = 162 \]

### Step 5: Solve for x
Now, we can solve for x by dividing both sides by 27:
\[ x = \frac{162}{27} = 6 \]

### Step 6: Find the Values of a, b, c, d, and e
Now that we have the value of x, we can find the actual values of a, b, c, d, and e:
- \( a = 2x = 2 \times 6 = 12 \)
- \( b = 3x = 3 \times 6 = 18 \)
- \( c = 5x = 5 \times 6 = 30 \)
- \( d = 8x = 8 \times 6 = 48 \)
- \( e = 9x = 9 \times 6 = 54 \)

### Step 7: Calculate the Average
To find the average of these numbers, we use the formula:
\[ \text{Average} = \frac{\text{Sum of the numbers}}{\text{Total number of numbers}} \]
The sum of the numbers is:
\[ 12 + 18 + 30 + 48 + 54 = 162 \]
The total number of numbers is 5. Therefore, the average is:
\[ \text{Average} = \frac{162}{5} = 32.4 \]

### Final Answer
The average of all these numbers is **32.4**.

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