The average of five numbers is 57.8. The average of the first and the second numbers is 77.5 and the fourth and fifth numbers is 46. What is the third number ?

Options

A. 45
B. 43
C. 42
D. Cannot be determined

Explanation

To find the third number given the average of five numbers and the averages of certain pairs, we can follow these steps:

### Step 1: Calculate the total sum of the five numbers.
The average of the five numbers is given as 57.8. Since the average is calculated by dividing the total sum by the number of items, we can find the total sum (S) of the five numbers:

\[ S = \text{Average} \times \text{Number of items} = 57.8 \times 5 = 289 \]

### Step 2: Set up the equation for the first and second numbers.
The average of the first and second numbers (let's denote them as \(x\) and \(y\)) is given as 77.5. Therefore, we can express this as:

\[ \frac{x + y}{2} = 77.5 \]

Multiplying both sides by 2 gives:

\[ x + y = 77.5 \times 2 = 155 \]

### Step 3: Set up the equation for the fourth and fifth numbers.
The average of the fourth and fifth numbers (let's denote them as \(p\) and \(q\)) is given as 46. Therefore, we can express this as:

\[ \frac{p + q}{2} = 46 \]

Multiplying both sides by 2 gives:

\[ p + q = 46 \times 2 = 92 \]

### Step 4: Substitute and solve for the third number.
Now we have the following equations:

1. \(x + y + z + p + q = 289\) (from Step 1)
2. \(x + y = 155\) (from Step 2)
3. \(p + q = 92\) (from Step 3)

We can substitute equations 2 and 3 into equation 1:

\[ 155 + z + 92 = 289 \]

Now, combine the known values:

\[ z + 247 = 289 \]

To find \(z\), subtract 247 from both sides:

\[ z = 289 - 247 = 42 \]

### Final Answer:
The third number is \(42\).

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