Mr. Smith's average annual income over the years 1966 and 1967 is x dollars. His average annual income in over the years 1968,1969 and 1970 is y dollars. What is his average annual income over the five continuous years 1966 through 1970?

Options

A. `2x//5 + 3y//5`
B. `x//2 + y//2`
C. `5(x + y)`
D. `5x//2 + 5y//2`

Answer

To find Mr. Smith's average annual income over the years 1966 through 1970, we can follow these steps:

### Step 1: Understand the given information
We know that:
- The average annual income from 1966 to 1967 is \( x \) dollars.
- The average annual income from 1968 to 1970 is \( y \) dollars.

### Step 2: Calculate the total income for the years 1966 and 1967
The average income for two years is given by:
\[ \text{Average} = \frac{\text{Sum of incomes}}{\text{Number of years}} \]
So, for the years 1966 and 1967:
\[ x = \frac{S_{1966} + S_{1967}}{2} \]
Multiplying both sides by 2 gives:
\[ S_{1966} + S_{1967} = 2x \]

### Step 3: Calculate the total income for the years 1968, 1969, and 1970
Similarly, for the years 1968, 1969, and 1970:
\[ y = \frac{S_{1968} + S_{1969} + S_{1970}}{3} \]
Multiplying both sides by 3 gives:
\[ S_{1968} + S_{1969} + S_{1970} = 3y \]

### Step 4: Calculate the total income over the five years
Now, we can find the total income over the five years (1966 to 1970) by adding the two sums:
\[ \text{Total income from 1966 to 1970} = (S_{1966} + S_{1967}) + (S_{1968} + S_{1969} + S_{1970}) = 2x + 3y \]

### Step 5: Calculate the average income over the five years
The average income over the five years is given by:
\[ \text{Average} = \frac{\text{Total income}}{\text{Number of years}} = \frac{2x + 3y}{5} \]

### Final Answer
Thus, Mr. Smith's average annual income over the years 1966 through 1970 is:
\[ \frac{2x + 3y}{5} \]