When you spend $1 at the grocery store, the store doesn’t keep all of your money as profit. Some of it is reinvested in the economy when the grocery store buys more merchandise from suppliers. In turn, the suppliers don’t keep all of that money as profit, either; they reinvest it in the economy when they buy more raw materials from other suppliers. The end result is that your $1 may have a larger effect on the economy. Consider the following situation.
In January, a factory invests $400,000 into the economy in the form of employee salaries. During February, assume that the employees reinvest in the economy by spending 80% of their salaries. Then, during March, 80% of what was spent by the employees is again reinvested in the economy by the merchants. And so on.
Part A: Find the amount of money that is invested during the first four months. Part B: Starting with January, the amount of money that is invested in the economy forms what kind of sequence?
Part C: Write a rule for the amount of money that is invested during Month n.
The total economic impact of the factory’s initial investment is the sum of all the investments and reinvestments over the months.
Part D: What is the total economic impact after the first four months?
Part A: To find the amount of money invested during the first four months, we need to calculate the total investment at the end of each month.
In January, the factory invests $400,000.
In February, assuming the employees spend 80% of their salaries, the amount reinvested is 80% of $400,000, which is 0.80 * $400,000 = $320,000.
In March, 80% of what was spent by the employees is reinvested, so the amount reinvested is 0.80 * $320,000 = $256,000.
In April, 80% of what was spent in March is reinvested, so the amount reinvested is 0.80 * $256,000 = $204,800.
Therefore, the amount of money invested during the first four months is $400,000 + $320,000 + $256,000 + $204,800 = $1,180,800.
Part B: To determine the sequence formed by the amount of money invested in the economy, we can observe that each month's investment is 80% of the previous month's investment. This forms a geometric sequence with a common ratio of 0.80.
Part C: The rule for the amount of money invested during Month n can be expressed as follows:
Investment(n) = Initial Investment * Common Ratio^(n-1)
Where:
- Investment(n) is the amount of money invested in Month n.
- Initial Investment is the starting investment, which is $400,000 in this case.
- Common Ratio is the ratio of reinvestment, which is 0.80 in this case.
- n is the month number.
Part D: To find the total economic impact after the first four months, we need to sum up the investments for each month:
Total Economic Impact = Investment(1) + Investment(2) + Investment(3) + Investment(4)
Substituting the values into the formula:
Total Economic Impact = $400,000 + $320,000 + $256,000 + $204,800
= $1,180,800
Therefore, the total economic impact after the first four months is $1,180,800.