Question:
If the revenues in 2009 were 3 million less, and the expenses for 2009 were $4 million more, then the average (arithmetic mean) annual profit for the 10 years shown would be approximately how much less?
If the revenues in 2009 were 3 million less, and the expenses for 2009 were $4 million more, then the average (arithmetic mean) annual profit for the 10 years shown would be approximately how much less?
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To find the change in an average, you don't need to calculate the original average. Simply calculate the total change in the sum of all values and divide it by the number of values. This is a significant time-saver.
Updated On: Oct 3, 2025
- $300,000
- $400,000
- $600,000
- $700,000
- $1,000,000
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Concept:
The question asks for the change in the average annual profit over a 10-year period due to a hypothetical change in the profit of a single year (2009). The key idea is that the change in the average is the total change in the sum divided by the number of items.
Step 2: Key Formula or Approach:
1. Calculate the change in the 2009 profit based on the given hypothetical changes in revenue and expenses.
2. The total change in the sum of profits for the 10 years is equal to the change in the 2009 profit.
3. Calculate the change in the average profit by dividing the total change by the number of years (10).
Step 3: Detailed Explanation:
First, let's calculate the change in the profit for the year 2009. The formula for profit is P = R - E.
The change in profit (ΔP) is related to the change in revenue (ΔR) and change in expenses (ΔE) by:
ΔP = ΔR - ΔE
According to the problem:
Revenues were $3 million less, so ΔR = -3,000,000.
Expenses were $4 million more, so ΔE = +4,000,000.
Now, calculate the change in the 2009 profit:
ΔP_2009 = (-3,000,000) - (+4,000,000) = -7,000,000
So, the profit in 2009 would decrease by $7 million.
This change in a single year's profit will affect the total sum of profits for the 10-year period. The total change in the sum is -$7,000,000.
The number of years is 10. The change in the average annual profit is:
Change in Average = (Total Change in Sum) / (Number of Years) = (-7,000,000) / 10 = -700,000
Step 4: Final Answer:
The average annual profit would be $700,000 less. This corresponds to option (D).
The question asks for the change in the average annual profit over a 10-year period due to a hypothetical change in the profit of a single year (2009). The key idea is that the change in the average is the total change in the sum divided by the number of items.
Step 2: Key Formula or Approach:
1. Calculate the change in the 2009 profit based on the given hypothetical changes in revenue and expenses.
2. The total change in the sum of profits for the 10 years is equal to the change in the 2009 profit.
3. Calculate the change in the average profit by dividing the total change by the number of years (10).
Step 3: Detailed Explanation:
First, let's calculate the change in the profit for the year 2009. The formula for profit is P = R - E.
The change in profit (ΔP) is related to the change in revenue (ΔR) and change in expenses (ΔE) by:
ΔP = ΔR - ΔE
According to the problem:
Revenues were $3 million less, so ΔR = -3,000,000.
Expenses were $4 million more, so ΔE = +4,000,000.
Now, calculate the change in the 2009 profit:
ΔP_2009 = (-3,000,000) - (+4,000,000) = -7,000,000
So, the profit in 2009 would decrease by $7 million.
This change in a single year's profit will affect the total sum of profits for the 10-year period. The total change in the sum is -$7,000,000.
The number of years is 10. The change in the average annual profit is:
Change in Average = (Total Change in Sum) / (Number of Years) = (-7,000,000) / 10 = -700,000
Step 4: Final Answer:
The average annual profit would be $700,000 less. This corresponds to option (D).
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