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During 2005, a company produced an average of 2,000 products per month. How many products will the company need to produce from 2006 through 2008 in order to increase its monthly average for the period from 2005 through 2008 by 200% over its 2005 average?
Let's break down what this problem is asking in plain English. The company wants to increase its monthly average by 200%. This is a common source of confusion, so let's be very clear about what this means.
When we say "increase by 200%," we mean the new value will be the original value PLUS 200% of the original value. Think of it like this: if you had \$100 and someone increased it by 200%, you'd have your original \$100 plus an additional \$200 (which is 200% of \$100), giving you \$300 total.
So if the 2005 average was 2,000 products per month, then:
Or more simply: New average = \(2000 \times (1 + 2) = 2000 \times 3 = 6000\) products/month
We need to find: How many products must be produced from 2006 through 2008 to achieve this new average?
Process Skill: TRANSLATE - Converting the percentage increase language into concrete numbers
From our translation above, we've determined that the target monthly average for the entire period 2005-2008 should be 6,000 products per month.
Let's verify this makes sense: If they averaged 2,000 per month in 2005, and we want the overall average to be 3 times higher, then 6,000 per month seems right.
Now we need to figure out the total production required for the entire period from 2005 through 2008.
First, let's count the months:
If we want an average of 6,000 products per month over 48 months, then:
Total production needed = \(6000 \times 48 = 288000\) products
We know what happened in 2005, so let's calculate that first:
2005 production = \(2000 \times 12 = 24000\) products
To find what we need from 2006-2008:
Production needed (2006-2008) = Total needed - 2005 production
Production needed (2006-2008) = \(288000 - 24000 = 264000\) products
The company needs to produce 264,000 products from 2006 through 2008.
Let's verify:
The answer is (D) 264,000.
Many students confuse "increase by 200%" with "increase to 200%" or think it means the new value becomes 200% of the original. The correct interpretation is that the new value = original + (200% of original) = original × 3. Students often calculate the new average as \(2000 \times 2 = 4000\) instead of \(2000 \times 3 = 6000\).
2. Misunderstanding the time period scopeStudents may incorrectly think they need to find the average for 2006-2008 only, rather than understanding that the 200% increase applies to the overall average for the entire period from 2005 through 2008. This leads to setting up the wrong equation structure.
3. Confusing what needs to be calculatedSome students may try to calculate the total production for all years (2005-2008) instead of specifically what's needed for 2006-2008 only, which is what the question is asking for.
Students may make computational mistakes when calculating \(6000 \times 48 = 288000\) or when subtracting \(288000 - 24000 = 264000\). These straightforward arithmetic errors can lead to selecting incorrect answer choices.
2. Incorrect month countingSome students might miscalculate the total number of months in the period 2005-2008, perhaps counting it as 36 months (thinking 3 years × 12) instead of the correct 48 months (4 years × 12), leading to wrong total production calculations.
No likely faltering points - once students have correctly calculated 264,000, the answer selection is straightforward as this value appears directly in choice (D).