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The total cost of producing item X

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The total cost of producing item X

by rsarashi » Sat Aug 19, 2017 11:08 pm

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The total cost of producing item X is equal to the sum of item X's overhead cost and production cost. If the production cost of producing X decreased by 5% in January, by what percent did the total cost of producing item X change in that same month?

(1) The overhead cost of producing item X increased by 13% in January.

(2) Before the changes in January, the overhead cost of producing item X was 5 times the production cost of producing item X.

OAC

I marked A

I was trying to do by letting the value.

Please explain.

Thanks
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by ceilidh.erickson » Sun Aug 20, 2017 10:22 am

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When we have a total value that is a SUM or DIFFERENCE of two component parts, we won't know the percent change to the total by just knowing the percent change to each part.

The easiest way to demonstrate is by testing numbers. Remember that whenever you pick numbers for DS, you need to test at least 2 different cases.
Let T = total cost
V = oVerhead cost (we shouldn't use "O" for a variable)
P = production cost
T = V + P

(1) The overhead cost of producing item X increased by 13% in January.

Case 1: easy numbers.
original V = 100 --> new (1.13V) = 113
original P = 100 --> new (0.95P) = 95
original T = 200 --> new total = 208
increase of 4%

Case 2: keep one element constant, change the other by orders of magnitude.
original V = 100 --> new (1.13V) = 113
original P = 10,000 --> new (0.95P) = 9,500
original T = 10,100 --> new total = 9,613
decrease of 3.87%

We get 2 different values in 2 different cases. Insufficient.

The reason that we got 2 different results is that we don't know the PROPORTION of overhead costs to production costs. If overhead is tiny and production costs huge (relatively speaking), then the 5% decrease in production will have way more impact on the total. If overhead is larger and production smaller, then the 13% increase will have a greater effect on the total.

Algebraically, this would look like:
T = V + P
let x = the percent change to the total:
xT = (1.13)(V) + (0.95)P
There is no way to simplify and solve for x here.

(2) Before the changes in January, the overhead cost of producing item X was 5 times the production cost of producing item X.

Now we have a proportion! This alone is not sufficient, since we don't know the change in overhead cost.

But if we put (1) and (2) together:
V = 5P

We can test cases:

Case 1: easy numbers.
original V = 100 --> new (1.13V) = 113
original P = 20 --> new (0.95P) = 19
original T = 120 --> new total = 132
increase of 10%

Case 2: shift order of magnitude (although we can't hold one constant, since we're given that V = 5P).
original V = 10,000 --> new (1.13V) = 11,300
original P = 2,000 --> new (0.95P) = 1,900
original T = 12,000 --> new total = 13,200
increase of 10%

Same result in both cases --> sufficient.

We can also represent this algebraically:
T = V + P
V = 5P
T = 5P + P --> T = 6P
xT = (1.13)(V) + (0.95)P
xT = (1.13)(5P) + (0.95)P
xT = 6.6P
We can solve for x --> 6.6P is 10% greater than 6P.

The answer is C.
Ceilidh Erickson
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Harvard Graduate School of Education
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by ceilidh.erickson » Sun Aug 20, 2017 10:25 am

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Here is another similar problem that deals with percent change in a sum/difference. I've posted 3 different solutions here: https://www.beatthegmat.com/og2016-q69-t ... tml#770961
Ceilidh Erickson
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Harvard Graduate School of Education
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by Jay@ManhattanReview » Sun Aug 20, 2017 10:11 pm

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rsarashi wrote:The total cost of producing item X is equal to the sum of item X's overhead cost and production cost. If the production cost of producing X decreased by 5% in January, by what percent did the total cost of producing item X change in that same month?

(1) The overhead cost of producing item X increased by 13% in January.

(2) Before the changes in January, the overhead cost of producing item X was 5 times the production cost of producing item X.

OAC

I marked A

I was trying to do by letting the value.

Please explain.

Thanks
Hi rsarashi,
I was trying to do by letting the value.
Since you got the answer A, and tried with plugging-in smart values for Overhead cost and Production cost, you may have assumed the same values for Overhead cost and Production cost, for example, $100 each. However, this is incorrect.

Case 1: Production cost = $100, and Overhead cost = $100
Thus, Total cost = $200

Costs by January, Production cost = $100 - 5% = $95, and Overhead cost = $100 + 13% = 113
Thus, Total cost = $95 + $113 = $208

Change in % of TC = [(208 - 200)/200] * 100 = 4%.

Case 2: Production cost = $100, and Overhead cost = $200
Thus, Total cost = $300

Costs by January, Production cost = $100 - 5% = $95, and Overhead cost = $200 + 13% = 226
Thus, Total cost = $95 + $226 = $321

Change in % of TC = [(321 - 300)/300] * 100 = 7%.

No unique answer. Statement 1 is insufficient.

Hope this helps!

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