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Formula for the total percentage change

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Formula for the total percentage change

by achandwa » Mon Nov 12, 2007 8:15 pm
If A and B are two entities whose individual percentage change is known. To find the overall percentage change in A+B, we can apply the following formula:

Total % change = [% change in A * proportion of A in (A+B)] + [% change in B * proportion of B in (A+B)]

For example: Consider the following problem from OG-11.
Problem:
In a horticultural experiment, 200 seeds were planted in plot I and 300 seeds were planted in plot II. If 57% of seeds in plot I germinated, and 42% of seeds in plot II germinated, then what percentage of the total number of planted seeds germinated?
Solution:
You could solve it by a 2 step method
Step 1:
57% of 200 = 114
42% of 300 = 126
Step 2:
Total % = (114+126)/(200+300) = 240/500 = 48%

Alternatively, you could just apply the formula:
total % germination = %germination in plot I * proportion of seeds in plot I + %germination in plot II * proportion of seeds in plot II
Proportion of 200 in (200+300=500) is 200/500=2/5 and that of 300 is 300/500=3/5
%germination in plot I = 57%
%germination in plot II = 42%
Thus, total % germination = 57*2/5 + 42*3/5 = (114+126) /5 = 240/5 = 48%

Consider the following Data sufficiency problem from Manhattan GMAT CAT.
Problem:
The total cost of producing item X is equal to the sum of item X's fixed cost and variable cost. If the variable cost of producing X decreased by 5% in January, by what percent did the total cost of producing item X change in January?
(1) The fixed cost of producing item X increased by 13% in January.
(2) Before the changes in January, the fixed cost of producing item X was 5 times the variable cost of producing item X.
Solution:
According to the question stem, total cost = fixed cost + variable cost
Ct = Cf + Cv
The question is asking for the percent change of the total cost of production of item X in January. Clearly if we knew the total cost of producing X before January and then in January, we could calculate the percent change. From the question, however, it doesn’t seem like we will be provided with this information.
(1) INSUFFICIENT: Since the total cost of production is also the sum of the fixed and variable costs, it would stand to reason that we should be able to calculate the percent change to the total cost if we knew the percent change of the fixed and variable costs. However, it is not that simple. We cannot simply average the percent change of the fixed and variable costs to find the percent change of the total cost of production. Two percents cannot be averaged unless we know what relative portions they represent.
(2) INSUFFICIENT: Lacking information about the percent change of the fixed cost, we cannot solve.
(1) AND (2) SUFFICIENT: Using the two statements, we not only have the percent changes of the fixed and variable percents, but we also know the relative portions they represent. If the fixed cost before January was five times that of the variable cost, we can calculate the percent change to the cost of production using a weighted average:

Percent change of Ct = [(5 × percent change of Cf) + (1 × percent change of Cv)]/6
Percent change of Ct = [(5 × 13%) + (1 × -5%)]/ 6 = 10%
The cost of production increased in January by 10%.
The correct answer is C.

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by beatthegmat » Mon Nov 12, 2007 8:23 pm
Thanks for this! I'm adding this to https://del.icio.us/beatthegmat/
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by shawoody » Sun Jan 25, 2009 4:42 pm
beatthegmat wrote:Thanks for this! I'm adding this to https://del.icio.us/beatthegmat/
Nice problem. Can you tell me where the divisor (6) came from in the final rephrasing?

Thanks.

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by mrsmarthi » Tue Feb 03, 2009 9:27 pm
shawoody wrote:
beatthegmat wrote:Thanks for this! I'm adding this to https://del.icio.us/beatthegmat/
Nice problem. Can you tell me where the divisor (6) came from in the final rephrasing?

Thanks.


This infact is a nice short cut. From the second statement, it is given that

Fixed Cost = 5 times Variable Cost.

Also in the Question stem, it is given that Production Cost = Fixed + variable cost which gives us 6 time Variable Cost.

Thus we have 6 in the denominator.