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
EdM in Mind, Brain, and Education
Harvard Graduate School of Education
