Sets for an overlapping sets problem it is best to use a double set matrix to organize the information and solve. Fill in the information in the order in which it is given

e = 2/12
e = 1/6

Once we know e, we can solve for s and f:

s = 1/2 – e
s = 1/2 – 1/6
s = 3/6 – 1/6
s = 2/6
s = 1/3
f = 1/4 – e
f = 1/4 – 1/6
f = 3/12 – 2/12
f = 1/12
We add up their individual rates to get a combined rate:
e + s + f =
1/6 + 1/3 + 1/12 =
2/12 + 4/12 + 1/12 = 7/12

Remembering that a rate is expressed in terms of treehouses/hour, this indicates that a smurf, an elf, and a fairy, working together, can produce 7 treehouses per 12 hours. Since we want to know the number of hours per treehouse, we must take the reciprocal of the rate. Therefore we conclude that it takes them 12 hours per 7 treehouses, which is equivalent to 12/7 of an hour per treehouse.

The correct answer is D.


9.
Rate is defined as distance divided by time.

Therefore:

The RATE of machine A =

The RATE of machine B =

The COMBINED RATE of machine A and machine B =

This expression can be simplified by eliminating the roots in the denominators as follows:

The question asks us for the time, t, that it will take both machines working together to finish one job.

Using the combined rate above and a distance of 1 job, we can solve for t as follows:

The correct answer is choice B.

9.
Since this is a work rate problem, we'll use the formula rate × time = work. Since we'll be calculating times, we'll use it in the form time = work / rate.

First let T₀ equal the time it takes to paint the houses under the speedup scenario. T₀ will equal the sum of the following two values:

1. the time it takes to paint y houses at a rate of x
2. the time it takes to paint (80 – y) houses at a rate of 1.25x. 

T0	=	y x	+	(80 – y) 1.25x

T0	=	1.25y 1.25x	+	(80 – y) 1.25x

T0	=	(80 + 0.25y) 1.25x

Then let T₁ equal the time it takes to paint all 80 houses at the steady rate of x.

T1 =

80 x

The desired ratio is

T0 T1

. This equals T0 times the reciprocal of T1.

T0 T1	=	80 + 0.25y 1.25x	×	x 80

T0 T1	=	80+0.25y 100

T0 T1	=	0.8 + 0.0025y

As a quick check, note that if y = 80, meaning they paint ALL the houses at rate x before bringing in the extra help, then T₀/T₁ = 1 as expected. 

The correct answer is B.

10. There are several ways to achieve sufficiency in solving this rate problem, so the question cannot be rephrased in a useful manner.

(1) INSUFFICIENT: This statement provides the difference between the number of hot dogs consumed by the third-place finisher (let’s call this t) and the number of hot dogs consumed by the winner (let’s call this w). We now know that w = t + 24, but this does not provide sufficient information to solve for w.

(2) INSUFFICIENT: The third-place finisher consumed one hot dog per 15 seconds. To simplify the units of measure in this problem, let’s restate this rate as 4 hot dogs per minute. Statement (2) tells us that the winner consumed 8 hot dogs per minute. This does not provide sufficient information to solve for w. 

(1) AND (2) SUFFICIENT: The rate of consumption multiplied by elapsed time equals the number of hot dogs consumed. This equation can be restated as time = hot dogs/rate. Because the elapsed time is equal for both contestants, we can set the hot dogs/rate for each contestant equal to one another:

w/8 = t/4
w = 2t

Substituting w – 24 for t yields

w = 2(w – 24)
w = 2w – 48
48 = w

The correct answer is C.

11.
This is a work problem. We can use the equation Work = Rate × Time (W = R × T) to relate the three variables Work, Rate, and Time. The question asks us to find the number of newspapers printed on Sunday morning. We can think of the “number of newspapers printed” as the “work done” by the printing press. So, the question is asking us to find the work done on Sunday morning, or W_sunday.

The printing press runs from 1:00 AM to 4:00 AM on Sunday morning, so T_sunday = 3 hours. Since W_sunday = R_sunday × T_sunday, or in this case W_sunday = R_sunday × 3, knowing the rate of printing, R_sunday, will allow us to calculate W_sunday (the number of newspapers printed on Sunday morning). Therefore, the rephrased question becomes: What is R_sunday?

(1) INSUFFICIENT: This statement tells us that R_saturday = 2R_sunday. While this relates Saturday’s printing rate to Sunday’s printing rate, it gives no information about the value of either rate.

(2) INSUFFICIENT: For Saturday morning, W_saturday = 4,000 and T_saturday = 4 hours. We can set up the following equation: