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

W_saturday = R_saturday × T_saturday

4,000 = R_saturday × 4

R_saturday = 1,000

This gives the rate of printing on Saturday morning, but fails to give any information about Sunday’s rate.

(1) AND (2) SUFFICIENT: Statement (1) tells us that R_saturday = 2R_sunday and statement (2) tells us that R_saturday = 1,000. Putting this information together yields:

R_saturday = 2R_sunday

1,000 = 2R_sunday

500 = R_sunday

The correct answer is C.

12.
To find the combined rate of Machines A and B, we combine their individual rates. If Machine A can fill an order of widgets in a hours, then in 1 hour it can fill of the order. By the same token, if Machine B can fill the order of widgets in b hours, then in 1 hour, it can fill of the order. So together in 1 hour, Machines A and B can fill of the order:

So in 1 hour, Machines A and B can complete of the order. To find the number of hours the machines need to complete the entire order, we can set up the following equation:

(fraction of order completed in 1 hour) x (number of hours needed to complete entire order) = 1 order.

If we substitute for the fraction of the order completed in 1 hour, we get:

, where x is the number of hours needed to complete the entire order. If we divide both sides by , we get:

In other words, it will take Machines A and B hours to complete the entire order working together at their respective rates.

The question stem tells us that a and b are both even integers. We are then asked whether a and b are equal. If they are equal, we can express each as 2z, where z is a non-zero integer, because they are even. If we replace a and b with 2z in the combined rate, we get:

So if a and b are equal, the combined rate of Machines A and B must be an integer (since z is an integer). We can rephrase the question as:

Is the combined rate of Machines A and B an integer?

Statement 1 tells us that it took 4 hours and 48 minutes for the two machines to fill the order (remember, they began at noon). This shows that the combined rate of Machines A and B is NOT an integer (otherwise, it would have taken the machines a whole number of hours to complete the order). So we know that a and b cannot be the same. Sufficient.

Statement 2 tells us that . Since both a and b must be positive (because they represent a number of hours), we can take the square root of both sides of the equation without having to worry about negative roots. Therefore, it must be true that a + b = 20. So it is possible that a = 10 and that b = 10, which would allow us to answer "yes" to the question. But it is also possible that a = 12 and b = 8 (or any other combination of positive even integers that sum to 20), which would give us a "no". Insufficient.

The correct answer is A: Statement 1 alone is sufficient, but statement 2 alone is not.

13. If water is rushing into tank 1 at x gallons per minute while leaking out at y gallons per minute, the net rate of fill of tank 1 is x – y. To find the time it takes to fill tank 1, divide the capacity of tank 1 by the rate of fill: z / (x – y).

We know that the rate of fill of tank 2 is y and that the total capacity of tank 2 is twice the number of gallons remaining in tank 1 after one minute. After one minute, there are x – y gallons in tank 1, since the net fill rate is x – y gallons per minute. Thus, the total capacity of tank 2 must be 2(x – y).

The time it takes to fill tank two then is	2(x – y) y	.

The question asks us if tank 1 fills up before tank 2.

We can restate the question: Is	z x – y	<	2(x – y) y	?

SUFFICIENT: We can manipulate zy < 2x² – 4xy + 2y²:


zy < 2x² – 4xy + 2y²
zy < 2(x² – 2xy + y²)
zy < 2(x – y)(x – y) (dividing by x – y is okay since x – y > 0)

zy x – y

<

2(x – y)

(dividing by y is okay since y > 0)

z x – y

<

2(x – y) y

This manipulation shows us that the time it takes to fill tank 1 is definitely longer than the time it takes to fill tank 2.

INSUFFICIENT: We can express this statement algebraically as: 1/2(z) > 2(x – y). We cannot use this expression to provide us meaningful information about the question.

The correct answer is A.

14.
From the question stem, we know that Bill's rate is 1 well per x! hours and Carlos's rate is 1 well per y! hours.
Therefore, their combined rate is .

Since q is the amount of time it takes Bill and Carlos to dig a well together, we can use the rate formula to find q in terms of x and y. We can rearrange the formula to isolate T: .

Since q is the amount of time (T) it takes the two men to dig 1 well together, the "distance" (D) here is 1 well. Therefore, . So we know that . 
The question then becomes: Is an integer?

Statement (1) tells us that . We now know that . We can substitute for y and simplify:

Is this sufficient to tell us whether q is an integer? Let's try some numbers. If x = 5, then . But if x = 2, then . So in one case we get an integer, in another case we get a fraction. Statement (1) alone is insufficient to answer the question.

Statement (2) tells us that y is a nonprime even number. This means y can be any even number other than 2. We cannot tell from this whether q is an integer. For example, if y = 4 and x = 2, then . But if y = 4 and x = 5, then . So in one case we get a fraction, in another we get an integer. Statement (2) alone is insufficient to answer the question.

If we take the statements together, we know that and that y is an even number greater than 2 (because we are dealing with rates here, we do not have to worry about zero or negative evens).

Let's begin by analyzing the denominator of q, the expression x + 1. Since y is even and , x must be odd. Therefore x + 1 must be even. If x + 1 is even, it must be the product of 2 and some integer (call it z) that is less than x.

Now let's analyze the numerator of q, the expression x!. Since x is greater than y, it must be greater than 2. This means x! will have both 2 and z as factors (remember, z is less than x).

Therefore both the 2 and z in x + 1 (the denominator of q) will cancel out with the 2 and z in x! (the numerator of q), leaving only the product of integers.

For example, if x = 5 and y = 4,

Therefore, if and if y is an even number greater than 2, then q will always be an integer.

The correct answer is C: BOTH statements TOGETHER are sufficient but NEITHER statement ALONE is sufficient.

15.

1/4/(1/4+1/8)=2/3