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As you can see, this problem requires nothing but arithmetic and a little bit of critical reasoning. Since it’s a Data Sufficiency problem, don’t worry about trying to solve all the way to a numerical final answer. Instead, let’s go through each of the two statements one by one.

First, what is given? There are 42 freshmen and seniors, but we don’t know exactly how many of each. Two unknowns, and one relation (equation). So we are looking for the statement(s) that can help to set up another equation if possible.

Statement (1): Be careful, as the wording is tricky here. To say that the group has more than four times as many seniors as freshmen only allows you to set up an inequality (not an equation). It could be that there are zero freshman and 42 seniors, or 8 freshmen 34 seniors, or anything in between.

Statement (2): By itself, this doesn’t narrow the field down either. Just saying that there are more than 7 freshmen leaves open all possibilities from 8 to 42 freshmen!

But now look again at the conclusions of the two statements. Statement (1) gives you a maximum of 8 freshmen. That’s because 9 freshmen would leave 33 seniors, which is more than four times 9. And Statement (2) gives you a minimum of 8 freshmen (the first whole number more than 7). Thus, together Statements (1) and (2) are sufficient.

Answer: C Both are sufficient, but neither one alone is sufficient.

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This is a typical problem dealing with units and ratios. Let’s use our number sense to quickly tackle this one.

First, the fact that the price of flour is \(w\) dollars per \(x\) pounds, means that whatever the final answer is, the \(w\) and \(x\) need to be on opposite parts of the fraction. That’s because \(w\) per \(x\) means \(w/x\). So either that, or its reciprocal will be in your final answer.

So that narrows it down to just two choices without much work! Either \(\frac{xy}{w}\) or \(\frac{w}{xy}\).

Finally, the question is asking for the cost of making one cake. So let’s what happens if we allow \(y\) to vary. Suppose \(y\) is small, like \(y=1\). Then it takes a whole pound of flour to make just 1 cake. But if \(y\) is larger, say \(y=4\), then that same one pound of flour goes much further, bringing the overall cost down per cake. As \(y\) increases, the cost per cake has to decrease. That tells you immediately that \(y\) must be on the bottom of the fraction (in order to get that kind of inverse relationship).

Answer: \(\frac{w}{xy}\)

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The usual way you’d have to work this out in a high school math class would be to use a formula that gets you the equation of a line from the given intercepts. But we don’t have to remember any kind of formula if you simply backsolve from the answer choices.

Take each answer in turn and see if it works. Very quickly you’ll see that \(-3x + 2y=6\) has the correct intercepts, and so it solves the problem!

Answer: \(-3x + 2y=6\)

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Want to avoid the algebra? Let’s pick some convenient numbers for the variables. Keep in mind that \(m \neq n\). So, let’s start with \(m=2\) and \(n=1\). Plugging those into the given equation, we get:

\(6x + 4y – 2y – 3x = 0\),

which simplifies to:

Now we could even plug in a number for \(x\) and work out \(y\) from that (to compare with the answer choices), but there’s no need on such a simple equation.

Answer: \(-\frac{3x}{2}\)

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What is given? JKLM is a square; P is the midpoint of KL.

Where do I need to end up? Determining whether triangle JQM is equilateral or not.

What info would be useful? Knowing all the angles, of course!

Helpful formulas? We’ll probably need the fact that all angles in a triangle add to 180 degrees and properties of parallel lines cut by a transversal, because frankly those concepts seem to be important in almost every one of these kinds of problems.

Let’s look at Statement (1). If angle KPQ is 90 degrees, then PQ would be parallel to KJ. That’s a great start, but doesn’t give enough info by itself to solve the problem. For instance, the angle JQM would vary depending on how long PQ is.

Now consider Statement (2). By itself, having angle JQP is nice, but just not sufficient. What if point Q is left or right of the midline? We’d have no definite way of finding the angles of triangle JQM.

However, if both Statements (1) and (2) are taken together, then you have KJ parallel to PQ, and angle JQP = 150. Then angle KJQ is equal to 30 (same-side interior angles). That makes angle MJQ equal to 60. But then because PQ is centered on the midline of the square, the other side is a perfect mirror image. And that gives you angle JMQ — 60 degrees as well. Finally, angle JQM must also be 60, and the triangle is guaranteed to be equilateral!

Answer: C Both Statements (1) and (2) TOGETHER are sufficient to answer the question asked; but NEITHER statement ALONE is sufficient.

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Both are sufficient, but neither one alone is sufficient.

There’s a lot to keep track of here, and some info is just not that important. For example, you don’t need to know that one pump is a “JQ” and other other is a “JT” pump, just that there are two types and they run at different rates. They could have been called “A” and “B” or “1” and “2” for all we care. But it is a good idea to jot down “JQ” and “JT” on your scratch paper to start organizing the rest of the data.

The JQ pump fills the tank in 72 hours. How much water is that? We don’t know. But you can say it’s 1 tank worth. So write “1 tank in 72 hrs.” in your JQ column.

Similarly, put “1 tank in 18 hrs.” in your JT column.

Now, it goes on to ask about filling up a half-full tank. So, alone the JQ would take 36 hours. But we have two JQ’s, which by themselves would cut that fill time to 18 hours.

Finally, the trickiest part, what happens when you add in the JT? By itself, it takes 9 hours to fill half the tank. Let’s bring in our number sense. Every time unit, the JQ’s are going to fill only half as much water as the JT, because the JT is pumping twice as fast. When the tank fills, two-thirds of the water was pumped in by the JT, and only a third of it by the two JT pumps.

So either way you look at it, 6 hours are needed — either one third of 18 hours, or 2/3 of 9 hours.

Answer: 6