## 11/3/11

### Test 8 Section 9 - #16 (page 861)

Tough one, right?  Let's start by drawing it:
So the area of ABED is 2/3. We don't have an equation for the area of a shape like that, so let's see if draw a few more lines and see if anything becomes more clear. I'm gonna draw a few more segments, and name another point, P, which is the midpoint of AD.
Alright, now we're doing it. Notice that ABED, which we know, is 3/4 of ABCD, which we want.

ABED, which has an area of 2/3, is 3/4 of what we want. Confusing, right? At this point, you can mess with the fractions*, or you can backsolve. Look at the choices. 3/4 of one of them is going to be 2/3.

(C) (8/9)(3/4) = 24/36 = 2/3.

* To "mess with" the fractions, divide 2/3 by 3/4. You get 8/9.

## 6/4/11

### Practice problem #14, page 340

This is a logic question really, but it's a pretty hard one by SAT standards.

You know that a(a) = 0, which means either a = 0, a = 0, or both. (You know this because when a product equals zero, that can only happen when at least one of the things being multiplied together is zero.)

Since we're given b, we know - a can't be zero. So a has to be 0. The other two conditions follow from that:

I    a = 0 (we know this is true)
II   < 0 (we know this is true because a = 0, and b is less than a)
III  - b > 0 (we know this is true because 0 - a negative number will always be positive)

## 5/17/11

### Test 4 Section 9 - #16 (page 613)

Plug in. Say x = 2. Then y = 2 + 1/2 = 2.5. Now try your conditions:

I. yx
Yep, that's true.

II. y is an integer.
Nope, not true. 2.5 is not an integer.

III. xy > x2
This is the tricky one, but it is true with our plug-in -- (2.5)(2) > 22. It will ALWAYS be true because y will always be slightly bigger than x.

So the answer is I and III are true. That's (D).

### Test 4 Section 9 - #15 (page 613)

Plug in to solve this one. (Note, it doesn't matter if the equation doesn't work with the numbers you pick, since you're not asked to solve. This is just to help you see how the equation should go.) Say the shortest side, x, is 6. Then the other two sides are 8 and 10. So the equation you'd set up is:

62 + 82 = 102

Or, in terms of x:

x2 + (x + 2)2 = (x + 4)2

That's choice (C).

### Test 4 Section 9 - #10 (page 611)

Tricky one. If Stacy is the 12th tallest, that means there are 11 people shorter than her. If she's the 12th shortest, that means there are 11 people taller than her. Note that, although it's not important for this question, Stacy is the median height in her class.

If there are 11 people taller, 11 people shorter, and Stacy, that's
11 + 11 + 1 = 23 people in the class. Choice (B).

### Test 4 Section 9 - #7 (page 611)

OK, you're going to have to do some tricky solving for this one. Start by solving for x:

x = ⅓
x½ = 3 (negative exponent means inverse)
x = 9 (square both sides to get rid of the ½ exponent)

You're also going to have to figure out y and z. The question says that y, so we can't go the obvious 42 route. Instead, we're going to have to say that z = 4 and y = 2. 24 = 16 as well.

So, x = 9, and z = 4. x + z = 13. That's choice (D).

### Test 4 Section 9 - #1 (page 609)

You can backsolve this one. Start with (C): If there are 8 girls on the bus, that means that there were 8 boys on the bus to start as well.

At the first stop, 4 boys got off and nobody got on, so now there are 4 boys on the bus and still 8 girls.

That's twice as many girls as there are boys (just like the question says) so (C) is our answer.

### Test 4 Section 6 - #15 (page 597)

Even though this is a grid-in question, we can actually plug-in to solve. That's because they don't ask us for the areas themselves, they ask us for the RATIO of the areas.

Say QS = 1, QV = 3, PT = 3, and PR = 4. That's going to mean that SV = 2, since it's the difference between QV and QS.

Our areas:

PST = ½(SV)(PT) = ½(2)(3) = 3
PQR = ½(QV)(PR) = ½(3)(4) = 6

So the ratio of ▵PST to ▵PQR = 3/6, or 1/2. You could also grid in .5.

### Test 4 Section 6 - #6 (page 594)

This one works kinda like a ratio problem. The key is to recognize that if the ratio of white eggs to brown eggs is 2 to 3, then the ratio of, say, white eggs to TOTAL eggs is 2 to 5. (You could also, of course, do brown eggs to TOTAL eggs 3 to 5. Potayto potahto.)

The total number of eggs in the basket has to be a multiple of 5. The only answer choice that isn't is (B) 12, so that's the answer.

### Test 4 Section 3 - #18 (page 585)

Big time plug-in here. Say x = 10, and z = 2. So the first shirt will cost \$10, and each shirt after that will cost \$10 - \$2 = \$8. Now say the customer buys n = 5 shirts. Those shirts would cost

\$10 for the first one
+ \$8 for the second
+ \$8 for the third
+ \$8 for the fourth
+ \$8 for the fifth
= \$42

So we're looking for the answer that gives us 42 when we plug in the number we chose above.

(A) x + (n - 1)(x - z)

We can translate that to
10 + (5 - 1)(10 - 2) = 10 + (4)(8) = 42.

That's our answer! Make sure, when you plug in, that you try the other answers too to make sure they don't work. In this case they don't, so we're done.

## 5/9/11

### Test 2 Section 2 - #15 (page 456)

This is a function question, which means you're going to need to follow directions very closely. Start by understanding the definition:

x▲ = (x + 1)(x - 1)

All that means is that whatever we find to the left of the ▲, we put it through the same machinery:

x▲ = (x + 1)(x - 1)
p▲ = (p + 1)(p - 1)
[dog]▲ = ([dog] + 1)([dog] - 1)

I'm having a bit of fun with the definition, but you get the point. Let's solve this problem:

6▲ - 5▲
= (6 + 1)(6 - 1) - (5 + 1)(5 - 1)
= (7)(5) - (6)(4)
= 35 - 24
= 11

So we're looking for the answer choice that gives us the same result. Which answer choice gives you 11? Try them all to make sure, but for the sake of saving space I'm just going to do out the correct one:

3▲ + 2▲
= (3 + 1)(3 - 1) + (2 + 1)(2 - 1)
= (4)(2) + (3)(1)
= 8 + 3
= 11

Nice. That's choice (B).

### Test 5 Section 2 - #15 (page 642)

As with any geometry question, your first step is to fill in EVERYTHING YOU KNOW.

Because angles A and C are equal, AB = BC. So ABC has sides of 5, 8 and 8, and therefore a perimeter of 21.

Because = 60, the third, unmarked angle in DEF must also be 60°, so DEF is equilateral. That means its sides are all 5, and its perimeter is 15.

21-15 = 6, so (C) is the answer.

### Test 5 Section 2 - #14 (page 641)

This one looks intimidating, but if you start plugging in you'll see the way pretty quickly.

Start with the lowest ordered pair you can use: (1,1). Yep, that works; it gives you 2(1) + 3(1) < 6, or 5 < 6. Note, though, that there aren't any others: (2,1) gives you 7, and (1,2) gives you 8. Both of those are bigger than 6, so the only ordered pair of positive integers that works is (1,1).

## 5/2/11

### Test 5 Section 4 - #8 (page 652)

This one is based on that all-important fact: parabolas are symmetrical! When they say f(b) = f(3), that's another way of saying that you have two points, (b,y) and (3,y), and the y-values are the same. So look at the graph, and see what that y-value is!

On the graph, I'm seeing the point (3,5), so our other point must be (b,5). The only other place this graph goes through y=5 is at (-1,5), so b must equal -1.

### Test 5 Section 2 - #17 (page 642)

It's important to note that when the SAT tells you that a line passes through the origin, they're giving you a point on the line: (0,0). So we know line l contains points (0,0) and (t, t+1).

Since we also know that it's perpendicular to the line y = -4k, we know line l has a slope of +1/4 (the negative reciprocal of the slope of the line it's perpendicular to).

We can use these two facts to solve for t. Set up the slope equation using the points we know:

1/4 = [t + 1 -0]/[t - 0]
1/4 = [t + 1]/[t]
t/4 = t + 1
t = 4t + 4
-3t = 4
t = -4/3