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MathVerbal

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The first thing we'll cover are AVERAGES. If the SAT goes after you in the easy or medium difficulty section, they're trying to confuse you about the number of factors you are dealing with as in this problem:

John takes a course. The average of two tests and a final will determine his grade. The final will be weighted double. John has made 86 and 88 on his two tests. What does he have to make on the final to average 90 in the course? What they do is they hide or try to camouflage the number of factors you're dealing with.

The final is weighted double; so, it has to be represented as a 2x or whatever you want to give as a variable. And the demoninator is 4 not 3 because you're weighting that double. But you see how they're trying to disguise what you have to consider. You have to remember that this is weighted double; so, you have to allow for that.

This brings up another point about the SAT. In this case, if you left a 3 in the denominator instead of a 4, trust me, they would have the answer that you come up with as a choice. That is one of the key things about the SAT. If you work a problem in a wrong direction, they'll have an answer waiting for you every time.

Let's look at PERCENTAGES. If I ask you what 1/4% is in decimals, what are you going to tell me? Most people say .25. What you want to remember, and the SAT loves to ask you this because they know with the tic-tic-tic of the clock, you're gonna look at that and say oh, that's .25.

One of the ways they ask this type of question is: 1/4% of 500 = ? You'll go what's .25 of 500? You punch it out on your calculator and come up with 125. But that's wrong!

Remember, if you have the percent sign, or the word percent, you've got to do something with the two decimal places. This is really: .25% of 500 = .0025 of 500 = 1.25

We're not through with the way they play with percentages yet. You're used to dealing with percentages, as in this problem below:

Typically to find out what 25% off is, you take 25% of the price of $400. Then you subtract the resulting $100 from $400. So, you have $300. Unfortunately, this method won't work on the SAT.

There is another way to do it, and that's to do your figuring on the percentage side. We start by saying 100% is the original price. In this case, if I want to take 25% off the price, I'm going to subtract .25 from that 1.00 and have .75. So, I could just multiply that $400 by .75 and get the correct price of $300. This way you're doing your adding and subtracting on the percentage side.

Now, what if I want to increase the price 25%. You add .25 to the 1.00 original price giving you 1.25. So, you multiply by 1.25, and it does the addition for you, giving you the correct answer of $500. Remember, if you are decreasing a price, you multiply by less than 100%. If you are increasing the price, you multiply by a number greater than 100%.

This is where a lot of students say, "Well that's cool, but I'm very comfortable with the my old way of doing these problems. Why do I want to fool with doing it a new way?" That's a good question. Unfortunately I have a good answer - the following question:

Now, on this problem, a car manufacturer increases the price of a model car 30% in 1980 from its original price in 1975, and in 1985, increased the price 50%. How much did they increase the price over the 10 years? You glance down there, and you look at the answers and say it's gotta be 80%. Sorry, that's too easy for a problem in the last few questions, which is where you'd find this one.

This problem illustrates three characteristics of how the SAT treats these problems. The first characteristic is that they didn't give you a starting price...they never do. That doesn't bother us, because we know how to work on the percentage side. What you do is work it like we just did - on the percentage side. You've got 100% to start with as the original price. You're increasing it 30%; so, after adding 30% to the 100% you multiply the original price by 1.30 which gives you 1.30.

This brings us to the second characteristic They want a 50% increase, but you are applying that increase to the new price of 1.30. Increase the price 50%, so you multiply the 1.30 by 1.50 and that leaves you with a total price now of 1.95.

This brings up the third characteristic. Notice that you have (d) as 95% and (e) as 195%. They ask for the total increase. You started with the original price of 1.00. Now we're at 1.95, so the net increase is .95. They ask this question two ways. If they ask it as they have here, how much have they increased the price in 10 years, then the answer is (d). They want to know how much it increased. The other way they ask that question is: "What is the total price after 10 years?" The answer there is, the total price, is 195%. Whether they want the total or they want the increase tells you whether you add or subtract that original 100%.

Now we come to a WORD PROBLEM. You want to remember that the numbers in a word problem really aren't very difficult. The words are what make word problems difficult. So it makes sense that the way to work word problems is to get the numbers out of the words. Think of it this way: If you leave the numbers in the words you are playing their game. If you get the numbers out of the words, you have the ball in your court, so to speak. For example:

You want to get the number out of the words as you read the problem. Michael Jordan made 40% of his first 15 shots. What does he have to average on the rest of his shots if he ends the game with a 50% average of his 30 shots? All you see there when you glance at it is shots, shots, shots. In this case, write down what the problem says.

He's made 40% of his first 15 shots. We write down .40 x 15 = 6. So, he's made 6. Where do you want to go? If he wants to make 50% of the 30 shots he takes for the whole game, then we write down .50 x 30 = 15. So what does he have to average? Well, how many does he have to make to get to 15? 9. How many shots does he have left to do it? 15. So, he's got to make 9 of 15 or 60%. See, once you set that up, that's an easy problem. That's the thing in doing a word problem. Take it apart and do it in components.

They ask two types of INEQUALITIES. This first is a stringout inequality.

Your teachers teach you to separate that and work it as two inequalities. I'm going to suggest on the SAT if you see one of these, leave it as one inequality. It helps keep things simple so you keep uniform in how you work it. Remember that what you do to one segment you do to all three segments.

So in this case you're going to subtract six across because you want to isolate that variable X in the middle. If you subtract six across the board from all of them, you're left with: -21 < 3X < 24 Now you divide through by three across the board and are left with: -7 < X < 8.

You're going to notice all the way through here we're going to be talking about most frequently is simplifying things. If you see an inequality that you have to solve on the SAT you're antenna needs to go up and you just need to start searching for a negative that you're going to be multiplying or dividing with because trust me, there will be one there.

Subtracting 6 - we have left -5X < 20 Then we divide by -5 and, remember when you multiply or divide by a negative number, you reverse the inequality. So we are left with: X > -4.

ABSOLUTE VALUES. Nobody has missed this since I started begging. Ergo I'm going to continue begging. When you have an absolute value, perform the operation within the brackets just like it was a parenthesis before you apply that absolute value. So in this case it's |7+6-18|=|-5|=5. It is not applying the absolute value across the board and saying 7+6+18=31. A lot of the things we go over you're going to say: "That looks obvious," but under the pressure of taking the SAT in that third hour, you will be surprised some of the stupid things you will do and some of the stupid ways you'll see things.

EXPONENTS. Exponents are fairly straightforward except for a couple of tripping points.

Number 1 is the easiest in that A to the m is A times itself m number of times.

Number 2 is where it starts to get a little bit interesting. Number 2 is 1 over A to the m equals A to the negative m.

Number 3 is the most important exponent rule on the test - by far! A to the m plus A to the n equals A to the m plus A to the n. We haven't lost it, this is very important because the SAT will come after you with problems like this (which is very similar to the one everyone missed on the test this one was on.)

Look at how dastardly they are. You go down here and look at the clue or cues they're giving you, as you glance from the problem to the answer choices, looking at the answers you say: "Oh, I know they think I'm going to multiply these exponents, but I know all you do is add the exponents. So you circle A, go merrily on your way, and never realize you missed it.

REMEMBER!!! If you are adding and subtracting the same number or variable with different exponents, you can't do a thing to the exponents. In the example, all you can do is factor it a bit: X2 (X + 1) You can take X squared out of it so it's X squared times the quantity X plus 1.

With number 4 you get to do what your impulse was in Number 3. A to the m times A to the n equals A to the m plus n. If you multiply the same number or variable with different exponents, you add the exponents.

The only time you multiply exponents is in number 5; when you're raising a power to a power.

Number 6, if you divide, you subtract. A to the M divided by A to the N equals A to the M minus N.

RADICALS. Anytime you have a fraction as an exponent you have a radical. For instance: if you've got A to the 1/2, that's the square root of A. A to the 2/3 is the cube root of A squared.

The square root of A times the square root of A is A. Just be very careful with that because the SAT loves, I don't mean likes, I mean loves to get rid of a square root that is a clue as to how to work the problem. For instance, the hypotenuse of an isosceles right triangle is side times the square root of 2. They love to give you a side that has square root of 2 in it so when you multiply the square root of two times three times the square root of 2, it comes up with 6. So when you look down to the answers it doesn't give you a clue as to how you got there.

SIMULTANEOUS EQUATIONS. Now simultaneous equations, you know how to do them by elimination or by substitution. But on the SAT, things are very different. You can't rely on all the methods and techniques from math class. Remember, this is largely a puzzle test. When you see simultaneous equations stacked up or side by side, you"ve got to realize the SAT is up to something. You must focus on what you are looking for, and then look at what information the SAT has given you to get there. To reiterate, simultaneous equations are very prominent on the SAT. You cannot treat them typically as you do in school. YOU MUST LOOK AT A NONTRADITIONAL METHOD FOR SOLVING THESE.

For instance, this example of a Quantitative Comparison problem:

So they give you x+2y=19, 2x+y=9. Under those conditions, which is greater? 3x+3y or 28. If you try to solve for x and y you'll fill up a page and, at the least, get tough fractions to work with. For some reason it just doesn't mix right. You must focus on the target - on 3x + 3y and how it compares to 28. Then look at 2x + y = 19 and x + 2y = 9 and see how you can manipulate those equations to get to your target.

This won't help you solve for x and y at all, but it will help you get the answer for the SAT. Just add the two equations together, and remember that technique because it shows up often. You may have to add the equations, subtract them, or manipulate them in some type of way which will get you to your target. Just remember to manipulate the equations in whatever manner you have to in order to get to your target.

Another type of simultaneous equation situation arises when the SAT puts them in the form of word problems. For instance:

Now, the majority of simultaneous equations that they ask you to solve are going to be in the form of a word problem like this. What you do with the word problem, as we said before, is get the numbers out of the words. In this case, identify who or what you're looking for, put everything else in terms of that, and then solve the problem.

In this problem, who are we looking for here? We're looking for Connie. So, put everybody in Connie's terms. Alex is twice as old as Connie which we denote with 2C. Connie is two years younger than Janice, so we denote that with C + 2. If their average age is 22, how old is Connie? So you set up your equation:

Now you substitute and get 2C + 2 + C + C = 66; 4c = 64; Connie is 16. So just identify who you're looking for, put the others in terms of that character or that individual, boom you've got the answer. They aren't hard, it's just a question of having a technique for doing it.

SCIENTIFIC NOTATION. This shouldn't be a problem for you since you can use your calculator. Just remember: 4 x 105 = 400,000; 7 x 104 = 70,000. You just count the number of zeros to the right of the number.

There is one particular situation pertaining to scientific notation which the SAT seems to enjoy presenting. This is where they ask you to multiply two numbers written in scientific notation. Separate the numbers and the powers ot ten. Multiply the numbers and treat the powers of ten the same as you would any other number mulitiplied by itself with different exponents. For example:

Again, the technique to remember is to separate and treat them separately. So you take 4 and 7 and multiply those together and multiply the result of that by, (just treat 10 as you would any other number with an exponent) 5 plus 6, equals 28 times 10 to the eleventh. Remember with Scientific Notation, you have to reduce the number you raising to the power of ten to a single decimal point.

SUPPLEMENTARY ANGLES share a line and total 180 degrees.

INTERSECTING LINES form vertical angles and opposite vertical angles that are equal.

FUNCTIONS & MADE UP FUNCTIONS This is an important part, because these are points they don't think you're going to get. You've dealt with functions, haven't you? A function is an equation written to determine coordinate values. You must be disciplined and make the substitutions indicated. For instance:

If you are asked what is f(2), you take 2 and substitute it in for x. If I ask you for f(g(2)), you take that 2 and plug it in the g(x) function, take that resulting 4 and substitute it for x in the f(x) function. If I ask you what is f(g(x)), you look carefully and see that all you have been given for g(x) is x + 2. So what you're doing is simply substituting.

What you will see three to four times per test are MADE UP FUNCTIONS. These are characterized by signs, symbols, or marks like squares, boxes, arrows, and the like. All you do with made-up functions is do the substitution they ask for. Ignore your math voice, just do what they say. Made-Up functions are easy points if you pay attention, focus and simply MAKE THE SUBSTITUTIONS INDICATED. Ignore what your logic or your "little math voice" tells you. Just do what the made-up function says. For example:

So for this one they say, for all integers a and b, a ^ b = 7. Another Quantitative Comparison where they ask: Which is a greater value, column A or column B, you mark C if they're equal, D if you can't tell. Remember to make the substitution they ask for. Don't try to reason it out mathematically, just make the substitution. So you put 4 in for a and 7 in for b. What does that equal? Get a little stubborn and say to yourself "What have they given me?" All you've been told is that a ^ b = 7. That's all you've got to go on, so 4 ^ 7 = 7!! That's right, under the terms you've been given, anything you put in for a and anything you put in for b will yield 7! Therefore the columns are equal.


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