SAT For Dummies

Part IV

Take a Number, Any Number: The Mathematics Sections

Chapter 12

Numb and Numbering: The Ins and Outs of Numbers and Operations

In This Chapter

 Identifying types of numbers and following the order of operations

 Calculating percents and working with ratios

 Figuring out rate/time/distance problems

 Eyeing radicals and absolute value

 Understanding sequences and sets

Once upon a time you could take care of all the numbers you needed for school purposes with ten fingers and, in a pinch, a couple of toes. Sadly, life has changed. For the SAT, you need to know what’s prime and what’s not, as well as how to calculate and manipulate percents, ratios, means, and the like. Not to mention sets and sequences! Never fear. Even though you’ve moved way beyond body-part math, this chapter tells you everything you need to know about numbers and operations, at least as they appear on the SAT.

Meeting the Number Families

Mathematics starts with numbers, which come in various “flavors.” You need to nibble on several types of numbers before you hit SAT day, so in this section I present a buffet of numbers.

You may be wondering why you need a vocabulary lesson to do well on SAT math. The fact of the matter is the SAT makers love to tuck these terms into the questions, as in “How many prime numbers are . . .” or “If the sum of three consecutive integers is 102, what is . . .” and the like. If you don’t know the vocabulary, you’re sunk before you start.

Check out this “menu” of toothsome (good-tasting) numbers:

 Whole numbers: Whole numbers aren’t very well named because they include 0, which isn’t a whole lot of anything. The whole numbers are the ones you (hopefully) remember from grade school: 0, 1, 2, 3, 4, 5, 6 . . . you get the idea. Whole numbers, by definition, don’t include fractions or decimals.

Whole numbers can be even or odd. Even numbers are divisible by 2, and odd numbers aren’t.

 Prime numbers: Prime numbers are divisible only by themselves and by 1. The first few prime numbers are 2, 3, 5, 7, 11, 13, 17, and 19. Zero and 1 aren’t prime numbers. They’re considered “special.” (The kids in grade school said that about me, too.) Two is the only even prime number. No negative number is ever prime because all negative numbers are divisible by –1.

One common misconception is that all odd numbers are prime. Don’t fall into that trap. Tons of odd numbers (9 and 15, for example) aren’t prime because they’re divisible by another number.

 Composite numbers: Anything that’s not prime or special is composite. If you can divide a number by some smaller number (other than 1) without getting a remainder, you have a composite number. A few composite numbers are 4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20, 21, and so on. (I could go on to add zillions more, but you get the idea.)

Speaking of divisibility, remembering these points will win you SAT points:

• All numbers whose digits add up to a multiple of 3 are also divisible by 3. For example, the digits of 789 add up to 24 (7 + 8 + 9 = 24); because 24 is divisible by 3, so is 789.

• Ditto for multiples of 9. If the digits of a number add up to a multiple of 9, you can divide the number itself by 9. For example, the digits of 729 add up to 18; because 18 is divisible by 9, so is 729.

• All numbers ending in 0 or 5 are divisible by 5.

• All numbers ending in 0 are also divisible by 10.

These divisibility rules work backward, too. Consider the number 365. It’s not even, so it can’t be divided by 2. Its digits add up to 14, which isn’t divisible by 3 or 9, so it’s not divisible by either 3 or 9. Because 365 ends in 5, it’s divisible by 5. Because it doesn’t end in 0, it’s not divisible by 10.

 Integers: The whole numbers and all their opposites — also known as negative ­numbers — are integers. The whole numbers go all the way up to infinity, but the ­integers are even more impressive. Integers reach infinity in both directions, as the number line in Figure 12-1 shows.

Figure 12-1:Integers go on forever and ever.

When you’re asked to compare integers, remember that the farther to the right a number is, the greater it is. For example, 3 is greater than –5. Also, –4 is greater than –20.

 Rational numbers: Numbers for whom a padded room without a view isn’t necessary. Just kidding. All integers are rational numbers. In addition, any number that can be written as a fraction — proper or improper — is a rational number. (In a proper fraction, the number on top is smaller than the number on the bottom, and in an improper fraction, the top number is greater than the bottom number.) Plus, any decimal that either ends, like 0.23 (the decimal for 23⁄100), or repeats like , the decimal for 1⁄6, or , the decimal for 1⁄7, is a rational number. The following are also rational: –2, 0.234, 735⁄13, .

 Irrational numbers: Irrational numbers are decimals that never end or repeat. Practically speaking, you need to worry about only two kinds of irrational numbers:

• Radicals (such as  and )

• π, which you’ve probably heard of because it appears in the formula for the area of a circle. (Like Mom’s apple dessert, π is in a class by itself.)

Every type of number I mention in this chapter is a real number. Right about now you’re probably wondering, “Are some numbers not real?” The bad news: Yes, and you may have to figure them out some day, perhaps at the college you’re sending your SAT scores to. The good news: That day isn’t today because every number on the SAT is real.

Getting Your Priorities Straight: Order of Operations

How many times has your mom told you to turn off the Wii and start on your homework because you “have to get your priorities straight.” I’m not going to comment on the annoyance that authority figures generate, especially when they’re right, but I am going to tell you that in math, priorities matter.

Consider the problem 3 + 4 × 2. If you add 3 + 4, which of course equals 7, and multiply by 2, you get 14. Nice answer, but wrong, because you forgot about Aunt Sally. “Aunt Sally,” or more accurately, “Please Excuse MDear Aunt Sally,” or PEMDAS, is a mnemonic (memory) device you can use to help you remember what mathematicians call order of operations. When faced with a multipart problem, just follow the order of operations that “Aunt Sally” calls for. Note the italicized letters in the following step list, which tells you what “Aunt Sally” really means:

1. Do everything in parentheses.

2. Calculate all exponents.

3. Multiply and divide, from left to right.

4. Add and subtract, from left to right.

Back to the sample problem, 3 + 4 × 2. No parentheses or exponents, so the first operations up are multiplication and division. Because there’s no division, you’re left with 4 × 2, which equals 8. Onward to addition and subtraction (in this problem, subtraction isn’t present, so forget about subtracting). Just add 3 to 8, at which point you arrive at 11, the correct answer.

Many calculators know the “Aunt Sally” rules, but on older ones, sometimes you have to input the numbers according to the “Aunt Sally” rules to ensure the right answer. Be sure to figure out which kind of calculator you have before test day.

Aunt Sally’s lonely, so here’s another chance to visit her:

The expression 20 – (40 ÷ 5 × 2) + 32 is equal to

(A) –5

(B) 7

(C) 10

(D) 13

(E) 25

The answer is (D). Start with what’s in the parentheses: 40 ÷ 5 × 2. Don’t fall into the trap of multiplying 5 × 2 first; proceed from left to right: 40 ÷ 5 = 8 and 8 × 2 = 16. Next, tackle the exponent: 32 = 9. At this stage, you have 20 – 16 + 9. Again, resist the temptation to start by adding; just go left to right (20 – 16 = 4 and then 4 + 9 = 13).

Playing Percentage Games

The SAT loves percentages, perhaps because math teachers who are sick of the question “Am I ever going to use this stuff in real life?” actually write the math portion of the exam. With percentages, the answer is yes if you’re taking out a loan (interest rates) or investing the earnings from your part-time job in mutual funds (still interest, but this time it’s a good thing). Percents represent how much of each hundred you’re talking about.

Taking a percentage of a number is a simple task if you’re using a calculator with a “%” button. Just hit the “%” and “×” buttons. For example, to find 60 percent of 35, multiply 60% by 35. The answer is 21. If you’re not blessed with such a calculator, you can turn a percent into a decimal by moving the decimal point two spaces to the left, as in 60% = 0.60. (Other examples of percents include 12.5% = 0.125, 0.4% = 0.004, and so on.) Or, turn the percent into a fraction. The “cent” in percent means hundred, so 60 percent = 60⁄100.

For more complicated problems, fall back on the formula you mastered in grade school:

Suppose you’re asked “40% of what number is 80?” The number you’re looking for is the number you’re taking the percent of, so x will go in the of space in the formula:

Now cross-multiply: 40x = 8,000. Dividing by 40 gives you x = 200.

A particularly annoying subtopic of percentages is a problem that involves a percent increase or decrease. A slight variation of the percentage formula helps you out with this type of problem. Here’s the formula and an example problem to help you master it:

The value of your investment in the winning team of the National Spitball League increased from $1,500 to $1,800 over several years. What was the percentage increase of the investment?

(A) 300

(B) 120

(C) 831⁄3

(D) 20

(E) 162⁄3

The correct answer is (D). The key here is that the number 1,800 shouldn’t be used in your formula. Before you can find the percent of increase, you need to find the amount of increase, which is 1,800 – 1,500 = 300. To find the percentage of increase, set up this equation:

Cross-multiply to get 1,500x = 30,000. Dividing tells you that x = 20 percent.

The SAT makers often try to confuse you by asking about something that doesn’t appear in the original question, as in this example:

At one point in the season, the New York Yankees had won 60 percent of their games. The Yanks had lost 30 times and never tied. (As you know, there are no ties in the world’s noblest sport, baseball. No crying either.) How many games had the team played?

(A) 12

(B) 18

(C) 50

(D) 75

(E) 90

The answer is (D). Did you find the catch? The winning percentage was 60 percent, but the question specified the number of losses. What to do? Well, because ties don’t exist, the wins and losses must have represented all the games played, or 100 percent. Thus the percentage of losses must be 100% – 60%, which is 40%. Put the formula to work:

As always, cross-multiply: 40x = 3,000, and x = 75.

Keeping It in Proportion: Ratios

After you know the tricks, ratios are some of the easiest problems to answer quickly. I call them “heartbeat” problems because you can solve them in one throb. Here are the points to remember:

 A ratio is written as  or of:to.

• The ratio of sunflowers to roses = .

• The ratio of umbrellas to heads = umbrellas:heads.

 A possible total is a multiple of the sum of the numbers in the ratio.

You may have to confront a proportion problem like this on the test:

At a party, the ratio of blondes to redheads is 4:5. What could be the total number of blondes and redheads at the party?

This one’s mega-easy. Just add the numbers in the ratio: 4 + 5 = 9. The total must be a multiple of 9, such as 9, 18, 27, 36, and so on. If this “multiple of” stuff is confusing, think of it another way: The sum must divide evenly into the total. That is, the total must be divisible by 9. Can the total, for example, be 54? Yes, 9 goes evenly into 54. Can it be 64? No, 9 doesn’t go evenly into 64.

Check out another example.

Trying to get Willie to turn down his stereo, his mother pounds on the ceiling and shouts. If she pounds seven times for every five times she shouts, which of the following can be the total number of poundings and shouts?

(A) 75

(B) 57

(C) 48

(D) 35

(E) 30

The correct answer is (C). Add the numbers in the ratio: 7 + 5 = 12. The total must be a multiple of 12. (It must be evenly divisible by 12.) Here, only 48, Choice (C), is evenly divisible by 12. Of course, 75 and 57 try to trick you by using the numbers 7 and 5 from the ratio.

Notice how carefully I’ve been asking what can be the possible total. The total can be any multiple of the sum. If a question asks you which of the following is the total, you have to answer, “It cannot be determined.” You know only which can be true.

Another ratio headache strikes when you’re given a ratio and a total and asked to find a ­specific term. To find a specific term, do the following, in order:

1. Add the numbers in the ratio.

2. Divide that sum into the total.

3. Multiply that quotient by each term in the ratio. (The quotient is the answer you get when you divide.)

4. Add the answers to double-check that they sum up to the total.

Pretty confusing stuff. Take it one step at a time. Look at this example problem:

Yelling at the members of his team, whose record was 0 for 21, the irate coach pointed his finger at each member of the squad, calling everyone either a “wimp” or a “slacker.” If he had 3 wimps for every 4 slackers, and every member of the 28-man squad was either a wimp or a slacker, how many wimps were there?

Here’s how to solve it:

1. Add the numbers in the ratio: 3 + 4 = 7.

2. Divide that sum into the total: 28⁄7 = 4.

3. Multiply that quotient by each term in the ratio: 4 × 3 = 12; 4 × 4 = 16.

4. Add to double-check that the numbers sum up to the total: 12 + 16 = 28.

Now you have all the information you need to answer a variety of questions: How many wimps were there? 12. How many slackers were there? 16. How many more slackers than wimps were there? 4. How many slackers would have to be kicked off the team for the number of wimps and slackers to be equal? 4. The SAT writers can ask all sorts of things, but if you have this information, you’re ready for anything they throw at you.

The SAT writers often throw in extra numbers that aren’t used at all to solve the problem. In the preceding example, the team’s not-quite-World-Series-quality 0 and 21 win/loss record is interesting but irrelevant in terms of the question you’re answering. Don’t get distracted by extra information.

Getting DIRTy: Time, Rate, and Distance

Time to dish the dirt, as in D.I.R.T. Distance IRate × Time. or D = RT. When the SAT throws a time, rate, and distance problem at you, use this formula. Make a chart with the formula across the top and fill in the spaces on the chart. Here’s an example to help you master this formula:

Jennifer drives 40 miles an hour for 21⁄2 hours. Her friend Ashley goes the same distance but drives at 11⁄2 times Jennifer’s speed. How many minutes longer does Jennifer drive than Ashley?

Don’t start making big, hairy formulas with xs and ys. Make the DIRT chart using the distance formula: Distance = Rate × Time.

When you fill in the 40 mph and 21⁄2 hours for Jennifer, you can calculate that she went 100 miles. Think of it this way: If she goes 40 mph for one hour, that’s 40 miles. For a second hour, she goes another 40 miles. In a half-hour, she goes 1⁄2 of 40, or 20 miles. (See? You don’t have to write down 40 × 21⁄2 and do all that pencil-pushing; use your brain, not your yellow No. 2 pencil or your calculator.) Add them together: 40 + 40 + 20 = 100. Jennifer drives 100 miles.

Because Ashley drives the same distance, fill in 100 under distance for her. She goes 11⁄2 times as fast. Uh-uh, put down that calculator. Use your brain! 1 × 40 is 40; 1⁄2 × 40 is 20. Add 40 + 20 = 60. Ashley drives 60 mph. Now this gets really easy. If she drives at 60 mph, she drives one mile a minute (60 minutes in an hour, 60 miles in an hour). Therefore, to go 100 miles takes her 100 minutes. Because your final answer is asked for in minutes, don’t bother converting this to hours; leave it the way it is.

Last step. Jennifer drives 21⁄2 hours. How many minutes is that? Do it the easy way, in your brain. One hour is 60 minutes. A second hour is another 60 minutes. A half-hour is 30 minutes. Add them together: 60 + 60 + 30 = 150 minutes. If Jennifer drives for 150 minutes and Ashley drives for 100 minutes, Jennifer drives 50 minutes more than Ashley.

Be careful to note whether the people are traveling in the same direction or opposite directions. Suppose you’re asked how far apart drivers are at the end of their trip. If you’re told that Jordan travels 40 mph east for 2 hours and Connor travels 60 mph west for 3 hours, they’re going in opposite directions. If they start from the same point at the same time, Jordan has gone 80 miles one way, and Connor has gone 180 miles the opposite way. They’re 260 miles apart. The trap answer is 100 because careless people (not you!) simply subtract 180 – 80.

Demonstrating the Value of Radicals

Knowing how to manipulate radicals can help you get around in Berkeley, California. Radical knowledge also helps with the SAT. In math-speak, a radical is a square root, as well as the symbol indicating square root, . The square root of number x, written , is the positive number which, multiplied by itself, gives you x. As a classic example, , because 3 × 3 = 9. If only radicals were always that easy. Unfortunately, most numbers have square roots that are decidedly not pretty. , for example, equals approximately 2.645751311.

The rules for multiplication and division of radicals are simple. Just multiply and divide the numbers normally:

Addition and subtraction are trickier. You can’t just add and subtract the numbers and plop the result under a square root sign. For example,  doesn’t equal . You can add or subtract radicals only if they have the same number under the symbol, so  is impossible as written. However, you can break down some radicals by factoring out a perfect square and simplifying it, so , and ; then, .

A few squares show up all the time on the SAT. Scan Table 12-1 so you’re familiar with these numbers when you see them.

Notice how the square of both x and –x is x2? Conveniently, when you multiply two negative numbers, the result is positive, as it is when you multiply two positive numbers. So the square of the same negative and positive number is always the same: (–8)2 = 64 and (+8)2 = 64.

Computing Absolute Value

Absolute value is a simple concept that’s annoyingly easy to mess up. Absolute value is the number, shorn of its positive or negative value. The symbol looks like a Superman phone booth without a roof. The absolute value of 3 is written , which equals 3; the ­absolute value of –3 is written , which also equals 3.

On the SAT, you may see a number or algebraic expression inside the absolute value symbol. If you do, follow these steps:

1. Simplify whatever is inside the absolute value symbol, if possible.

2. If the answer is negative, switch it to positive.

Some people have the (incorrect) idea that absolute value changes subtraction to addition. Nope. If you’re working with , don’t change the quantity to 3 + 4. Calculate whatever is inside the absolute value symbols first, , and only then change the result to a positive number, in this case 1.

Finding the Pattern

Math sometimes involves recognizing patterns and seeing where those patterns lead. The SAT occasionally asks you to play mathematician with two types of patterns: arithmetic and geometric. The math word for pattern, by the way, is sequence.

Check out this arithmetic sequence: 2, 5, 8, 11, 14, . . . Notice how each number is obtained by adding 3 to the previous number? In an arithmetic sequence, you always add or subtract the same number to the previous term to get the next term. Another example of an arithmetic sequence is 80, 73, 66, 59, . . . In this one, you’re subtracting 7 from the previous term.

A geometric sequence is similar to an arithmetic sequence, but it works by multiplication or division. In the sequence 2, 6, 18, 54, . . . every term is multiplied by 3 to get the next term. In 100, 50, 25, 121⁄2, . . . each term is divided by 2 to get the next term.

Often, the best way to solve these problems is just to make a list and follow the pattern. However, if the test writers ask you for something like the 20th term of the sequence, this process can take forever. Each type of sequence has a useful formula, which is worth memorizing if you have the time and the room in your head:

 For an arithmetic sequence, the nth term = the first term + (n – 1)d, where d is the difference between terms in the sequence. In the sequence 2, 5, 8, 11, 14, the difference between terms is +3, because you add 3 each time. What would be the 20th term? Take 2, the first term, and add 3 19 times, so it’s 2 + 19(3) = 2 + 57 = 59.

 For a geometric sequence, the nth term = the first term × r(n – 1), where r is the ratio of one term to the next. Huh? Well, you probably remember that taking something to a power (that’s what the exponent stands for) means multiplying it by itself a bunch of times. For example, 4 to the 5th power = 4 × 4 × 4 × 4 × 4, which equals 1,024. (Powers get big really fast.) You can do powers on most calculators by using either the “yx” or the “^” button. On mine, I do 4 to the 5th by typing: 4 “yx” 5 = .

Check out this sequence: 2, 6, 18, 54. The ratio is 3 because you multiply by 3 each time. To find the 10th term (the 20th would be way too big to handle), take 2 × 39 (that’s 3 to the 9th power). 39 = 19,683, and 2 × 19,683 = 39,366, so that’s the answer.

To find the nth term, you always use n – 1, no matter what kind of sequence it is. That’s because n – 1 is how many steps it takes to get from the first term to the nth term.

As if your life weren’t tough enough, the SAT folks often hide these sequences inside a word problem, such as the following:

The bacteria population in my day-old wad of chewing gum doubles every 3 hours. If there are 100 bacteria at 12:00 noon on Friday, how many bacteria will be present at midnight of the same day?

(A) 200

(B) 300

(C) 800

(D) 1,600

(E) 409,600

The right answer is (D). To solve this problem, make a chart. Because the population doubles every 3 hours, count off 3-hour intervals, doubling as you go:

12:00 (noon) = 100 bacteria

3:00 p.m. = 200 bacteria

6:00 p.m. = 400 bacteria

9:00 p.m. = 800 bacteria

12:00 (midnight) = 1,600 bacteria

And here’s another example in which the formulas come in handy:

Author A, an extraordinarily fast writer who zips through a chapter a day, gets paid $100 for her first chapter, $200 for her second, $300 for her third, and so on. Author B, also a member of the chapter-a-day club, gets paid $1 for his first chapter, $2 for his second, $4 for his third, $8 for his fourth, and so on. On the 12th day,

(A) Author A is paid $76 more.

(B) Author B is paid $24 more.

(C) They are paid the same amount.

(D) Author A is paid $1,178 more.

(E) Author B is paid $848 more.

The correct answer is (E). Author A’s plan is an arithmetic sequence, increasing by $100 each time, so on the 12th day she’s paid 100 + 11(100) = 100 + 1,100 = $1,200. Author B’s plan is a geometric sequence, multiplied by 2 each time, so on the 12th day, he’s paid 1 × 211 = 1 × 2,048 = $2,048. So Author B is paid $848 more.

Setting a Spell

A set is just a collection of things — shrunken heads, leftover hockey pucks, Barbie outfits, whatever. In math, a set is a collection of elements, usually numbers, which you find inside brackets: { . . . }. For example, the set of whole numbers less than 6 is a set with six elements: {0, 1, 2, 3, 4, 5}. Some sets go on forever, and three dots at the end tell you so. The set of positive odd numbers is {1, 3, 5, 7, . . . } because it reaches infinity. A set may have nothing inside of it; this is the “empty set,” and it’s written either { } or (more commonly) .

For the SAT, you need to know about two specific things when it comes to sets — the union and the intersection of sets. The union of two sets is just the two sets put together; thus, the union of {1, 2, 3} and {5, 7, 8} is {1, 2, 3, 5, 7, 8}.

Even if something shows up in both sets, it shows up only once in the union. Thus, the union of {2, 3, 4} and {3, 4, 5} is {2, 3, 4, 5}, not {2, 3, 4, 3, 4, 5}. The following steps help you find the number of elements in the union of two sets:

1. Add up the number of elements in each set.

2. Subtract the number of elements that show up in both.

In the preceding example, 3 + 3 = 6; but because 3 and 4 show up in both sets, you have to subtract 2. The union has 4 elements. The intersection of two sets, on the other hand, contains only those elements that show up in both of them. The intersection of {1, 2, 4} and {4, 6, 7} is {4}; the intersection of {3, 5, 7} and {2, 4, 6} is , also known as “empty.”