SAT For Dummies

Part IV

Take a Number, Any Number: The Mathematics Sections

Chapter 13

Practicing Problems in Numbers and Operations

In This Chapter

 Trying your hand at SAT questions involving numbers and operations

 Figuring out which problems give you the most trouble

That old saying, “Practice makes perfect,” is annoying yet true. In this chapter, I hit you with two sets of numbers and operations questions along with explanations of the answers. After you practice each question in the first set, check your answers and read the explanations for any questions you answered incorrectly. (The answers immediately follow each question. Use a piece of paper to cover the answers as you work.) If you’re confused about any point, turn back to Chapter 12 for more details on the kind of problem that’s stumping you. The second set is set up like the real test: You do all the problems and then check your work with the answer key that follows the last question.

Set 1: Trying Out Some Guided Questions

1. If you invest $2,000 for one year at 5% annual interest, the total amount you would have at the end of the year would be

(A) $100

(B) $2,005

(C) $2,100

(D) $2,500

(E) $3,000

Solve the question like this: 5% = 0.05, so 5% of 2,000 = 0.05 × 2,000 = 100. But wait! Before you choose 100 as your answer, remember that you still have the $2,000 that you originally invested, so you now have 2,000 + 100 = $2,100. You can also solve this problem using the is/of method I discuss in Chapter 12. You’re basically being asked, “What is 5% of 2000?” so

you write . Cross-multiplying gives you 100x = 10,000, so x = $100. Of course, you

still need to add in the original $2,000 to get your answer, so Choice (C) is correct.

2. Which number is an element of the set of prime numbers but not of the set of odd numbers?

(A) 0

(B) 1

(C) 2

(D) 3

(E) 9

Because 2 is the only prime number that isn’t odd, Choice (C) is correct.

3. 100 percent of 99 subtracted from 99 percent of 100 equals

(A) –1

(B) 0

(C) 0.99

(D) 1

(E) 1.99

Keep in mind that 100 percent of anything is itself, so 100 percent of 99 is 99. Ninety-nine ­percent of 100 equals 0.99 × 100 = 99 (not a big surprise because percent means “out of one hundred”). And 99 – 99 = 0, so Choice (B) is the correct answer.

4. The tenth number of the sequence 50, 44.5, 39, 33.5, . . . would be

(A) –4

(B) 0.5

(C) 1

(D) 1.5

(E) 6

The numbers decrease by 5.5 every time. The simplest way to do this problem is to continue the pattern: 50, 44.5, 39, 33.5, 28, 22.5, 17, 11.5, 6, 0.5. You can also use the following formula to find the tenth term: the nth term = the first term + (n – 1)d, where d is the difference between terms in the sequence. Therefore, 50 + 9(–5.5) = 50 – 49.5 = 0.5. Hooray for Choice (B), the correct answer.

5. If E represents the set of even numbers and N represents the set of numbers divisible by 9, which number is in the intersection of E and N?

(A) 99

(B) 92

(C) 66

(D) 54

(E) 9

An element is in the intersection of two sets only if it’s in both of them. You can go through the choices until you find the right one: 99 isn’t even; 92 isn’t divisible by 9; 66 isn’t divisible by 9; 54 is even and divisible by 9; 9 isn’t even. Thus, 54 is the only one that works, and Choice (D) is the right answer.

6. The first three elements of a geometric sequence are 1, 2, and 4. What is the eighth element of the sequence?

(A) 14

(B) 16

(C) 29

(D) 128

(E) 256

The formula for geometric sequences tells you that the answer is 1 × 27 = 1 × 128 = 128. (Remember that in this formula the exponent is one less than the number of the term you’re being asked for.) Three cheers for Choice (D).

7. The expression 3– 4 + 5(8⁄2) equals

(A) –27

(B) –15

(C) 5

(D) 22

(E) 25

Aunt Sally to the rescue! (See Chapter 12 for the lowdown on my favorite relative.) First, do the operation in parenthesis, 8⁄2 = 4, and then calculate 32, which equals 9. That leaves you with 9 – 4 + 5(4). Next, multiply 5 × 4 = 20. Now the expression is 9 – 4 + 20. You have a trap to avoid: Did you see it? Don’t do addition before subtraction; just go left to right: 9 – 4 = 5, and 5 + 20 = 25. Give it up for Choice (E).

8. Which of the following numbers is rational?

(A) π

(B) 0.12112111211112 . . .

(C)

(D)

(E)

To do this problem, you need to remember the definitions of rational and irrational numbers. π is irrational by definition. (Yes, it’s worth memorizing this fact.) The number 0.12112111211112 . . . is irrational because the decimal never ends or repeats. (For those of you who are still awake, it doesn’t repeat because the number of 1s keeps increasing.) All radicals are irrational if the number underneath isn’t a perfect square: So  and  are both irrational. However, because , it’s rational. Choice (D) is correct.

9. Given that there are 30 days in April, the ratio of rainy days to sunny days during the month of April could not be

(A) 5:3

(B) 3:2

(C) 5:1

(D) 4:1

(E) 3:7

The rule for ratios states that the total must be divisible by the sum of the numbers in the ratio. Because 5 + 3 = 8, and 30 isn’t divisible by 8, Choice (A) is correct. Just to be thorough, of course, you should check that all the other possible sums do go into 30. (They do, I promise. But check anyway!)

10. At a sale, a shirt normally priced at $60 was sold for $48. What was the percentage of the discount?

(A) 12%

(B) 20%

(C) 25%

(D) 48%

(E) 80%

Use the percentage formula,  but, as always, be extra careful. The problem asks for

the percentage of the discount, so don’t just plug in 48. Instead, first figure out the amount

of the discount, which was 60 – 48 = 12. Using 12, write , where p is the percentage

of the discount. Cross-multiplying, you get 1,200 = 60p, and p = 20. You can still get the right answer using 48. If you use 48 in the formula, you get 80%. Because the shirt now costs 80% of what it used to, the discount is 100% – 80% = 20%. Choice (B) is correct either way.

Set 2: Practicing Some Questions on Your Own

Note: Two questions (2 and 6) are grid-ins. On the blank grids in this section, write and bubble in your answers. (See Chapter 11 for the proper way to bubble in your answers for grid-in questions.)

1. The total number of even three-digit numbers is

(A) 49

(B) 100

(C) 449

(D) 450

(E) 500

2. Evaluate .

3. A disease is killing the fish in a certain lake. Every 8 days, half of the fish in the lake die. If there are 1,000 fish alive on March 3, how many are still alive on March 27?

(A) 0

(B) 100

(C) 125

(D) 250

(E) 500

4. If a number n is the product of two distinct primes, x and y, how many factors does n have, including 1 and itself?

(A) 2

(B) 3

(C) 4

(D) 5

(E) 6

5. Which number is 30% greater than 30?

(A) 27

(B) 30.9

(C) 33

(D) 36

(E) 39

6. A recipe for French toast batter calls for 1⁄2 teaspoon of cinnamon for every 5 eggs. How many teaspoons of cinnamon would be needed if a restaurant made a huge batch of batter using 45 eggs?

7. Which of the following is not equivalent to ?

(A)

(B)

(C)

(D)

(E)

8. Janice wrote down all the numbers from 11 to 20. Darren wrote down all the positive numbers less than 30 that are divisible by 6. How many numbers are in the union of their two lists?

(A) 2

(B) 12

(C) 14

(D) 15

(E) 16

9. Elena drove for one hour at 60 miles per hour, and for half an hour at 30 miles per hour. Returning home along the same route, she maintained a constant speed. If the journey home took the same total amount of time as the original drive, what was her speed on the journey home?

(A) 40 miles per hour

(B) 42 miles per hour

(C) 45 miles per hour

(D) 50 miles per hour

(E) 54 miles per hour

10. In the correctly solved addition problem above, A, B, and C all stand for different numbers from 1 to 9. The value of C must be

(A) 8

(B) 7

(C) 6

(D) 5

(E) 4

Answers to Set 2

1. D. Counting all the even three-digit numbers would take a really long time, so try to figure out this question logically. The three-digit numbers start with 100 and end with 999. How many numbers do you have? It’s 900, not 899. (Yes, there is a formula you can use here: Subtract the numbers and add 1. Works every time.) How many of these numbers are even? Well, because even and odd numbers alternate on this list, half of them are even, and half are odd. So you have 450 of each type. Choice (D) is right.

2. 4. When doing an absolute value problem, treat the absolute value symbols as parentheses when trying to figure out the order of operations. Because this problem has a bunch of parentheses and absolute values, work from the inside out:

3. C. On March 3, 1,000 fish are alive. On March 11, 500 fish are alive. On March 19, 250 fish are left. And on March 27, 125 fish are left. Choice (C) is correct.

4. C. Prime numbers have only two factors: 1 and themselves. Pretend in your problem that x = 5 and y = 7. Then n = 5 × 7 = 35. The factors of 35 are 1, 5, 7, and 35. Because you can’t break down 5 or 7, there are no other factors. As long as you pick prime numbers for x and y, you’ll always get four factors for n. Choice (C) is correct.

5. E. Solve it like this: 30% of 30 = 0.30 × 30 = 9. Because the answer is 30% greater than 30, add 30 + 9 = 39. Three cheers for Choice (E).

6. 4.5 or 9⁄2. If you set up a ratio, you’d write .

Cross-multiplying gives you 22.5 = 5x, and x = 4.5. You can also reason as follows: The number of eggs was multiplied by 9, so the amount of cinnamon should be, too. Okay, 1⁄2 × 9 = 41⁄2 = 9⁄2 or 4.5. The answer is 4.5 or 9⁄2. (Don’t grid in 41⁄2.)

7. B. You could use a calculator to figure out what each choice equals, but this problem gives you a chance to practice working with radicals. Start with . This doesn’t equal  because you can’t multiply a whole number and a radical. In order to multiply these, you must turn 2 into ; now, . On to (B): It’s “illegal” to add radicals that don’t have the same number inside (penalty = 5 to 10 years of multiplication tables). Also, there’s no way to break down  or , because no perfect square goes into either one. So you’re stuck on this one. In (C), . In (D), you can break down  to . Then  (remember that  has an “invisible” 1 in front of it). And you saw in (A) that . For (E), use the same trick as in (A): Change 2 into , so . Bottom line: They all equal , except for (B). If you check it out on a calculator, , but . Thus, (B) is correct.

8. B. Janice’s list has 10 numbers: {11, 12, 13, 14, 15, 16, 17, 18, 19, 20}. Darren’s list has 4 numbers: {6, 12, 18, 24}. Now, don’t fall into the trap of thinking that there are 14 numbers in the union; even though 12 and 18 show up in both sets, you’re not allowed to count them twice in the union. The total number of elements in the union is 14 – 2 = 12. Thus, (B) is correct.

9. D. The original trip took 11⁄2 hours. Elena traveled 60 miles plus half of 30, which is 60 + 15 or 75 miles. And 75 divided by 11⁄2 equals 50. Choice (D) is correct.

Hitting a vocabulary Homer

Had enough math for the moment? Take a TV break. Have you ever seen anyone less hirsute (hairy) than Homer Simpson? This famous gourmand (someone who loves to eat and drink a large quantity) isn’t exactly a fan of gourmet cooking (featuring excellent or high-quality food and drink). Homer spends so much time gobbling down doughnuts that he’s made himself rotund, or obese, and not at all agile or lithe. (The first two mean “fat” and the second two mean “graceful in movement.”) Homer’s also gullible (he’ll believe anything) and indolent (don’t wake him up when he’s “working”). Two words you’ll never use to describe Homer: svelte (fashion-model thin) and emaciated (thin to the point of starvation).

10. A. We know that 

A must be an even number because you get it by adding C + C, but A can’t be 2 because then the two numbers wouldn’t add up to something bigger than 1,000. So A = 4, 6, or 8.

Now look at the tens column. The sum of B + B can’t be 5 unless you carried a “1” from the ones column. That means that C + C = 14, 16, or 18, so C = 7, 8, or 9.

What about BB + B +1 (you carried, remember) gives you 5. So B could be either 2 or 7, because 7 +7 + 1 = 15. But if B = 7, then the hundreds column makes no sense. (Try it and you’ll see why.) So B must be 2. Because B = 2, A must = 6 to make the hundreds column work, and that makes C = 8. Check the original problem: 

It works! Choice (A) is correct.