Cracking the SSAT & ISEE - The Princeton Review 2019
SSAT math
The SSAT
INTRODUCTION
This section will provide you with a review of all the math that you need to know to do well on the SSAT. When you get started, you may feel that the material is too easy. Don’t worry. The SSAT measures your basic math skills, so although you may feel a little frustrated reviewing things you have already learned, basic review is the best way to improve your score.
We recommend that you work through these math sections in order, reading each section and then doing each set of drills. If you have trouble with one section, mark the page so you can come back later to go over it again. Keep in mind that you shouldn’t breeze over pages or sections just because they look familiar. Take the time to read over all of the Math sections, so you’ll be sure to know all the math you’ll need!
Lose Your Calculator!
You will not be allowed to use a calculator on the SSAT. If you have developed a habit of reaching for your calculator whenever you need to add or multiply a couple of numbers, follow our advice: put your calculator away now and take it out again after the test is behind you. Do your math homework assignments without it, and complete the practice sections in this book without it. Trust us, you’ll be glad you did.
Write It Down
Do not try to do math in your head. You are allowed to write in your test booklet. You should write in your test booklet. Even when you are just adding a few numbers together, write them down and do the work on paper. Writing things down will not only help eliminate careless errors but also give you something to refer to if you need to check over your work.
One Pass, Two Pass
Within any Math section you will find three types of questions:
· Those you can answer easily without spending too much time
· Those that, if you had all the time in the world, you could do
· Some questions that you have absolutely no idea how to tackle
When you work on a Math section, start out with the first question. If you think you can do it without too much trouble, go ahead. If not, save it for later. Move on to the second question and decide whether or not to do that one. In general, the questions in each Math section are in a very rough order of difficulty. This means that earlier questions tend to be somewhat easier than later ones. You will likely find yourself answering more questions toward the beginning of the sections and leaving more questions blank toward the end.
Don’t Get Stuck
Make sure you don’t spend too much time working on one tough question; there might be easier questions left in the section.
Once you’ve made it all the way through the section, working slowly and carefully to do all the questions that come easily to you, go back and try some of the ones that you think you can do but will take a little longer. You should pace yourself so that time will run out while you’re working on the second pass through the section. By working this way, you’ll know that you answered all the questions that were easy for you. Using a two-pass system is a smart test-taking strategy.
Guesstimating
Sometimes accuracy is important. Sometimes it isn’t.
Which of the following fractions is less than ?
(A)
(B)
(C)
(D)
(E)
Before making any kind of calculation, think about this question. It asks you to find a fraction smaller than . Even if you’re not sure which one is actually smaller, you can certainly eliminate some wrong answers.
Start simple: is less than 1, right? Are there any fractions in the choices that are greater than 1? Get rid of (D) and (E).
Some Things Are Easier Than They Seem
Guesstimating, or finding approximate answers, can help you eliminate wrong answers and save lots of time.
Look at (C). equals 1. Can it be less than ? Eliminate (C). Already, without doing any math, you have a 50 percent chance of guessing the right answer.
Here’s another good example.
A group of three men buys a one-dollar raffle ticket that wins $400. If the one dollar that they paid for the ticket is subtracted and the remainder of the prize money is divided equally among the men, how much will each man receive?
(A) $62.50
(B) $75.00
(C)$100.00
(D)$133.00
(E)$200.00
This isn’t a terribly difficult question. To solve it mathematically, you would take $400, subtract $1, and then divide the remainder by three. But by using a little bit of logic, you don’t have to do any of that.
The raffle ticket won $400. If there were four men, each one would have won about $100 (actually slightly less because the problem tells you to subtract the $1 price of the ticket, but you get the idea). So far so good?
However, there weren’t four men; there were only three. This means fewer men among whom to divide the winnings, so each one should get more than $100, right? Look at the choices. Eliminate (A), (B), and (C).
Two choices left. Choice (E) is $200, half of the amount of the winning ticket. If there were three men, could each one get half? Unfortunately not. Eliminate (E). What’s left? The right answer!
Guesstimating also works very well with some geometry questions, but just to give you something you can look forward to, we’ll save that for the Geometry review.
WORKING WITH CHOICES
In Chapter 2, Fundamental Math Skills for the SSAT & ISEE, we reviewed the concepts that will be tested on the SSAT tests. However, the questions in those practice drills were slightly different from the ones that you will see on your exam. Questions on test day are going to give you five answers to choose from. And as you’ll soon see, there are many benefits to working with multiple-choice questions.
For one, if you really mess up calculating the question, chances are your choice will not be among the ones given. Now you have a chance to go back and try that problem again more carefully. Another benefit is that you may be able to use the information in the choices to help you solve the problems (don’t worry; we’ll tell you how soon).
A Tip About Choices
Notice that the choices are often in numerical order.
We are now going to introduce to you the type of multiple-choice questions you will see on the SSAT. Each one of the following questions will test some skill that we covered in the Fundamental Math Skills chapter. If you don’t see how to solve the question, take a look back at Chapter 2 for help.
Math Vocabulary
1.Which of the following is the greatest even integer less than 25 ?
(A)26
(B)24.5
(C)22
(D)21
(E)0
The first and most important thing you need to do on this—and every—problem is to read and understand the question. What important vocabulary words did you see in the question? There is “even” and “integer.” You should always underline the important words in the questions. This way you will make sure to pay attention to them and avoid careless errors.
Now that we understand that the question is looking for an even integer, we can eliminate any answers that are not even or an integer. Cross out (B) and (D). We can also eliminate (A) because 26 is greater than 25 and we want a number less than 25. Now all we have to do is ask which is greater—0 or 22. (C) is the right answer.
Try it again.
Set A = {All multiples of 7}
Set B = {All odd numbers}
2.All of the following are members of both set A and set B above EXCEPT
(A) 7
(B)21
(C)49
(D)59
(E)77
Did you underline the words multiples of 7 and odd ? Because all the choices are odd, you can’t eliminate any that would not be in Set B, but only (D) is not a multiple of 7. So (D) is the right answer.
The Rules of Zero
3.x, y, and z stand for three distinct numbers, where xy = 0 and yz = 15. Which of the following must be true?
(A)y = 0
(B)x = 0
(C)z = 0
(D)xyz = 15
(E)It cannot be determined from the information above.
Remember the Rules of Zero
Zero is even. It’s neither positive nor negative, and anything multiplied by 0 = 0.
Because x times y is equal to zero, and x, y, and z are different numbers, we know that either x or y is equal to zero. If y was equal to zero, then y times z should also be equal to zero. Because it is not, we know that it must be x that equals zero. Choice (B) is correct.
The Multiplication Table
4.Which of the following is equal to 6 × 5 × 2 ?
(A)60 ÷ 3
(B)14 × 7
(C)2 × 2 × 15
(D)12 × 10
(E)3 × 3 × 3 × 9
6 × 5 × 2 = 60 and so does 2 × 2 × 15. Choice (C) is correct.
Working with Negative Numbers
5.7 — 9 is the same as
(A) 7 — (—9)
(B) 9 — 7
(C) 7 + (—9)
(D)—7 — 9
(E)—9 — 7
Remember that subtracting a number is the same as adding its opposite. Choice (C) is correct.
Don’t Do More Work Than You Have To
When looking at answer choices, start with what’s easiest for you; work through the harder ones only when you have eliminated all the others.
Order of Operations
6.9 + 6 ÷ 2 × 3 =
(A) 7
(B) 9
(C)10
(D)13
(E)18
Remember your PEMDAS rules? Left to right; Since this problem has no parentheses (P) or exponents (E), you can proceed to MD (multiplication and division). Finally, perform the addition. The correct answer is (E).
Factors and Multiples
7.What is the sum of the prime factors of 42 ?
(A)18
(B)13
(C)12
(D)10
(E) 7
How do we find the prime factors? The best way is to draw a factor tree. Then we will see that the prime factors of 42 are 2, 3, and 7. Add them up and we get 12, (C).
Factors Are Small; Multiples Are Large
The factors of a number are always equal to or less than that number. The multiples of a number are always equal to or greater than that number. Be sure not to confuse the two!
Fractions
8.Which of the following is less than ?
(A)
(B)
(C)
(D)
(E)
When comparing fractions, you have three choices. You can find a common denominator and then compare the fractions (such as when you add or subtract fractions). You can also change the fractions to decimals. If you have memorized the fraction-to-decimal chart in Fundamentals (Chapter 2), you probably found the right answer without too much difficulty. It’s (A). Or, if you remember the Bowtie method, you can compare answers that way too!
Percents
9.Thom’s CD collection contains 15 jazz CDs, 45 rap albums, 30 funk CDs, and 60 pop albums. What percent of Thom’s CD collection is funk?
(A)10%
(B)20%
(C)25%
(D)30%
(E)40%
First we need to find the fractional part that represents Thom’s funk CDs. He has 30 out of a total of 150. We can reduce to . As a percent, is 20%, (B).
Exponents
10.2^{6} =
(A)2^{3}
(B)3^{2}
(C)4^{2}
(D)4^{4}
(E)8^{2}
Expand 2^{6} out and multiply to find that it equals 64. Choice (E) is correct.
Elementary Level
You shouldn’t expect to see exponents or roots on your tests.
Square Roots
11.The square root of 75 falls between what two integers?
(A)5 and 6
(B)6 and 7
(C)7 and 8
(D)8 and 9
(E)9 and 10
If you have trouble with this one, try using the choices and work backward. As we discussed in Fundamentals (Chapter 2), a square root is just the opposite of squaring a number. So let’s square the choices. Then we find that 75 falls between 8^{2} (64) and 9^{2} (81). Choice (D) is correct.
Simple Algebraic Equations
12.11x = 121. What does x = ?
(A) 2
(B) 8
(C)10
(D)11
(E)12
Remember, if you get stuck, use the choices and work backward. Each one provides you with a possible value for x. Start with the middle choice and replace x with it. 11 × 10 = 110. That’s too small. Now we know that not only is (C) incorrect, but also that (A) and (B) are incorrect because they are smaller than (C). The correct choice is (D).
Solve for X
13.If 3y + 17 = 25 — y, then y =
(A)1
(B)2
(C)3
(D)4
(E)5
Just as above, if you get stuck, use the choices. The correct answer is (B).
The Case of the Mysteriously Missing Sign
If there is no operation sign between a number and a variable (letter), the operation is multiplication.
Percent Algebra
14.25% of 30% of what is equal to 18 ?
(A) 1
(B) 36
(C)120
(D)240
(E)540
If you don’t remember the math conversion table, look it up in Fundamentals (Chapter 2). You can also use the choices and work backward. Start with (C) and find out what 25% of 30% of 120 is (9). The correct answer is (D).
Percent
Percent means “out of 100,” and the word of in a word problem tells you to multiply.
Geometry
15.BCDE is a rectangle with a perimeter of 44. If the length of BC is 15, what is the area of BCDE ?
(A)105
(B)15
(C)17
(D)14
(E)It cannot be determined.
Don’t Cut Corners: Estimate Them
Don’t forget to guesstimate on geometry questions! This is a quick way to make sure you’re not making calculation errors.
From the perimeter, we can find that the sides of the rectangle are 7 and 15. So the area is 105, (A).
16.If the perimeter of this polygon is 37, what is the value of x + y ?
(A) 5
(B) 9
(C)10
(D)16
(E)20
The sum of x and y is equal to the perimeter of the polygon minus the lengths of the sides we know. So (C) is correct.
Word Problems
17.Emily is walking to school at a rate of 3 blocks every 14 minutes. When Jeff walks at the same rate as Emily and takes the most direct route to school, he arrives in 56 minutes. How many blocks away from school does Jeff live?
(A) 3
(B) 5
(C) 6
(D) 9
(E)12
This is a proportion question because we have two sets of data that we are comparing. Set up your fractions.
We know that we must do the same thing to the top and bottom of the first fraction to get the second fraction. Notice that the denominator of the second fraction (56) is 4 times the denominator of the first fraction (14). Therefore, the numerator of the second fraction must be 4 times the numerator of the first fraction (3).
So Jeff walks 12 blocks in 56 minutes. This makes (E) the correct answer.
18.Half of the 30 students in Mrs. Whipple’s first-grade class got sick on the bus on the way back from the zoo. Of these students, of them were sick because they ate too much cotton candy. The rest were sick because they sat next to the students who ate too much cotton candy. How many students were sick because they sat next to the wrong student?
(A) 5
(B)10
(C)15
(D)20
(E)25
This is a really gooey fraction problem. Because we’ve seen the word of, we know we have to multiply. First we need to multiply by 30, the number of students in the class. This gives us 15, the number of students who got sick. Now we have another of, so we must multiply the fraction of students who ate too much cotton candy, , by the number of students who got sick, 15. This gives us 10. So then the remainder, those who were unlucky in the seating plan, is 15 — 10 or 5, (A).
19.A piece of rope is 18 inches long. It is cut into 2 unequal pieces. The longer piece is twice as long as the shorter piece. How long, in inches, is the shorter piece?
(A) 2
(B) 6
(C) 9
(D)12
(E)18
Again, if you are stuck for a place to start, go to the choices. Because we are looking for the length of the shorter rope, we can eliminate any choice that gives us a piece equal to or longer than half the rope. That gets rid of (C), (D), and (E). Now if we take one of the pieces, we can subtract it from the total length of the rope to get the length of the longer piece. For (B), if 6 is the length of the shorter piece, we can subtract that from 18 and know that the length of the longer piece must be 12. 12 is double 6, so we have the right answer.
PRACTICE DRILL 1—MULTIPLE CHOICE
When you are done, check your answers in Chapter 9. Don’t forget to time yourself!
Remember to time yourself during this drill!
1.The sum of five consecutive positive integers is 30. What is the square of the largest of the five positive integers?
(A)25
(B)36
(C)49
(D)64
(E)81
2.How many factors does the number 24 have?
(A) 2
(B) 4
(C) 6
(D) 8
(E)10
3.If 12 is a factor of a certain number, what must also be factors of that number?
(A)2 and 6 only
(B)3 and 4 only
(C)12 only
(D)1, 2, 3, 4, and 6
(E)1, 2, 3, 4, 6, and 24
4.What is the smallest number that can be added to the number 1,024 to produce a result divisible by 9 ?
(A)1
(B)2
(C)3
(D)4
(E)6
5.Which of the following is a multiple of 3 ?
(A) 2
(B) 6
(C)10
(D)14
(E)16
6.Which of the following is NOT a multiple of 6 ?
(A)12
(B)18
(C)23
(D)24
(E)42
7.Which of the following is a multiple of both 3 and 5 ?
(A)10
(B)20
(C)25
(D)45
(E)50
8.A company’s profit was $75,000 in 1972. In 1992, its profit was $450,000. The profit in 1992 was how many times as great as the profit in 1972 ?
(A) 2
(B) 4
(C) 6
(D)10
(E)60
9.Joanna owns one-third of the pieces of furniture in the apartment she shares with her friends. If there are 12 pieces of furniture in the apartment, how many pieces does Joanna own?
(A) 2
(B) 4
(C) 6
(D) 8
(E)12
10.A tank of oil is one-third full. When full, the tank holds 90 gallons. How many gallons of oil are in the tank now?
(A)10
(B)20
(C)30
(D)40
(E)50
11.Tigger the Cat sleeps three-fourths of every day. In a four-day period, he sleeps the equivalent of how many full days?
(A)
(B)
(C)1
(D)3
(E)4
12.Which of the following has the greatest value?
(A) +
(B) −
(C) ÷
(D) ×
(E) × 2
13. + + + + + =
(A)
(B) 1
(C) 6
(D) 3
(E)12
14.The product of 0.34 and 1,000 is approximately
(A) 3.50
(B) 35
(C) 65
(D)350
(E)650
15.2.398 =
(A)2 × × ×
(B)2 + + +
(C)2 + + +
(D) + +
(E)None of the above
Stop. Check your time for this drill:
Don’t forget to check your answers in Chapter 9.
HOW DID YOU DO?
That was a good sample of some of the kinds of questions you’ll see on the SSAT. Now there are a few things to check other than your answers. Remember that taking the test involves much more than just getting answers right. It’s also about guessing wisely, using your time well, and figuring out where you’re likely to make mistakes. Once you’ve checked to see what you’ve gotten right and wrong, you should then consider the points that follow to improve your score.
Time and Pacing
How long did it take you to do the 15 questions? 15 minutes? It’s okay if you went a minute or two over. However, if you finished very quickly (in fewer than 10 minutes) or slowly (more than 20 minutes), look at any problems that may have affected your speed. Which questions seriously slowed you down? Did you answer some quickly but not correctly? Your answers to these questions will help you plan which and how many questions to answer on the SSAT.
Question Recognition and Selection
Did you use your time wisely? Did you do the questions in an order that worked well for you? Which kinds of questions were the hardest for you? Remember that every question on the SSAT, whether you know the answer right away or find the question confusing, is worth one point, and that you don’t have to answer all the questions to get a good score. In fact, because of the guessing penalty, skipping questions can actually raise your score. So depending on your personal speed, you should concentrate most on getting as many questions you find easy or sort-of easy right as possible, and worry about harder problems later. Keep in mind that in Math sections, the questions generally go from easiest to hardest throughout. Getting the questions you know you know the answers to right takes time, but you know you can solve them—so give yourself that time!
POE and Guessing
Did you actively look for wrong answers to eliminate, instead of just looking for the right answer? (You should.) Did you physically cross off wrong answers to keep track of your POE? Was there a pattern to when guessing worked (more often when you could eliminate one wrong answer, and less often when you picked simpler-looking over harder-looking answers)?
Be Careful
Did you work problems out on a separate piece of paper? Did you move too quickly or skip steps on problems you found easier? Did you always double-check what the question was asking? Often students miss questions that they know how to do! Why? It’s simple—they work out problems in their heads or don’t read carefully. Work out every SSAT math problem on the page. Consider it a double-check because your handwritten notes confirm what you’ve worked out in your head.
PRACTICE DRILL 2—MULTIPLE CHOICE—UPPER LEVEL ONLY
While doing the next drill, keep in mind the general test-taking techniques we’ve talked about: guessing, POE, order of difficulty, pacing, and working on the page and not in your head. At the end of the section, check your answers. But don’t stop there: Investigate the drill thoroughly to see how and why you got your answers wrong. And check your time. You should be spending about one minute per question on this drill. When you are done, check your answers in Chapter 9. Don’t forget to time yourself!
Remember to time yourself during this drill!
1.How many numbers between 1 and 100 are multiples of both 2 and 7 ?
(A) 6
(B) 7
(C) 8
(D) 9
(E)10
2.What is the smallest multiple of 7 that is greater than 50 ?
(A) 7
(B)49
(C)51
(D)56
(E)63
3.2^{3} × 2^{3} × 2^{2} =
(A)64
(B) 2^{8}
(C) 2^{10}
(D) 2^{16}
(E) 2^{18}
4.For what integer value of m does 2m + 4 = m^{3} ?
(A)1
(B)2
(C)3
(D)4
(E)5
5.One-fifth of the students in a class chose recycling as the topic for their science projects. If four students chose recycling, how many students are in the class?
(A) 4
(B)10
(C)16
(D)20
(E)24
6.If 6x — 4 = 38, then x + 10 =
(A) 7
(B)10
(C)16
(D)17
(E)19
7.If 3x — 6 = 21, then what is x ÷ 9 ?
(A)0
(B)1
(C)3
(D)6
(E)9
8.Only one-fifth of the chairs in a classroom are in working order. If three additional working chairs are brought in, there are 19 working seats available. How many chairs were originally in the room?
(A)16
(B)19
(C)22
(D)80
(E)95
9.If a harvest yielded 60 bushels of corn, 20 bushels of wheat, and 40 bushels of soybeans, what percent of the total harvest was corn?
(A)50%
(B)40%
(C)33%
(D)30%
(E)25%
10.At a local store, an item that usually sells for $45 is currently on sale for $30. By what percent is that item discounted?
(A)10%
(B)25%
(C)33%
(D)50%
(E)66%
11.Which of the following is most nearly 35% of $19.95 ?
(A) $3.50
(B) $5.75
(C) $7.00
(D) $9.95
(E)$13.50
12.Of the 50 hotels in the Hilltop Hotels chain, 5 have indoor swimming pools and 15 have outdoor swimming pools. What percent of all Hilltop Hotels have either an indoor or an outdoor swimming pool?
(A)40%
(B)30%
(C)20%
(D)15%
(E) 5%
13.For what price item does 40% off equal a $20 discount?
(A) $50.00
(B)$100.00
(C)$400.00
(D)$800.00
(E)None of the above
14.A pair of shoes is offered on a special blowout sale. The original price of the shoes is reduced from $50 to $20. What is the percent change in the price of the shoes?
(A)60%
(B)50%
(C)40%
(D)25%
(E)20%
15.Lisa buys a silk dress regularly priced at $60, a cotton sweater regularly priced at $40, and four pairs of socks regularly priced at $5 each. If the dress and the socks are on sale for 20% off the regular price and the sweater is on sale for 10% off the regular price, what is the total amount of her purchase?
(A) $90.00
(B) $96.00
(C)$100.00
(D)$102.00
(E)$108.00
16.Thirty percent of $17.95 is closest to
(A) $2.00
(B) $3.00
(C) $6.00
(D) $9.00
(E)$12.00
17.Fifty percent of the 20 students in Mrs. Schweizer’s third-grade class are boys. If 90 percent of these boys ride the bus to school, which of the following is the number of boys in Mrs. Schweizer’s class who ride the bus to school?
(A) 9
(B)10
(C)12
(D)16
(E)18
18.On a test with 25 questions, Marc scored an 88 percent. How many questions did Marc answer correctly?
(A)22
(B)16
(C)12
(D) 4
(E) 3
19.Four friends each pay $5 for a pizza every Friday night. If they were to start inviting a fifth friend to come with them and still bought the same pizza, how much would each person then have to pay?
(A) $1
(B) $4
(C) $5
(D)$20
(E)$25
20.A stop sign has 8 equal sides of length 4. What is its perimeter?
(A) 4
(B) 8
(C)12
(D)32
(E)It cannot be determined from the information given.
21.If the perimeter of a square is 56, what is the length of each side?
(A) 4
(B) 7
(C) 14
(D) 28
(E)112
22.The perimeter of a square with a side of length 4 is how much less than the perimeter of a rectangle with sides of length 4 and width 6 ?
(A)0
(B)2
(C)4
(D)6
(E)8
23.What is the perimeter of an equilateral triangle, one side of which measures 4 inches?
(A)12 inches
(B)8 inches
(C)6 inches
(D)4 inches
(E)It cannot be determined from the information given.
24.x =
(A) 8
(B) 30
(C) 50
(D) 65
(E)180
25.If b = 45, then v^{2} =
(A)32
(B)25
(C)16
(D) 5
(E)It cannot be determined from the information given.
26.One-half of the difference between the number of degrees in a square and the number of degrees in a triangle is
(A) 45
(B) 90
(C)180
(D)240
(E)360
27.If the area of a square is equal to its perimeter, what is the length of one side?
(A) 1
(B) 2
(C) 4
(D) 8
(E)10
28.The area of a rectangle with width 4 and length 3 is equal to the area of a triangle with a base of 6 and a height of
(A) 1
(B) 2
(C) 3
(D) 4
(E)12
29.Two cardboard boxes have equal volume. The dimensions of one box are 3 × 4 × 10. If the length of the other box is 6 and the width is 4, what is the height of the second box?
(A) 2
(B) 5
(C)10
(D)12
(E)24
30.If the area of a square is 64p^{2}, what is the length of one side of the square?
(A)64p^{2}
(B)64p
(C) 8p^{2}
(D) 8p
(E) 8
31.If AB = 10 and AC = 15, what is the perimeter of the figure above?
(A)25
(B)35
(C)40
(D)50
(E)It cannot be determined from the information given.
32.If ABCD, shown above, is a rectangle, what is the value of w + x + y + z ?
(A) 90°
(B)150°
(C)180°
(D)190°
(E)210°
33.What is the area of the figure above if all the angles shown are right angles?
(A)38
(B)42
(C)50
(D)88
(E)96
34.In the figure above, the length of side AB of square ABCD is equal to 4 and the circle has a radius of 2. What is the area of the shaded region?
(A)4 — π
(B)16 — 4π
(C)8 + 4π
(D)4π
(E)8π
35.The distance between points A and B in the coordinate plane above is
(A) 5
(B) 6
(C) 8
(D) 9
(E)10
Stop. Check your time for this drill:
Don’t forget to check your answers in Chapter 9.
Ratios
A ratio is like a recipe. It tells you how much of each ingredient goes into a mixture.
For example:
To make punch, mix two parts grape juice with three parts orange juice.
This ratio tells you that for every two units of grape juice, you will need to add three units of orange juice. It doesn’t matter what the units are; if you were working with ounces, you would mix two ounces of grape juice with three ounces of orange juice to get five ounces of punch. If you were working with gallons, you would mix two gallons of grape juice with three gallons of orange juice. How much punch would you have? Five gallons.
To work through a ratio question, first you need to organize the information you are given. Do this using the Ratio Box.
In a club with 35 members, the ratio of boys to girls is 3:2. To complete your Ratio Box, fill in the ratio at the top and the “real value” at the bottom.
Then look for a “magic number” that you can multiply by the ratio total to get the real value total. In this case, the magic number is 7. That’s all there is to it!
PRACTICE DRILL 3—RATIOS
Remember to time yourself during this drill!
1.In a jar of lollipops, the ratio of red lollipops to blue lollipops is 3:5. If only red lollipops and blue lollipops are in the jar and if the total number of lollipops in the jar is 56, how many blue lollipops are in the jar?
(A)35
(B)28
(C)21
(D) 8
(E) 5
2.At Jed’s Country Hotel, there are three types of rooms: singles, doubles, and triples. If the ratio of singles to doubles to triples is 3:4:5, and the total number of rooms is 36, how many doubles are there?
(A) 4
(B) 9
(C)12
(D)24
(E)36
3.Matt’s Oak Superstore has exactly three times as many large oak desks as small oak desks in its inventory. If the store sells only these two types of desks, which could be the total number of desks in stock?
(A)10
(B)13
(C)16
(D)18
(E)25
4.In Janice’s tennis club, 8 of the 12 players are right-handed. What is the ratio of right-handed to left-handed players in Janice’s club?
(A)1:2
(B)1:6
(C)2:1
(D)2:3
(E)3:4
5.One-half of the 400 students at Booth Junior High School are girls. Of the girls at the school, the ratio of those who ride a school bus to those who walk is 7:3. What is the total number of girls who walk to school?
(A) 10
(B) 30
(C) 60
(D)120
(E)140
6.A pet goat eats 2 pounds of goat food and 1 pound of grass each day. When the goat has eaten a total of 15 pounds, how many pounds of grass will it have eaten?
(A) 3
(B) 4
(C) 5
(D)15
(E)30
Stop. Check your time for this drill:
Don’t forget to check your answers in Chapter 9.
Averages
There are three parts to every average problem: total, number, and average. Most SSAT problems will give you two of the three pieces and ask you to find the third. To help organize the information you are given, use the Average Pie.
The Average Pie organizes all of your information visually. It makes it easier to see all of the relationships between the pieces of the pie.
· TOTAL = (# of items) × (Average)
· # of items =
· Average =
For example, if your friend went bowling and bowled three games, scoring 71, 90, and 100, here’s how you would compute her average score using the Average Pie.
To find the average, you would simply write a fraction that represents , in this case .
The math becomes simple. 261 ÷ 3 = 87. Your friend bowled an average of 87.
Practice working with the Average Pie by using it to solve the following problems.
Average
When you see the word average, draw an Average Pie.
PRACTICE DRILL 4—AVERAGES
Remember to time yourself during this drill!
1.The average of 3 numbers is 18. What is 2 times the sum of the 3 numbers?
(A)108
(B) 54
(C) 36
(D) 18
(E) 6
2.If Set M contains 4 positive integers whose average is 7, then what is the largest number that Set M could contain?
(A) 6
(B) 7
(C)18
(D)25
(E)28
3.An art club of 4 boys and 5 girls makes craft projects. If the boys average 2 projects each and the girls average 3 projects each, what is the total number of projects produced by the club?
(A) 14
(B) 23
(C) 26
(D) 54
(E)100
4.If a class of 6 students has an average grade of 72 before a seventh student joins the class, then what must the seventh student’s grade be to raise the class average to 76 ?
(A)100
(B) 92
(C) 88
(D) 80
(E) 76
5.Catherine scores an 84, 85, and 88 on her first three exams. What must she score on her fourth exam to raise her average to an 89 ?
(A)99
(B)97
(C)93
(D)91
(E)89
Don’t forget to check your answers in Chapter 9.
PERCENT CHANGE—UPPER LEVEL ONLY
There is one special kind of percent question that shows up on the SSAT: percent change. This type of question asks you to find what percent something has increased or decreased. Instead of taking the part and dividing it by the whole, you will take the difference between the two numbers and divide it by the original number. Then, to turn the fraction to a percent, divide the numerator by the denominator and multiply by 100.
For example:
The number of people who watched Empire last year was 3,600,000. This year, only 3,000,000 are watching the show. By approximately what percent has the audience decreased?
(The difference is 3,600,000 — 3,000,000.)
The fraction reduces to , and as a percent is 17%.
PRACTICE DRILL 5—PERCENT CHANGE
Remember to time yourself during this drill!
1.During a severe winter in Ontario, the temperature dropped suddenly to 10 degrees below zero. If the temperature in Ontario before this cold spell occurred was 10 degrees above zero, by what percent did the temperature drop?
(A) 25%
(B) 50%
(C)100%
(D)150%
(E)200%
% change = × 100
2.Fatty’s Burger wants to attract more customers by increasing the size of its patties. From now on Fatty’s patties are going to be 4 ounces larger than before. If the size of its new patty is 16 ounces, by approximately what percent has the patty increased?
(A)25%
(B)27%
(C)33%
(D)75%
(E)80%
Stop. Check your time for this drill:
Don’t forget to check your answers in Chapter 9.
Plugging In
The SSAT will often ask you questions about real-life situations in which the numbers have been replaced with variables. One of the easiest ways to tackle these questions is with a powerful technique called Plugging In.
Mark is two inches taller than John, who is four inches shorter than Terry. If t represents Terry’s height in inches, then in terms of t, an expression for Mark’s height is
(A)t + 6
(B)t + 4
(C)t + 2
(D)t
(E)t — 2
The problem with this question is that we’re not used to thinking of people’s heights in terms of variables. Have you ever met someone who was t inches tall?
Take the Algebra Away, and Arithmetic Is All That’s Left
When you Plug In for variables, you won’t need to write equations and won’t have to solve algebra problems. Doing simple arithmetic is always easier than doing algebra.
Whenever you see variables used in the question and in the choices, just Plug In a number to replace the variable.
1. Choose a number for t.
2. Using that number, figure out Mark’s and John’s heights.
3. Put a box around Mark’s height because that’s what the question asked you for.
4. Plug your number for t into the choices and choose the one that gives you the number you found for Mark’s height.
Here’s How It Works
For Terry’s height, let’s pick 60 inches. This means that t = 60.
Remember, there is no right or wrong number to pick. 50 would work just as well.
But given that Terry is 60 inches tall, now we can figure out that, because John is four inches shorter than Terry, John’s height must be (60 — 4), or 56 inches.
The other piece of information we learn from the problem is that Mark is two inches taller than John. If John’s height is 56 inches, that means Mark must be 58 inches tall.
Here’s what we’ve got:
Terry |
60 inches = t |
John |
56 inches |
Mark |
58 inches |
Now, the question asks for Mark’s height, which is 58 inches. The last step is to go through the choices substituting 60 for t and choose the one that equals 58.
(A) t + 6 |
60 + 6 = 66 |
ELIMINATE |
(B) t + 4 |
60 + 4 = 64 |
ELIMINATE |
(C) t + 2 |
60 + 2 = 62 |
ELIMINATE |
(D) t |
60 |
ELIMINATE |
(E) t — 2 |
60 — 2 = 58 |
PICK THIS ONE! |
After reading this explanation, you may be tempted to say that Plugging In takes too long. Don’t be fooled. The method itself is often faster and (more importantly) more accurate than regular algebra. Try it out. Practice. As you become more comfortable with Plugging In, you’ll get even quicker and better results. You still need to know how to do algebra, but if you do only algebra, you may have difficulty improving your SSAT score. Plugging In gives you a way to break through whenever you are stuck. You’ll find that having more than one way to solve SSAT math problems puts you at a real advantage.
PRACTICE DRILL 6—PLUGGING IN
Elementary Level
While some of these problems may seem harder than what you will encounter, Plugging In and Plugging In The Answers (the next section) are great strategies. Be sure you understand how to use these strategies on your test.
1.At a charity fund-raiser, 200 people each donated x dollars. In terms of x, what was the total number of dollars donated?
(A)
(B)200
(C)
(D)200 + x
(E)200x
2.If 10 magazines cost d dollars, then in terms of d, how many magazines can be purchased for 3 dollars?
(A)
(B)30d
(C)
(D)
(E)
3.The zoo has four times as many monkeys as lions. There are four more lions than there are zebras at the zoo. If z represents the number of zebras in the zoo, then in terms of z, how many monkeys are there in the zoo?
(A)z + 4
(B)z + 8
(C)4z
(D)4z + 16
(E)4z + 4
Occasionally, you may run into a Plugging In question that doesn’t contain variables. These questions usually ask about a percentage or a fraction of some unknown number or price. This is the one time that you should Plug In even when you don’t see variables in the answer!
Also, be sure you Plug In good numbers. Good doesn’t mean right because there’s no such thing as a right or wrong number to Plug In. A good number is one that makes the problem easier to work with. If a question asks about minutes and hours, try Plugging In 30 or 60, not 128. Also, whenever you see the word percent, Plug In 100!
4.The price of a suit is reduced by half, and then the resulting price is reduced by 10%. The final price is what percent of the original price?
(A) 5%
(B)10%
(C)25%
(D)40%
(E)45%
5.On Wednesday, Miguel ate one-fourth of a pumpkin pie. On Thursday, he ate one-half of what was left of the pie. What fraction of the entire pie did Miguel eat on Wednesday and Thursday?
(A)
(B)
(C)
(D)
(E)
6.If p pieces of candy costs c cents, then in terms of p and c, 10 pieces of candy will cost
(A) cents.
(B) cents.
(C)10pc cents.
(D) cents.
(E)10 + p + c cents.
7.If J is an odd integer, which of the following must be true?
(A)(J ÷ 3) > 1
(B)(J — 2) is a positive integer.
(C)2 × J is an even integer.
(D)J^{2} > J
(E)J > 0
8.If m is an even integer, n is an odd integer, and p is the product of m and n, which of the following is always true?
(A)p is a fraction.
(B)p is an odd integer.
(C)p is divisible by 2.
(D)p is between m and n.
(E)p is greater than zero.
Stop. Check your time for this drill:
Don’t forget to check your answers in Chapter 9.
Plugging In The Answers (PITA)
Plugging In The Answers is similar to Plugging In. When you have variables in the choices, you Plug In. When you have numbers in the choices, you should generally Plug In the Answers. The only time this may get tricky is when you have a question that asks for a percent or fraction of some unknown number.
Plugging In the Answers works because on a multiple-choice test, the right answer is always one of the choices. On this type of question, you can’t Plug In any number you want because only one number will work. Instead, you can Plug In numbers from the choices, one of which must be correct. Here’s an example.
Nicole baked a batch of cookies. She gave half to her friend Lisa and six to her mother. If she now has eight cookies left, how many did Nicole bake originally?
(A) 8
(B)12
(C)20
(D)28
(E)32
See what we mean? It would be hard to just start making up numbers of cookies and hope that eventually you guessed correctly. However, the number of cookies that Nicole baked originally must be either 8, 12, 20, 28, or 32 (the five choices). So pick one—always start with (C)—and then work backward to determine whether you have the right choice.
Let’s start with (C): Nicole baked 20 cookies. Now work through the events listed in the question.
She had 20 cookies—from (C)—and she gave half to Lisa. That leaves Nicole with 10 cookies.
What next? She gives 6 to her mom. Now she’s got 4 left.
Keep going. The problem says that Nicole now has 8 cookies left. But if she started with 20—(C)—she would only have 4 left. So is (C) the right answer? No.
No problem. Choose another choice and try again. Be smart about which choice you pick. When we used the number in (C), Nicole ended up with fewer cookies than we wanted her to have, didn’t she? So the right answer must be a number larger than 20, the number we took from (C).
The good news is that the choices in most Plugging In The Answers questions go in order, so you can choose the next larger or smaller number—you just pick either (B) or (D), depending on which direction you’ve decided to go.
Back to Nicole and her cookies. We need a number larger than 20. So let’s go to (D)—28.
Nicole started out with 28 cookies. The first thing she did was give half, or 14, to Lisa. That left Nicole with 14 cookies.
Then she gave 6 cookies to her mother. 14 — 6 = 8. Nicole has 8 cookies left over. Keep going with the question. It says, “If she now has eight cookies left…” She has eight cookies left and, voilà—she’s supposed to have 8 cookies left.
What does this mean? It means you’ve got the right answer! Pick (D) and move on.
If (D) had not worked, and you were still certain that you needed a number larger than (C), you also would be finished. Since you started with the middle, (C), which didn’t work, and then you tried the next larger choice, (D), which didn’t work either, you could pick the only choice bigger than (C) that was left—in this case (E)—and be done.
This diagram helps illustrate the way you should move through the choices.
To wrap up, Plugging In The Answers should always go the following way:
1. Start with (C). This number is now what you are working with.
2. Work the problem. Go through the problem with that number, using information to help you determine if it is the correct answer.
3. If (C) doesn’t work, try another answer. Remember to think logically about which choice you should check next.
4. Once you find the correct answer, STOP.
PRACTICE DRILL 7—PLUGGING IN THE ANSWERS
Remember to time yourself during this drill!
1.Ted can read 60 pages per hour. Naomi can read 45 pages per hour. If both Ted and Naomi read at the same time, how many minutes will it take them to read a total of 210 pages?
(A) 36
(B) 72
(C)120
(D)145
(E)180
2.If the sum of y and y + 1 is greater than 18, which of the following is one possible value for y ?
(A)—10
(B) —8
(C) 2
(D) 8
(E) 10
3.Kenny is 5 years older than Greg. In 5 years, Kenny will be twice as old as Greg is now. How old is Kenny now?
(A) 5
(B)10
(C)15
(D)25
(E)35
4.Three people—Paul, Sara, and John—want to put their money together to buy a $90 radio. If Sara agrees to pay twice as much as John, and Paul agrees to pay three times as much as Sara, how much must Sara pay?
(A)$10
(B)$20
(C)$30
(D)$45
(E)$65
5.Four less than a certain number is two-thirds of that number. What is the number?
(A) 1
(B) 6
(C) 8
(D)12
(E)16
Stop. Check your time for this drill:
Don’t forget to check your answers in Chapter 9.
GEOMETRY
Guesstimating: A Second Look
Guesstimating worked well back in the introduction when we were just using it to estimate or “ballpark” the size of a number, but geometry problems are undoubtedly the best place to guesstimate whenever you can.
Let’s try the next problem. Remember, unless a particular question tells you otherwise, you can safely assume that figures are drawn to scale.
A circle is inscribed in square PQRS. What is the area of the shaded region?
(A)16 — 6π
(B)16 — 4π
(C)16 — 3π
(D)16 — 2π
(E)16π
Elementary Level
This question is harder than what you will encounter, but it’s a good idea to learn how guesstimating can help you!
Wow, a circle inscribed in a square—that sounds tough!
It isn’t. Look at the picture. What fraction of the square looks like it is shaded? Half? Three-quarters? Less than half? In fact, about one-quarter of the area of the square is shaded. You’ve just done most of the work necessary to solve this problem.
Try these values when guesstimating:
π ≈ 3+
= 1.4
= 1.7
Now, let’s just do a little math. The length of one side of the square is 4, so the area of the square is 4 × 4 or 16.
So the area of the square is 16, and we said that the shaded region was about one-fourth of the square. One-fourth of 16 is 4, right? So we’re looking for a choice that equals about 4. Let’s look at the choices.
(A)16 — 6π
(B)16 — 4π
(C)16 — 3π
(D)16 — 2π
(E)16π
This becomes a little complicated because the answers include π. For the purposes of guesstimating, and in fact for almost any purpose on the SSAT, you should just remember that π is a little more than 3.
Let’s look back at those answers.
(A) 16 — 6π |
is roughly equal to |
16 — (6 × 3) = —2 |
(B) 16 — 4π |
is roughly equal to |
16 — (4 × 3) = 4 |
(C) 16 — 3π |
is roughly equal to |
16 — (3 × 3) = 7 |
(D) 16 — 2π |
is roughly equal to |
16 — (2 × 3) = 10 |
(E) 16π |
is roughly equal to |
(16 × 3) = 48 |
Now let’s think about what these answers mean.
Choice (A) is geometrically impossible. A figure cannot have a negative area. Eliminate it.
Choice (B) means that the shaded region has an area of about 4. Sounds pretty good.
Choice (C) means that the shaded region has an area of about 7. The area of the entire square was 16, so that would mean that the shaded region was almost half the square. Possible, but doubtful.
Choice (D) means that the shaded region has an area of about 10. That’s more than half the square and in fact, almost three-quarters of the entire square. No way; cross it out.
Finally, (E) means that the shaded region has an area of about 48. What? The whole square had an area of 16. Is the shaded region three times as big as the square itself? Not a chance. Eliminate (E).
At this point you are left with only (B), which we feel pretty good about, and (C), which seems a little large. What should you do?
Pick (B) and pat yourself on the back because you chose the right answer without doing a lot of unnecessary work. Also, remember how useful it was to guesstimate and make sure you do it whenever you see a geometry problem, unless the problem tells you that the figure is not drawn to scale!
Weird Shapes
Whenever the test presents you with a geometric figure that is not a square, rectangle, circle, or triangle, draw a line or lines to divide that figure into the shapes that you do know. Then you can easily work with shapes you know all about.
Shaded Regions—Middle and Upper Levels Only
Sometimes geometry questions show you one figure inscribed in another and then ask you to find the area of a shaded region inside the larger figure and outside the smaller figure (like the problem at the beginning of this section). To find the areas of these shaded regions, find the area of the outside figure and then subtract from that the area of the figure inside. The difference is what you need.
ABCE is a rectangle with a length of 10 and width of 6. Points F and D are the midpoints of AE and EC, respectively. What is the area of the shaded region?
(A)25.5
(B)30
(C)45
(D)52.5
(E)It cannot be determined from the information given.
The first step is to find the area of the rectangle. If you multiply the length by the width, you’ll find the area is 60. Now we find the area of the triangle that we are removing from the rectangle. Because the height and base of the triangle are parts of the sides of the rectangle, and points D and F are half the length and width of the rectangle, we know that the height of the triangle is half the rectangle’s width, or 3, and the base of the triangle is half the rectangle’s length, or 5. Using the formula for area of a triangle, we find the area of the triangle is 7.5. Now we subtract the area of the triangle from the area of the rectangle. 60 — 7.5 = 52.5. The correct choice is (D). Be careful not to choose (E) just because the problem looks tricky!
Functions—Middle and Upper Levels Only
In a function problem, an arithmetic operation is defined and then you are asked to perform it on a number. A function is just a set of instructions written in a strange way.
# x = 3x(x + 1)
On the left there is usually a variable with a strange symbol next to or around it.
In the middle is an equals sign.
On the right are the instructions. These tell you what to do with the variable.
# x = 3x(x + 1) |
What does # 5 equal? |
# 5 = (3 × 5)(5 + 1) |
Just replace each x with a 5! |
Here, the function (indicated by the # sign) simply tells you to substitute a 5 wherever there was an x in the original set of instructions. Functions look confusing because of the strange symbols, but once you know what to do with them, they are just like manipulating an equation.
Sometimes more than one question will refer to the same function. The following drill, for example, contains two questions about one function. In cases such as this, the first question tends to be easier than the second.
PRACTICE DRILL 8—FUNCTIONS
Remember to time yourself during this drill!
Questions 1 and 2 refer to the following definition.
For all real numbers n, $n = 10n — 10.
1.$7 =
(A)70
(B)60
(C)17
(D) 7
(E) 0
2.If $n = 120, then n =
(A) 11
(B) 12
(C) 13
(D)120
(E)130
Questions 3—5 refer to the following definition.
For all real numbers d and y, d ¿ y = (d × y) — (d + y).
[Example: 3 ¿ 2 = (3 × 2) — (3 + 2) = 6 — 5 = 1]
3.10 ¿ 2 =
(A)20
(B)16
(C)12
(D) 8
(E) 4
4.If K (4 ¿ 3) = 30, then K =
(A)3
(B)4
(C)5
(D)6
(E)7
5.(2 ¿ 4) × (3 ¿ 6) =
(A)(9 ¿ 3) + 3
(B)(6 ¿ 4) + 1
(C)(5 ¿ 3) + 4
(D)(8 ¿ 4) + 2
(E)(9 ¿ 4) + 3
Stop. Check your time for this drill:
Don’t forget to check your answers in Chapter 9.
Charts and Graphs
Charts
Chart questions are simple, but you must be careful. Follow these three steps and you’ll be well on the way to mastering any chart question.
1. Read any text that accompanies the chart. It is important to know what the chart is showing and what scale the numbers are on.
2. Read the question.
3. Refer to the chart and find the specific information you need.
Don’t Be in Too Big of a Hurry
When working with charts and graphs, make sure you take a moment to look at the chart or graph, figure out what it tells you, and then go to the questions.
If there is more than one question about a single chart, the later questions will tend to be more difficult than the earlier ones. Be careful!
Here is a sample chart.
Club Membership by State, 2010 and 2011 |
||
State |
2010 |
2011 |
California |
300 |
500 |
Florida |
225 |
250 |
Illinois |
200 |
180 |
Massachusetts |
150 |
300 |
Michigan |
150 |
200 |
New Jersey |
200 |
250 |
New York |
400 |
600 |
Texas |
50 |
100 |
There are many different questions that you can answer based on the information in this chart. For instance:
What is the difference between the number of members who came from New York in 2010 and the number of members who came from Illinois in 2011 ?
This question asks you to look up two simple pieces of information and then do a tiny bit of math.
First, the number of members who came from New York in 2010 was 400.
Second, the number of members who came from Illinois in 2011 was 180.
Finally, look back at the question. It asks you to find the difference between these numbers. 400 — 180 = 220. Done.
The increase in the number of members from New Jersey from 2010 to 2011 was what percent of the total number of members in New Jersey in 2010 ?
You should definitely know how to do this one! Do you remember how to translate percentage questions? If not, go back to Fundamental Math Skills (Chapter 2).
In 2010, there were 200 club members from New Jersey. In 2011, there were 250 members from New Jersey. That represents an increase of 50 members. To determine what percent that is of the total amount in 2010, you will need to ask yourself, “50 (the increase) is what percent of 200 (the number of members in 2010)?”
Translated, this becomes:
With a little bit of simple manipulation, this equation becomes:
50 = 2g
and
25 = g
So from 2010 to 2011, there was a 25% increase in the number of members from New Jersey. Good work!
Which state had as many club members in 2011 as a combination of Illinois, Massachusetts, and Michigan had in 2010 ?
First, take a second to look up the number of members who came from Illinois, Massachusetts, and Michigan in 2010 and add them together.
200 + 150 + 150 = 500
Which state had 500 members in 2011? California. That’s all there is to it!
Graphs
Some questions will ask you to interpret a graph. You should be familiar with both pie and bar graphs. These graphs are generally drawn to scale (meaning that the graphs give an accurate visual impression of the information) so you can always guess based on the figure if you need to.
The way to approach a graph question is exactly the same as the way to approach a chart question. Follow the same three steps.
1. Read any text that accompanies the graph. It is important to know what the graph is showing and what scale the numbers are on.
2. Read the question.
3. Refer back to the graph and find the specific information you need.
This is how it works.
The graph in Figure 1 shows Emily’s clothing expenditures for the month of October. On which type of clothing did she spend the most money?
(A)Shoes
(B)Shirts
(C)Socks
(D)Hats
(E)Pants
This one should be simple. You can look at the pieces of the pie and identify the largest, or you can look at the amounts shown in the graph and choose the largest one. Either way, the answer is (A) because Emily spent more money on shoes than on any other clothing items in October.
Emily spent half of her clothing money on which two items?
(A)Shoes and pants
(B)Shoes and shirts
(C)Hats and socks
(D)Socks and shirts
(E)Shirts and pants
Again, you can find the answer to this question two different ways. You can look for which two items together make up half the chart, or you can add up the total amount of money Emily spent ($240) and then figure out which two items made up half (or $120) of that amount. Either way is just fine, and either way the right answer is (B), shoes and shirts.
PRACTICE DRILL 9—CHARTS AND GRAPHS
Remember to time yourself during this drill!
Questions 1-3 refer to the following summary of energy costs by district.
District |
1990 |
1991 |
A |
400 |
600 |
B |
500 |
700 |
C |
200 |
350 |
D |
100 |
150 |
E |
600 |
800 |
(All numbers are in thousands of dollars.)
1.In 1991, which district spent twice as much on energy as District A spent in 1990 ?
(A)A
(B)B
(C)C
(D)D
(E)E
2.Which district spent the most on energy in 1990 and 1991 combined?
(A)A
(B)B
(C)D
(D)E
(E)It cannot be determined from the information given.
3.The total increase in energy expenditure in these districts, from 1990 to 1991, is how many dollars?
(A) $800
(B) $1,800
(C) $2,400
(D) $2,600
(E)$800,000
Questions 4 and 5 refer to Figure 2, which shows the number of compact discs owned by five students.
4.Carl owns as many CDs as which two other students combined?
(A)Abe and Ben
(B)Ben and Dave
(C)Abe and Ed
(D)Abe and Dave
(E)Ben and Ed
5.Which one student owns one-fourth of the CDs accounted for in Figure 2 ?
(A)Abe
(B)Ben
(C)Carl
(D)Dave
(E)Ed
Questions 6-8 refer to Matt’s weekly time card, shown below.
6.If Matt’s hourly salary is $6, what were his earnings for the week?
(A) $6
(B)$14
(C)$21
(D)$54
(E)$84
7.What is the average number of hours Matt worked on the days he worked during this particular week?
(A) 3
(B) 3.5
(C) 4
(D) 7
(E)14
8.The hours that Matt worked on Monday accounted for what percent of the total number of hours he worked during this week?
(A) 3.5
(B)20
(C)25
(D)35
(E)50
Stop. Check your time for this drill:
Don’t forget to check your answers in Chapter 9.
PRACTICE DRILL 10—MIDDLE AND UPPER LEVELS ONLY
When you are done, check your answers in Chapter 9. Don’t forget to time yourself!
Middle Level
Some of the questions here are harder than what you will see on your test. Still, give them all a try. The skills you have learned should help you do well on most of the questions.
Remember to time yourself during this drill!
1.If p is an odd integer, which of the following must be an odd integer?
(A)p^{2} + 3
(B)2p + 1
(C)p ÷ 3
(D)p — 3
(E)2(p^{2})
2.If m is the sum of two positive even integers, which of the following CANNOT be true?
(A)m < 5
(B)3m is odd
(C)m is even
(D)m^{3} is even
(E)m ÷ 2 is even
3.The product of b and a^{2} can be written as
(A)(ab)^{2}
(B)
(C)2a × b
(D)
(E)
4.Damon has twice as many records as Graham, who has one-fourth as many records as Alex. If Damon has d records, then in terms of d, how many records do Alex and Graham have together?
(A)
(B)
(C)
(D)
(E)2d
5.x^{a} = (x^{3})^{3}
What is the value of a × b ?
(A)17
(B)30
(C)48
(D)45
(E)72
6.One six-foot Italian hero serves either 12 children or 8 adults. Approximately how many sandwiches do you need to feed a party of 250, 75 of whom are children?
(A)21
(B)24
(C)29
(D)30
(E)32
7.Liam and Noel are traveling from New York City to Dallas. If they traveled of the distance on Monday and of the distance that remained on Tuesday, what percentage of the trip do they have left to travel?
(A)25%
(B)30%
(C)40%
(D)50%
(E)80%
8. of a bag of potato chips contains 10 grams of fat. Approximately how many grams of fat are in of that same bag of chips?
(A)5.5
(B)6.5
(C)7.5
(D)8.5
(E)9.5
9.Students in Mr. Greenwood’s history class are collecting donations for a school charity drive. If the total number of students in the class, x, donated an average of y dollars each, in terms of x and y, how much money was collected for the drive?
(A)
(B)xy
(C)
(D)
(E)2xy
10.If e + f is divisible by 17, which of the following must also be divisible by 17 ?
(A)(e × f) — 17
(B)e + (f × 17)
(C)(e × 17) + f
(D)(e + f ) / 17
(E)(e × 3) + (f × 3)
11.Joe wants to find the mean number of pages in the books he has read this month. The books were 200, 220, and 260 pages long. He read the 200 page book twice, so it will be counted twice in the mean. If he reads one more book, what is the fewest number of pages it can have to make the mean no less than 230 ?
(A)268
(B)269
(C)270
(D)271
(E)272
12.Sayeeda is a point guard for her basketball team. In the last 3 games, she scored 8 points once and 12 points in each of the other two games. What must she score in tonight’s game to raise her average to 15 points?
(A)28
(B)27
(C)26
(D)25
(E)15
13.What is the greatest common factor of (3xy)^{3} and 3x^{2}y^{5} ?
(A)xy
(B)3x^{2}y^{5}
(C)3x^{2}y^{3}
(D)27x^{3}y^{3}
(E)27x^{5}y^{8}
14.The town of Mechanicville lies due east of Stillwater and due south of Half Moon Crescent. If the distance from Mechanicville to Stillwater is 30 miles, and from Mechanicville to Half Moon Crescent is 40 miles, what is the shortest distance from Stillwater to Half Moon Crescent?
(A) 10
(B) 50
(C) 70
(D)100
(E)It cannot be determined from the information given.
15.PQRS is a square with an area of 144. What is the area of the shaded region?
(A) 50
(B) 72
(C)100
(D)120
(E)It cannot be determined from the information given.
16.PO and QO are radii of the circle with center O. What is the value of x ?
(A)30
(B)45
(C)60
(D)90
(E)It cannot be determined from the information given.
17.What is the value of x ?
(A)360
(B)100
(C) 97
(D) 67
(E)It cannot be determined from the information given.
18.ABC is an equilateral triangle. What is the perimeter of this figure?
(A)4 + 2π
(B)4 + 4π
(C)8 + 2π
(D)8 + 4π
(E)12 + 2π
19.What is the perimeter of this figure?
(A)120
(B) 44
(C) 40
(D) 36
(E)It cannot be determined from the information given.
20.How many meters of police tape are needed to wrap around a rectangular crime scene that measures 6 meters wide by 28 meters long?
(A) 34 meters
(B) 68 meters
(C) 90 meters
(D)136 meters
(E)168 meters
21.Billy Bob’s Beans are currently packaged in cylindrical cans that contain 9 servings. The cans have a height of 20 cm and a diameter of 18 cm. Billy Bob wants to introduce a new single-serving can. If he keeps the height of the can the same, what should the diameter of the single-serving can be?
(A)3
(B)3
(C)4.5
(D)6
(E)6
Stop. Check your time for this drill:
Don’t forget to check your answers in Chapter 9.
MATH REVIEW
Make sure you can confidently answer all of the following questions before you take your test.
1.Is zero an integer?
2.Is zero positive or negative?
3.What operation do you perform to find a sum?
4.What operation do you perform to find a product?
5.What is the result called when you divide?
6.Is 312 divisible by 3 ?
Is 312 divisible by 9 ?
(Actually, dividing isn’t fair—use your divisibility rules!)
7.What does the “E” in PEMDAS stand for?
8.Is 3 a factor of 12 ?
Is 12 a factor of 3 ?
9.Is 3 a multiple of 12 ?
Is 12 a multiple of 3 ?
10.What is the tens digit in the number 304.275 ?
11.What is the tenths digit in the number 304.275 ?
12.2^{3} =
13.In “math language,” the word percent means ______________.
14.In “math language,” the word of means ______________.
15.In a Ratio Box, the last column on the right is always the ______________.
16.Whenever you see a problem involving averages, draw the ______________.
17.When a problem contains variables in the question and in the answers, I will ______________.
18.To find the perimeter of a square, I ______________ the length(s) of ______________ side(s).
19.To find the area of a square, I ______________ the length(s) of ______________ sides(s).
20.There are ______________ degrees in a straight line.
21.A triangle has ______________ angles, which total ______________ degrees.
22.A four-sided figure contains ______________ degrees.
23.An isosceles triangle has ______________ equal sides; a(n) ______________ triangle has three equal sides.
24.The longest side of a right triangle is called the ______________ and is located opposite the ______________.
25.To find the area of a triangle, I use the formula ______________.
Don’t forget to check your answers in Chapter 9.