Cracking the SSAT & ISEE  The Princeton Review 2019
ISEE math
The ISEE
INTRODUCTION
This section will provide you with a review of all the math that you need to do well on the ISEE. When you get started, you may feel that the material is too easy. Don’t worry. This test measures your basic math skills, so although you may feel a little frustrated reviewing things you have already learned, this type of basic review is undoubtedly the best way to improve your score.
Taking the Lower Level ISEE?
You can skip the section on quantitative comparison (this page).
Lose Your Calculator!
You will not be allowed to use a calculator on the ISEE. 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 don’t take it out again until the test is behind you. Do your homework assignments without it, and complete the practice sections of 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 will also give you something to refer back 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 in a short period of time
· those that you can do given enough time
· 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 it is one of the first type and you think you can do it without too much trouble, go ahead. If not, mark it and save it for later. Move on to the second question and decide whether or not to do that one.
Once you’ve made it all the way through the section, working slowly and carefully to answer all the questions that come easily to you, go back and try some of those that you think you can answer but will take you a little longer. You should pace yourself so that time will run out while you’re working on the second pass through the section. Make sure you save the last minute to bubble in an answer for any question you didn’t get to. Working this way, you’ll know that you answered all the questions that were easy for you. Using a twopass system is good, smart testtaking.
Guesstimating
Sometimes accuracy is important. Sometimes it isn’t.
Which of the following fractions is less than ?
(A)
(B)
(C)
(D)
Some Things Are Easier Than They Seem
Guesstimating, or finding approximate answers, can help you eliminate wrong answers and save lots of time.
Without doing a bit 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).
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 onedollar 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
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 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). What’s left? The right answer!
Guesstimating Geometry
Now that you’ve seen a couple examples that used guesstimating in arithmetic and word problems, you will see how we can also guesstimate geometry problems.
Let’s try the problem below. Remember that unless a particular question tells you that a figure is not drawn to scale, you can safely assume that the figure is drawn to scale.
Don’t Forget to Guesstimate!
Guesstimating works best on geometry questions. Make sure you use your common sense, combined with POE, to save time and energy.
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)16π
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? Threequarters? Less than half? In fact, about onequarter of the area of the square is shaded. You’ve nearly guesstimated the answer!
Now, let’s do a tiny bit of math. The length of one side of the square is 4, so the area of the square is 4 × 4 or 16.
Try These Values When Guesstimating:
π ≈ 3 + = 1.4 = 1.7
So the area of the square is 16 and we said that the shaded region was about onefourth of the square. Onefourth 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)16π
This could get a little complicated since the answers include π. However, since you’re guesstimating, you should just remember that π is just 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)16π is roughly equal to (16 × 3) = 48
Now, let’s think about what these answers mean.
Since we guesstimated that the shaded region’s area is roughly 4, (B) must be correct…and it is! Pat yourself on the back because you chose the right answer without doing a lot of unnecessary work. Unless the problem tells you that the figure is not drawn to scale, remember how useful guesstimating on geometry problems can be!
Working With Choices
In Chapter 2, Fundamental Math Skills for the SSAT & ISEE, we reviewed the concepts that the ISEE will be testing on the Lower, Middle, and Upper Level tests. However, the questions in the practice drills were slightly different from those that you will see on your exam. The ones on the exam are going to give you four answers from which to choose, not five.
There are many benefits to working with multiplechoice questions. For one, if you really mess up calculating the question, chances are your answer will not be among those given. Now you have a chance to go back and try that problem again more carefully. Another benefit, which this chapter will explore in more depth, is that you may be able to use the information in the choices to help you solve the problems.
We are now going to introduce you to the type of multiplechoice questions you will see on the ISEE. Each one of the questions on the pages that follow 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
Notice that the choices are often in either ascending or descending numerical order.
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. So (C) is the right answer.
Try it again.
Set A = {All multiples of 7}
Set B = {All odd numbers}
2.Which of the following is NOT a member of both Set A and Set B above?
(A) 7
(B)21
(C)49
(D)59
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.
Remember the Rules of Zero
Zero is even. It’s neither + nor —, and anything multiplied by 0 = 0.
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
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 Case of the Mysteriously Missing Sign
If there is no operation sign between a number and a variable (letter), the operation is multiplication.
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
6 × 5 × 2 = 60 and so does 2 × 2 × 15. Choice (C) is correct.
Don’t Do More Work Than You Have To
When looking at answer choices, start with what’s easy for you; work through the harder ones only when you have eliminated all of the others.
Working with Negative Numbers
5.7 — 9 is the same as
(A) 7 — (—9)
(B) 9 — 7
(C) 7 + (—9)
(D)—7 — 9
Remember that subtracting a number is the same as adding its opposite. Choice (C) is correct.
Order of Operations
6.9 + 6 × 2 ÷ 3 =
(A) 7
(B) 9
(C)10
(D)13
Remember your PEMDAS rules? The multiplication comes first. The correct answer is (D).
Factors and Multiples
7.What is the sum of the prime factors of 42 ?
(A)18
(B)13
(C)12
(D)10
Remember!
1 is NOT a prime number.
How do we find the prime factors? The best way is to draw a factor tree. Then we see that the prime factors of 42 are 2, 3, and 7. Add them up and we get 12, (C).
Fractions
8.Which of the following is less than ?
(A)
(B)
(C)
(D)
When comparing fractions, you have two choices. You can find a common denominator and then compare the fractions (such as when you add or subtract them). You can also change the fractions to decimals. If you have completed and memorized the fractiontodecimal charts in the Fundamentals chapter (this page and this page), you probably found the right answer without too much difficulty. It’s (A).
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%
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)4^{2}
(C)3^{2}
(D)8^{2}
Expand 2^{6} out and we can multiply to find that it equals 64. Choice (D) is correct.
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
If you have trouble with this one, use the choices and work backward. As we discussed in the Fundamentals chapter, 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.
Basic Algebraic Equations
12.11x = 121. What does x = ?
(A) 2
(B) 8
(C)10
(D)11
Remember, if you get stuck, use the choices and work backward. Each one provides you with a possible value for x. Start with a middle choice and replace x with it. 11 × 10 = 110. That’s too small. Now we know not only that (C) is the incorrect answer, but also that (A) and (B) are incorrect because they are smaller than (C). The correct answer is (D).
Solve for X—Upper Level Only
13.3y + 17 = 25 — y. What does y = ?
(A)1
(B)2
(C)3
(D)4
Just as on the previous question, if you get stuck, use the choices. The correct answer is (B).
Percent Algebra—Upper Level Only
14.25% of 30% of what is equal to 18 ?
(A) 1
(B) 36
(C)120
(D)240
Percent means out of 100, and the word of in a word problem tells you to multiply.
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).
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) 17
(C) 15
(D) 14
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
x + y is equal to the perimeter of the polygon minus the lengths of the sides we know. Choice (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 42 minutes. How many blocks away from school does Jeff live?
(A)3
(B)5
(C)6
(D)9
This is a proportion question because we have two sets of data we are comparing. Set up your fractions.
Because we know that we must do the same thing to the top and the bottom of the first fraction to get the second fraction, and because 14 × 3 = 42, we must multiply 3 × 3 to get 9.
So Jeff walks 9 blocks in 42 minutes. Choice (D) is correct.
18.Half of the 30 students in Mrs. Whipple’s firstgrade 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
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 is the shorter piece?
(A) 2
(B) 6
(C) 9
(D)12
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) and (D). 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. In (B), if 6 is the length of the shorter piece, we can subtract it from 18 and now we know the length of the longer piece is 12. And 12 is twice the length of 6, so we have the right answer.
PRACTICE DRILL 1—LOWER LEVEL
Time yourself on this drill. When you are done, check your answers in Chapter 17.
Remember to time yourself during this drill!
1.How many factors does the number 24 have?
(A)2
(B)4
(C)6
(D)8
2.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
3.Which of the following is a multiple of 3 ?
(A) 2
(B) 6
(C)10
(D)14
4.Which of the following is NOT a multiple of 6 ?
(A)12
(B)18
(C)23
(D)24
5.Which of the following is a multiple of both 3 and 5 ?
(A)10
(B)20
(C)25
(D)45
6.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
7.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
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
9.Joanne owns onethird 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 Joanne own?
(A)2
(B)4
(C)6
(D)8
10.A tank of oil is onethird 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
11.Ginger the dog sleeps threefourths of every day. In a fourday period, she sleeps the equivalent of how many full days?
(A)
(B)
(C)1
(D)3
12.Which of the following has the greatest value?
(A) +
(B) 
(C) ÷
(D) ×
13. + + + + + =
(A)
(B)1
(C)3
(D)6
14.The product of 0.34 and 1,000 is approximately
(A) 3.50
(B) 35
(C) 65
(D)350
15.2.398 =
(A)2 × × ×
(B)2 + + +
(C)2 + + +
(D) + +
Stop. Check your time for this drill:
How Did You Do?
That was a good sample of the kinds of questions you’ll see on the ISEE. 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 following 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), your pacing is off. Take a look at any problems that may have affected your speed. Were there any questions that seriously slowed you down? Did you answer some quickly but not correctly? In general, don’t just look to see what you got right, but rather how you got it right.
Question Recognition and Selection
Did you use your time wisely? Did you do the questions in an order that worked well for you? Did you get stuck on one problem and spend too much time on it? Which kinds of questions were hardest for you? Remember that on the ISEE you must answer every question, but you don’t have to work on every problem. Every question on the ISEE, whether you find it easy or hard, is worth one point, and there is no penalty for wrong answers. You should concentrate most on getting all the questions you find easy or sortof easy right, and worry about doing problems you find harder later. Keep in mind that questions generally go from easiest to hardest throughout the section. Getting the early questions right takes time, but you know you can do it, so give yourself that time! If you don’t have time for a question or can’t guess wisely, pick a “letter of the day” (the same letter for every problem you can’t do), fill it in, and move on. Because there is no penalty for wrong answers, guessing can only help your score.
POE and Guessing
Did you actively look for wrong answers to eliminate, rather than 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 simplerlooking over harderlooking numbers)?
Write It Down
Did you work out the practice questions? Did you move too quickly or skip steps on problems you found easier? Did you always doublecheck what the question was asking? Students frequently 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 ISEE math problem on a piece of paper. Consider it a doublecheck because your handwritten notes confirm what you’ve worked out in your head.
PRACTICE DRILL 2—MULTIPLE CHOICE—MIDDLE AND UPPER LEVELS ONLY
While doing the next drill, keep in mind the general testtaking techniques we’ve talked about: guessing, POE, order of difficulty, pacing, choosing a letteroftheday for problems that stump you, and working on the page and not in your head. When you are done, check your answers in Chapter 17. 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.
Remember to time yourself during this drill!
1.How many numbers between 1 and 100, inclusive, are both prime and a multiple of 4 ?
(A) 0
(B)12
(C)20
(D)25
2.How many factors do the integers 24 and 81 have in common?
(A)1
(B)2
(C)3
(D)4
3.If the final total of a dinner bill—after including a 25% tip—is $50, what was the cost of the dinner before including the tip?
(A)$12.50
(B)$25.00
(C)$37.50
(D)$40.00
4.How many numbers between 1 and 100 are multiples of both 2 and 7 ?
(A)6
(B)7
(C)8
(D)9
5.2^{3} × 2^{3} × 2^{3} =
(A)2^{6}
(B)2^{9}
(C)2^{27}
(D)8^{9}
6.For what integer value of m does 2m + 4 = m^{3} ?
(A)1
(B)2
(C)3
(D)4
7.If 6x — 4 = 38, then x + 10 =
(A) 7
(B)10
(C)16
(D)17
8.What is the smallest multiple of 7 that is greater than 50 ?
(A) 7
(B)49
(C)51
(D)56
9.Onefifth 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
10.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%
11.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%
12.Which of the following is most nearly 35% of $19.95 ?
(A)$3.50
(B)$5.75
(C)$7.00
(D)$9.95
13.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%
14.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)$25
15.If the perimeter of a square is 56, what is the length of each side?
(A) 4
(B) 7
(C)14
(D)28
16.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
Lower and Middle levels can stop here. Don’t forget to check your answers!
Upper Level Only
17.If b = 45, then v^{2} =
(A)32
(B)25
(C)16
(D) 5
18.Onehalf 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
19.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
20.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
21.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
22.If the area of a square is 64p^{2}, what is the length of one side of the square?
(A)64p^{2}
(B)8p^{2}
(C)64p
(D)8p
23.If AB = 10 and AC = 15, what is the perimeter of the figure above?
(A)25
(B)35
(C)40
(D)50
24.If ABCD is a rectangle, what is the value of w + x + y + z ?
(A) 90
(B)150
(C)180
(D)190
25.What is the area of the figure above if all the angles shown are right angles?
(A)38
(B)42
(C)50
(D)88
26.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)168 meters
27.The distance between points A and B in the coordinate plane above is
(A) 5
(B) 6
(C) 8
(D)10
28.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
29.What is the value of x ?
(A)360
(B)100
(C) 97
(D) 67
30.ABC is an equilateral triangle. What is the perimeter of this figure?
(A)4 + 2π
(B)4 + 4π
(C)8 + 2π
(D)8 + 4π
31.In trapezoid LMNO, line LO has a length of 20. What is the area of the trapezoid?
(A) 48
(B) 64
(C) 88
(D)112
When you’re finished, check your answers in Chapter 17.
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 to 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.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
2.In Janice’s tennis club, 8 of the 12 players are righthanded. What is the ratio of righthanded to lefthanded players in Janice’s club?
(A)1:2
(B)1:6
(C)2:1
(D)2:3
3.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
Stop. Check your time for this drill:
Check your answers in Chapter 17.
Averages
There are three parts to every average problem: total, number, and average. Most ISEE 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 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.
Get used to working with the Average Pie by using it to solve the following problems.
PRACTICE DRILL 4—AVERAGES
Remember to time yourself during this drill!
1.The average of 3 numbers is 18. What is two times the sum of the 3 numbers?
(A)108
(B) 54
(C) 36
(D) 18
2.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
3.Catherine scores 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
4.If a class of 6 students has an average grade of 72 before a seventh student joins the class, what must the seventh student’s grade be to raise the class average to 76 ?
(A)100
(B) 92
(C) 88
(D) 80
Check your answers in Chapter 17.
More Practice: Lower Level
5.Anna ate 2 doughnuts on Monday, Wednesday, and Friday and ate 4 doughnuts on Tuesday and Thursday. She did not eat any doughnuts on Saturday or Sunday. What is the average number of doughnuts that Anna ate each day of the week?
(A)2.0
(B)2.5
(C)2.8
(D)3.0
6.Merry drove 350 miles from New Orleans to Houston in 7 hours. She then drove 240 miles from Houston to Dallas in 4 hours. What was her approximate average rate of speed, in miles per hour (mph), for the entire trip?
(A)50.0 mph
(B)53.6 mph
(C)55.0 mph
(D)60.0 mph
7.The ticket price to a school’s spring musical production is $6. The auditorium has a seating capacity of 300. After having spent $550 on stage production and $250 on advertising, how much profit did the school make, assuming the show was sold out?
(A) $700
(B) $800
(C) $900
(D)$1,000
Check your answers in Chapter 17.
More Practice: Middle and Upper Levels
8.Michael scored an average of 24 points over his first 5 basketball games. How many points must he score in his 6th game to average 25 points over all 6 games?
(A)24
(B)25
(C)30
(D)36
9.Dwan measured a total of 245 inches of rainfall in his hometown over one week. During the same week the previous year, his hometown had a total of 196 inches. How many more inches was the daily amount of rainfall for the week this year than the week last year?
(A)6
(B)7
(C)8
(D)9
10.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
When you’re finished, don’t forget to check your answers in Chapter 17!
Percent Change—Upper Level Only
There is one special kind of percent question that shows up on the ISEE: percent change. This type of question asks you to find by 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 The Voice 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
% change = × 100
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) 50%
(B)100%
(C)150%
(D)200%
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%
Check your answers in Chapter 17.
Plugging In
The ISEE will often ask you questions about reallife situations for 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 Bernal. If b represents Bernal’s height in inches, then in terms of b, an expression for Mark’s height is
(A)b + 6
(B)b + 4
(C)b + 2
(D)b — 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 b inches tall?
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 b.
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 b into the choices and choose the one that gives you the number you found for Mark’s height.
Here’s How It Works
For Bernal’s height, let’s pick 60 inches. This means that b = 60. Remember, there is no right or wrong number to pick. 50 would work just as well.
But given that Bernal is 60 inches tall, now we can figure out that, because John is four inches shorter than Bernal, 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.
So here’s what we’ve got.
Bernal 60 inches = b
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 b, and choose the one that equals 58.
(A) b +6 60 + 6= 66 ELIMINATE
(B) b + 4 60 + 4 = 64 ELIMINATE
(C) b + 2 60 + 2 = 62 ELIMINATE
(D) b — 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 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 ISEE 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 ISEE math problems puts you at a real advantage.
PRACTICE DRILL 6—PLUGGING IN
1.At a charity fundraiser, 200 people each donated x dollars. In terms of x, what was the total number of dollars that was donated?
(A)
(B)200x
(C)
(D)200 + x
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.
2.If 10 magazines cost d dollars, how many magazines can be purchased for 3 dollars?
(A)
(B) 30d
(C)
(D)
Don’t worry about timing yourself on this drill. Focus on the strategy. Plug In for each question so you learn how to use the technique.
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)4z
(B)z + 4
(C)4z + 16
(D)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 30 or 60, not 128. Also, whenever you see the word percent, Plug In 100!
More Practice: Lower Level
Lower level students can stop here and check answers in Chapter 17. Middle and Upper level students should keep on drilling!
4.There were 6 pairs of earrings sold at a price of y dollars each. In terms of y, what is the total amount of money for which these earrings were sold?
(A)6 + y
(B)6y
(C)6y
(D)6 + 6y
5.If p pieces of candy costs c cents, 10 pieces of candy will cost
(A) cents
(B) cents
(C) cents
(D) 10pc cents
Lower level students can stop here and check answers in Chapter 17. The rest of you, keep going!
More Practice: Middle Level
6.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 > 0
7.On Monday, Sharon ate onehalf of a fruit tart. On Tuesday, Sharon then ate onefourth of what was left of the tart. What fraction of the tart did Sharon eat on Monday and Tuesday?
(A)
(B)
(C)
(D)
More Practice: Middle and Upper Levels
8.The price of a suit is reduced by 20%, and then the resulting price is reduced by another 10%. The final price is what percent off of the original price?
(A)20%
(B)25%
(C)28%
(D)30%
9.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 greater than zero.
Middle level students can stop here and check their answers in Chapter 17. Upper level students have more math fun ahead!
More Practice: Upper Level
10.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
11.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.
12.Anthony has twice as many baseball cards as Keith, who has onethird as many baseball cards as Ian. If Keith has k baseball cards, how many baseball cards do Anthony and Ian have together?
(A)
(B)
(C)
(D)
13.The product of b and a^{2} can be written as
(A)(ab)^{2}
(B)
(C)2a × b
(D)
14.x^{a} = (x^{3})^{3}
y^{b} =
What is the value of a × b ?
(A)17
(B)30
(C)48
(D)72
15.Hidden Glen Elementary school is collecting donations for a school charity drive. The total number of students in Mr. Greenwood’s history class donate an average of y dollars each. The same number of students in Ms. Norris’s science class donate an average of z dollars each. In terms of y and z, what is the average amount of donations for each student from both classes?
(A)
(B)
(C)(y + z)
(D)2(y + z)
16.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}
Upper level students, it’s time to check your answers in Chapter 17.
Plugging In The Answers (PITA)
Plugging In the Answers is similar to Plugging In. When variables are in the choices, plug in. When numbers are in the choices, Plug In the Answers.
Plugging In the Answers works because on a multiplechoice 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
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, or 28 (the four choices). So pick one—start with either (B) or (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 and she gave half to Lisa. That leaves Nicole with 10 cookies. Then, she gave 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) right? No.
No problem. Pick 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 consecutive order, so it makes it easier to pick the next larger or smaller number, depending on which direction you’ve decided to go. 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 eight cookies left over. Keep going with the question. It says, “If she now has eight cookies left….” She has 8 cookies left and, voilà—she’s supposed to have 8 cookies left.
What does this mean? It means you’ve got the right answer!
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)72
(B)120
(C)145
(D)180
2.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
3.Four less than a certain number is twothirds of that number. What is the number?
(A)1
(B)6
(C)8
(D)12
More Practice: Lower Level
4.There are 12 more girls than boys in a classroom. If there are 30 total students in the classroom, how many girls are there in the classroom?
(A)9
(B)12
(C)20
(D)21
5.Victor, Jonathan, and Russell buy a home theater system. Victor pays twice as much as Jonathan, and Victor pays half as much Russell. If the home theater system costs $560, how much does Jonathan pay?
(A) $60
(B) $80
(C)$100
(D)$120
Lower level students can stop here and check answers in Chapter 17. The rest of you, keep going!
More Practice: Middle and Upper Levels
6.Adam is half as old as Bob and three times as old as Cindy. If the sum of their ages is 40, what is Bob’s age?
(A)6
(B)12
(C)18
(D)24
7.If 70x + 33y = 4,233, and x and y are positive integers, x could be which of the following values?
(A)42
(B)47
(C)55
(D)60
8.The sum of three positive integers is 9 and their product is 24. If the smallest of the integers is 2, what is the largest?
(A)4
(B)6
(C)8
(D)9
9.Lori is 15 years older than Carol. In 10 years, Lori will be twice as old as Carol. How old is Lori now?
(A)5
(B)12
(C)20
(D)25
10.A group of people are sharing equally the $30 cost of renting a car. If an additional person joined the group, each person would owe $1 less. How many people are in the group currently?
(A)5
(B)6
(C)10
(D)12
Check your answers in Chapter 17.
GEOMETRY
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 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 the area of the figure inside. The difference is what you need.
ABCE is a rectangle with a length of 10 and a 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
The first step is to find the area of the rectangle. Multiply the length by the width and find that the area of the rectangle 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 the 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 answer is (D).
Extra Practice
1.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
2.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 — 4 π
(C)8 + 4 π
(D)4 π
Functions
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
2.If $n = 120, then n =
(A) 11
(B) 12
(C) 13
(D)120
Questions 35 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
4.If K (4 ¿ 3) = 30, then K =
(A)3
(B)4
(C)5
(D)6
5.(2 ¿ 4) × (3 ¿ 6) =
(A)(9 ¿ 3) + 3
(B)(6 ¿ 4) + 1
(C)(5 ¿ 3) + 4
(D)(8 ¿ 4) + 2
When you’re finished, check your answers in Chapter 17.
Charts and Graphs
Charts
Chart questions usually do not involve much computation, 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 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, 2012 and 2013 

State 
2012 
2013 
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 2012 and the number of members who came from Illinois in 2013 ?
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 2012 was 400.
Second, the number of members who came from Illinois in 2013 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 2012 to 2013 was what percent of the total number of members in New Jersey in 2012 ?
You should definitely know how to do this one! Do you remember how to translate percentage questions? If not, go back to Chapter 2.
In 2012, there were 200 club members from New Jersey. In 2013, 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 2012, you need to ask yourself, “50 (the increase) is what percent of 200 (the number of members in 2012)?”
Translated, this becomes:
With a little bit of simple manipulation, this equation becomes:
50 = 2g
and
25 = g
So from 2012 to 2013, there was a 25% increase in the number of members from New Jersey. Good work!
Which state had as many club members in 2013 as a combination of Illinois, Massachusetts, and Michigan had in 2012 ?
First, take a second to look up the number of members who came from Illinois, Massachusetts, and Michigan in 2012 and add them together.
200 + 150 + 150 = 500
Which state had 500 members in 2013? 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
This one is easy. 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
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 
2013 
2014 
A 
400 
600 
B 
500 
700 
C 
200 
350 
D 
100 
150 
E 
600 
800 
(All numbers are in thousands of dollars.) 
1.In 2014, which district spent twice as much on energy as District A spent in 2013 ?
(A)A
(B)B
(C)C
(D)E
2.Which district spent the most on electricity in 2013 and 2014 combined?
(A)A
(B)B
(C)D
(D)E
3.The total increase in energy expenditure in these districts, from 2013 to 2014, is how many dollars?
(A) $800
(B) $1,800
(C) $2,600
(D)$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
5.Which one student owns onefourth of the CDs accounted for in Figure 2 ?
(A)Abe
(B)Ben
(C)Carl
(D)Dave
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)$14
(B)$21
(C)$54
(D)$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
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
Check your answers in Chapter 17. You can make a chart to see how you’ve been doing!
QUANTITATIVE COMPARISON—MIDDLE AND UPPER LEVELS ONLY
Quant Comp: Same Book, Different Cover
Quantitative comparison is a type of question—one slightly different from the traditional multiplechoice questions you’ve seen so far—that tests exactly the same math concepts you have learned so far in this book. There is no new math for you to learn here, just a different approach for this type of question.
You will see a total of 17 quant comp questions in one of your ISEE Math sections.
Lower Level Test Takers
The ISEE’s Lower Level test does not include quantitative comparison questions, so you can skip this section.
The Rules of the Game
In answering a quant comp question, your goal is very simple: Determine which column is larger and choose the appropriate answer. There are four possible answers.
(A)means that Column A is always greater.
(B)means that Column B is always greater.
(C)means that Column A is always equal to Column B.
(D)means that A, B, or C are not always true.
So that you can use POE in quant comp, where there are no choices written out for you, we suggest that you write “A B C D” next to each question. Then when you eliminate an answer, you can cross it off.
Don’t Do Too Much Work
Quant comp is a strange, new question type for most students. Don’t let it intimidate you, however. Always keep your goal in mind: to figure out which column is larger. Do you care how much larger one column is? We hope not.
Here’s a good example.
Column A 
Column B 
2 × 4 × 6 × 8 
3 × 5 × 7 × 9 
Test takers who don’t appreciate the beauty of quant comp look at this one and immediately start multiplying. Look carefully, however, and compare the numbers in both columns.
They Look Different, but the Math Is the Same
This section will introduce you to quantitative comparison, a different type of question from the “regular” multiplechoice questions you’ve seen so far. Don’t worry—these questions test your knowledge of exactly the same math skills you have already learned in this chapter.
Of the first numbers in each column, which is larger, 2 or 3 ?
Next, look at the second number in each column. Which is larger, 4 or 5 ?
Now, look at the third numbers. Which is larger, 6 or 7 ?
Finally, look at the fourth numbers. Which is larger, 8 or 9 ?
In each case, column B contains larger numbers. Now, when you multiply larger numbers together, what happens? You guessed it—even larger numbers!
Which column is larger? Without doing a single bit of multiplication you know that (B) is the right answer. Good work!
(D) Means Different
Choice (D) is useful when the relationship between the columns can change. You may have to choose (D) when you have variables in a quant comp problem. For example:
Column A 
Column B 
g + 12 
h — 7 
Which column is larger here depends entirely on what g and h equal, and the problem doesn’t give you that information. This is a perfect time to choose (D).
But be careful and don’t be too quick to choose (D) when you see a variable.
Column A 
Column B 
g + 12 
g — 7 
With one small change, the answer is no longer (D). Because the variables are the same here, you can determine that no matter what number is represented by g, column A will always be larger. So in this case the answer is (A).
One valuable thing to remember is that when a quant comp question contains no variables and no unknown quantities, the answer cannot be (D).
Column A 
Column B 
6 × 3 × 4 
4 × 6 × 3 
Even if you somehow forget how to multiply (don’t worry, you won’t forget), someone somewhere knows how to multiply, so you can get rid of (D).
By the way, look quickly at the last example. First, you eliminate (D) because there are no variables. Do you need to multiply? Nope! The columns contain exactly the same numbers, just written in a different order. What’s the answer? You got it: (C)!
PRACTICE DRILL 10—QUANT COMP—MIDDLE AND UPPER LEVELS ONLY
Remember to time yourself during this drill!
(A) means that Column A is always greater
(B) means that Column B is always greater
(C) means that Column A is always equal to Column B
(D) means that A, B, or C are not always true
Column A 
Column B 

1. 
17 × 3 
17 × 2 + 17 
2. 

3. 
b + 80 
b + 82 
Rob is two inches shorter than Matt. Joel is four inches taller than Matt. 

4. 
Rob’s height 
Joel’s height 
5. 
16^{3} 
4^{6} 
Kimberly lives two miles from school. Jennifer lives four miles from school. 

6. 
The distance from Kimberly’s house to school 
The distance from Kimberly’s house to Jennifer’s house 
Check your answers in Chapter 17.
Quant Comp Plugging In
Think back to the Algebra section. Plugging In helped you deal with variables, right? The same technique works on quant comp questions. There are some special rules you’ll need to follow to make sure you can reap all the benefits that Plugging In has to offer you in the Quantitative Comparison section.
Column A 
Column B 
x 
x^{2} 
Follow these three simple steps, and you won’t go wrong.
Step 1: Write “A B C D” next to the problem.
Step 2: Plug In an “easy” number for x. By easy number, we mean a nice simple integer, like 3. When you Plug In 3 for x in the above example, column A is 3 and column B is 9, right? Think about the choices and what they mean. Column B is larger, so can the correct answer be (A)? No, eliminate it. Can the correct answer be (C)? No, you can get rid of that one, too!
Step 3: Plug In a “weird” number for x. A weird number might be a little harder to define, but it is something that most test takers won’t think of—for instance, zero, one, a fraction, or a negative number. In this case, try plugging in 1. Column A is 1 and column B is also 1. So the columns can be equal. Now look at the choices you have left. Choice (B) means that column B is always greater. Is it? No. Cross off (B) and pick (D).
Weird Numbers
For your second Plug In, try something weird:
Zero
One
Negative
Extreme
Fraction
Remember, if you get one result from Plugging In a number and you get a different result by Plugging In another number, you have to pick (D). But don’t think too much about these questions, or you’ll end up spending a lifetime looking for the perfect “weird” number. Just remember that you always have to Plug In twice on quant comp questions.
PRACTICE DRILL 11—QUANT COMP
Remember to time yourself during this drill!
(A) means that Column A is always greater
(B) means that Column B is always greater
(C) means that Column A is always equal to Column B
(D) means that A, B, or C are not always true
Column A 
Column B 

x > 1 

1. 
x 
x^{2} 
b is an integer and —1 < b < 1. 

2. 

3. 
p gallons 
m quarts 
x is a positive integer. 

4. 

w is an integer less than 4. p is an integer greater than 10. 

5. 
pw 
w 
6. 
4c + 6 
3c + 12 
When you’re finished, check your answers in Chapter 17.
PRACTICE DRILL 12—QUANT COMP—MIDDLE AND UPPER LEVELS ONLY
Do this drill in three parts. Questions 1—11, 12—28, and 29—45. When you’re done with each set, check your progress in Chapter 17. Don’t forget to time yourself!
Remember to time yourself during this drill!
(A) means that Column A is always greater
(B) means that Column B is always greater
(C) means that Column A is always equal to Column B
(D) means that A, B, or C are not always true
Column A 
Column B 

1. 
The total cost of 3 plants that cost $4 each 
The total cost of 4 plants that cost $3 each 
2. 
30(1 — 2n) 
30 — 2n 
The product of 3 integers is 48. 

3. 
The smallest of the 3 integers 
1 
4. 
(x + y) (x — y) 
x^{2} — y^{2} 
5. 
(7 — 4) × 3 — 3 
0 
Line m is the graph of y = x + 4. 

6. 
Slope of line m 
Slope of line l that is perpendicular to line m 
The price of a pair of shoes is $100. The price is increased by 20%. Nobody buys it, so the price is then reduced by 20%. 

7. 
The final price of the pair of shoes after reductions 
$100 
8. 
9. 
10. 
11. 
Note: Figure not drawn to scale. 

12. 
Circumference of Circle P 
Area of Circle Q 
A 6sided number die, numbered 1 to 6, is rolled. 

13. 
Probability that the number rolled is prime. 
a and b are integers. a + b = 5 

14. 
a 
b 
15. 
− 
Set A: {all prime numbers} Set B: {all positive multiples of 5 less than 50} Set C: intersection of Sets A and B 

16. 
Number of elements in Set C 
1 
17. 
× 
+ 
a > 0 b < 0 

18. 
— (ab) 
—ab 
Set A: {1, 3, 8, 11, 15} Set B: {2, 4, 8, 9, 10, 20} 

19. 
Median of Set A 
Median of Set B 
20. 
Sum of all consecutive integers between 1 and 10, inclusive 
5(11) 
21. 
2^{3} + 2^{3} + 2^{3} 
2^{9} 
22. 
7(x — 3) 
21 — 7x 
23. 
The smallest positive factor of 25 multiplied by biggest positive factor of 16 
40 
24. 
Probability of a fair penny having heads face up on two consecutive flips 
Probability of a fair penny having heads face up on three consecutive flips 
Note: Volume of a right cylinder: V = πr^{2}h 

25. 
Volume of Cylinder A 
Total Volume of Cylinders B and C 
26. 
x 
60 
27. 
The percent increase from 1 to 2 
The percent decrease from 2 to 1 
28. 
The average (arithmetic mean) of 4, 6, 8, and 10 
The median of 4, 6, 8, and 10 
x > 0 y > 0 

29. 
30. 
(567.83) (0.40) 
(40) (5.6783) 
Meredith has 7 pairs of purple shoes, 2 pairs of red shoes, and 1 pair of white shoes. She chooses one pair of shoes at random. 

31. 
Probability of not picking a red pair of shoes 
32. 
Total cost of 10 shirts at $8 each 
Total cost of 20 shirts at $4.50 each 
33. 
x^{3} 
x^{2} = 36 

34. 
x 
—6 
35. 
Largest positive factor of 16 
Smallest positive multiple of 16 
36. 
Perimeter of square ABCD 
Perimeter of rectangle MNOP 
E is the midpoint of side AD. 

37. 
Area of ΔBDE 
12 
38. 
4^{12} 
64^{4} 
A hat contains blue and red tickets. The ratio of blue tickets to red tickets is 3:5. 

39. 
The fractional part of all the tickets in the hat that are blue 
Luke travels from Providence to Boston at an average speed of 50 miles per hour without stopping. He returns to Providence along the same route at an average speed of 60 miles per hours without stopping. 

40. 
Luke’s average speed for the entire trip 
55 miles per hour 
41. 
The slope of the line 12x — 4y = 16 
The slope of the line containing points (—3, 6) and (3, 12) 
42. 
A rectangle with sides y and z has an area of 36. 

43. 
The length of y 
The length of z 
44. 
The number of nonnegative even integers less than 10 
4 
Triangle ABC is isosceles. Figure not drawn to scale. 

45. 
The area of triangle ABC 
4 
Check your answers in Chapter 17.
MATH REVIEW
Make sure you can confidently answer all of the following questions before you take the ISEE.
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.
20.There are __________ degrees in a straight line.
21.A triangle has __________ angles, which total __________ degrees.
22.A foursided 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: __________.
Check your answers in Chapter 17.