SSAT & ISEE Prep 2023 - Princeton Review 2023

ISEE math
The ISEE

Taking the Primary/Lower Level ISEE?

You can skip the section on quantitative comparison (this page).

INTRODUCTION

This section will provide you with a review of the math strategy 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 important for ensuring that you don’t make preventable mistakes. Primary 2 and Primary 3 students can stop after the Primary Basics portion of the chapter. Primary 4 students may find some of the Lower Level content useful.

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 if you're taking a paper-based test, and we recommend that you do so. If you are taking an online ISEE, use scratch paper. You'll need to provide your own at home, but test centers will provide it for you. You may have 4 pieces of scratch paper for the full test. 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 it 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 in your test booklet or flag it on your screen 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 fill 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 two-pass system is good, smart test-taking.

PRIMARY BASICS

The Math sections on the Primary Levels will test Number Sense and Operations (including counting, addition and subtraction, coin value, fractions and decimals, and basic equations), Geometric Concepts (including shape similarities, symmetry, rotation, and basic 3D figures), Measurement (including rulers, tool use, ordering, unit conversion, and perimeter), Algebraic Concepts (including symbols, patterns, number sentences, equivalence, and proportions), and Data Analysis and Probability (including bar graphs, charts, and trends). Let’s review some concepts!

Coin Value

Coins can be a little bit tricky, since the smallest coin is worth more than some of the bigger coins!

Penny = 1¢

Nickel = 5¢

Dime = 10¢

Quarter = 25¢

Half-Dollar = 50¢

Measuring Tools

Measuring different things requires different tools! Would you measure water the same way you measure your height?

Measuring Tape

Measures length, distance, or circumference, such as your height or the length of a room or the size of your wrist

Measuring Cup

Measures liquids or tiny things that can be poured, such as water or sand

Ruler

Measures straight distances, such as the length of a pencil

Scale

Measures how heavy something is, such as your weight or the weight of an apple

Thermometer

Measures temperature

Units and Conversions

Some questions will expect you to know the correct units for a particular measurement or the conversion from one type of unit to another:

·  Inches, feet, yards, centimeters, meters, and kilometers are used for length or distance, including perimeter

·  Square inches (in²), square feet (ft²), square yards (yd²), and square meters (m²) are used for two-dimensional area, such as the size of a piece of paper or a plot of land.

·  Milliliters, liters, cups, and quarts are used for amounts of liquid

·  Milligrams, grams, kilograms, ounces and pounds are used for weight

1 kilometer = 1,000 meters

1 meter = 100 centimeters

1 foot = 12 inches

1 liter = 1,000 milliliters

1 kilogram = 1,000 grams

1 gram = 1,000 milligrams

Symmetry

Shapes that can be folded in half with both halves matching exactly have line symmetry. The figures below have symmetry:

These figures do not:

Number Sentences (Equations)

A number sentence is a mathematical sentence that shows the relationship between numbers using mathematical symbols. Some useful things to keep in mind:

+Use the plus sign to represent "add," "sum," "more," or combining or including amounts.

—Use the minus sign to represent “subtract” or things going away or being taken away.

×Use the multiplication sign to represent “times” or “by."

÷Use the division sign to represent “divide” or “per."

=The equals sign goes between two balanced sides of the number sentence.

Fractions

Fractions are a way of expressing the relationship between a part and a whole. You encounter fractions all the time—you just may not realize it! If someone makes a sandwich, cuts it down the middle into two pieces, and gives one piece to you and one piece to your friend, you each have a fractional part of the sandwich!

Now check out how these concepts are tested on the ISEE Primary Levels.

Primary Basics Drill

Number Sense and Operations — Primary 2

1.Shiva has 18 pieces of pizza. She gives 7 to Anna and 4 to Brendan. How many pieces of pizza does Shiva have left?

(A) 7

(B) 11

(C) 22

(D) 29

Number Sense and Operations — Primary 3

2.Which shows the largest amount?

(A) $53.35

(B) $56.26

(C) $53.53

(D) $56.62

Number Sense and Operations — Primary 4

3.Colleen has a collection of friendship bracelets. She gives of her bracelets to her sister Patty and of the bracelets to her friend Linda. If Colleen gave Patty 1 bracelet, how many bracelets does Colleen have left in her collection?

(A) 3

(B) 4

(C) 5

(D) 8

Geometric Concepts — Primary 2

4.Gianna is building a wooden table top, but she forgot to make one of the pieces.

Which one of the following shapes did she forget to make?

Geometric Concepts — Primary 3

5.Wayne notices pictures of four animals on a poster. Which animal has line symmetry?

Geometric Concepts — Primary 4

6.Dante follows a path on the coordinate grid below. Starting at point (4, 2) he goes 3 spaces right and 5 spaces up. At which point does Dante land?

(A) R

(B) T

(C) U

(D) W

Measurement — Primary 2

7.Carissa is measuring her pencil.

How long is the pencil?

(A) 4 inches

(B) 5 inches

(C) 7 inches

(D) 8 inches

Measurement — Primary 3

8.Sheila measures a hallway in her house and finds that it is 6 meters long. How long is it in centimeters?

(A) 6 centimeters

(B) 60 centimeters

(C) 600 centimeters

(D) 6,000 centimeters

Measurement — Primary 4

9.Chris discovers that a brick weighs about 3 kilograms. Which object weighs about 3 grams?

(A) A garbage truck

(B) A penny

(C) A cotton ball

(D) A gallon of milk

Algebraic Concepts — Primary 2

10.Jared and Daniella each have a collection of balloons as shown below. Then 9 of the balloons pop.

Jared

Daniella

Which number sentence shows the total number of unpopped balloons left?

(A) 16 — 9 = 7

(B) 16 — 12 = 4

(C) 12 + 16 — 9 = 19

(D) 12 + 16 + 9 = 37

Algebraic Concepts — Primary 3

11.Antoine has 37 chocolate chip cookies and 24 snickerdoodle cookies in his bakery. He sells 11 cookies. How many cookies does he have left?

(A) 2

(B) 40

(C) 50

(D) 72

Algebraic Concepts — Primary 4

12.Use the following key to answer the question below.

= ?

= 5

= 8

In the equation (9—) × = 30, what is the value of ?

(A) 2

(B) 3

(C) 4

(D) 6

Questions 13 and 14 refer to the following graph.

A fruit stand owner is taking inventory at the end of the day. She counts how many of each fruit she has left and makes this graph.

Data Analysis and Probability — Primary 2

13.Which statement is true based on the graph?

(A) There are 16 bananas.

(B) There are 7 strawberries.

(C) The type of fruit that there are the least of is bananas.

(D) There are the same number of apples and mangoes.

Data Analysis and Probability — Primary 3

14.Based on the graph, how many total pieces of fruit were left at the end of the day?

(A) 22

(B) 27

(C) 65

(D) 66

Data Analysis and Probability — Primary 4

Jayden timed himself washing his hands for 8 consecutive days.

15.He got faster each day for several days and then slower. On which day did he get slower?

(A) July 10

(B) July 11

(C) July 13

(D) July 14

Check your answers in Chapter 17. For detailed explanations to these Primary Basics practice questions, check your online student tools!

When You Are Done

Don’t forget to check your answers in Chapter 17, this page.

LOWER, MIDDLE, AND UPPER LEVELS

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. If you have a few extra seconds, you can evaluate the remaining two answer choices. Which one's easier to work with? Look at (B). reduces to , which is bigger than . You might also think about the fact that would be , so must be bigger. The answer is (A).

Here’s another good example.

A group of three people 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 group, how much will each person 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 people, 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 people; there were only three. This means fewer people 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.

Guesstimating also works very well with some geometry questions, but we’ll save that for the Geometry section.

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.

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 necessarily, though. Look at the picture. What fraction of the square looks like it is shaded? Half? Three-quarters? Less than half? Looks like about one-quarter 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⁺

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) 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. In this chapter, we'll look at how to apply test strategy to those math concepts.

There are many benefits to working with multiple-choice 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 multiple-choice 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 least odd integer greater than 26 ?

(A) 29

(B) 28

(C) 27.5

(D) 25

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 "odd" and "integer". You should always underline, highlight, or jot down (depending on your testing modality) important words from the questions so that you avoid careless errors. Eliminate (B) because 28 is even. 27.5 is not an integer, so eliminate (C) as well. 25 is odd, but it is not greater than 26, so eliminate (D). The correct answer is (A). Even though the least odd integer greater than 26 is 27, that's not an answer choice. The question is only asking about the numbers that are given to you.

2.Which of the following is NOT a factor of 36 and a multiple of 3 ?

(A) 36

(B) 18

(C) 4

(D) 3

Did you mark the words factor and multiple? Did you note that this is a NOT question? Eliminate the three answer choices that ARE factors of 36 and multiples of 3. A factor is a number that divides into a number, and a multiple is that number multiplied by something else. 36 is a factor of 36 since 36 × 1 = 36, and it is also a multiple of 3; eliminate (A). 18 × 2 = 36, so it is a factor of 36. 3 × 6 = 18, so it is a multiple of 3; eliminate (B). 4 × 9 = 36, so 4 is a factor of 36. However, 4 is not a multiple of 3. The correct answer is (C).

The Rules of Zero

3.If ab = 16 and cd = 0, and a, b, c, and d represent four distinct integers, which of the following must be true?

(A) a = 4

(B) c = 0

(C) d = 0

(D) abcd = 0

While a, c, and d each could equal what they do in the answer choices, they do not have to equal those amounts. However, if cd = 0, anything multiplied by 0 equals 0 as well. Therefore, abcd must equal 0. The correct answer is (D).

Remember the Rules of Zero

Zero is even. It’s neither + nor —, and anything multiplied by 0 = 0.

The Multiplication Table

4.Which of the following is equal to 8 × 4 × 3 ?

(A) 96 ÷ 3

(B) 32 + 12

(C) 4 × 4 × 6

(D) 24 × 2

Solve 8 × 4 × 3 from left to right. 8 × 4 = 32, and 32 × 3 = 96. Eliminate (A) because 96 ÷ 3 is definitely less. 32 + 12 is too small as well, so eliminate (B). 24 × 2 is also too small, so the correct answer must be (C). 4 × 4 = 16, and 16 × 6 = 96, but you didn't have to do that work thanks to POE!

The Case of the Mysteriously Missing Sign

If there is no operation sign between a number and a variable (letter), the operation is multiplication.

Working with Negative Numbers

5.10 — 12 is equivalent to

(A) 10 — (—12)

(B) 12 — 10

(C) 10 + (—12)

(D) —10 — 12

10 — 12 = —2. Remember that two negatives are the same as a positive, so (A) is the same as 10 + 12. Eliminate (A). 12 — 10 = 2, not —2, so eliminate (B) as well. Keep (C) because adding a negative number is the same as subtracting. Choice (D) equals —22, so eliminate this choice as well. The correct answer is (C).

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.

Order of Operations

6.14 + 18 ÷ 2 × 3 — 7 =

(A) —483

(B) —4

(C) 34

(D) 41

Remember PEMDAS, so tackle multiplication and division before addition and subtraction. Work left to right within each group, so 18 ÷ 2 = 9, and 9 × 3 = 27. Now the equation reads 14 + 27 — 7. Do the addition and subtraction from left to right as well: 14 + 27 = 41, and 41 — 7 = 34. The correct answer is (C).

Factors and Multiples

7.What is the difference of the distinct prime factors of 54 ?

(A) —5

(B) 1

(C) 3

(D) 11

First, note the key words "difference" and "distinct". Then find the distinct prime factors by breaking down 54 into a factor tree. 54 = 9 × 6. Break 9 down into 3 × 3, and 6 into 2 × 3. Therefore, 2 and 3 are the distinct prime factors of 54. Now, find the difference: 3 — 2 = 1. The correct answer is (B).

Factors Are Few; Multiples Are Many

The factors of a number are always equal to or less than that number and there’s a limited quantity of them. The multiples of a number are always equal to or greater than that number and there are infinitely many of them. Be sure not to confuse the two!

Fractions

8.Which of the following is greater than ?

(A)

(B)

(C)

(D)

Eliminate (A) because is equivalent to . Now try to eliminate others. Eliminate (D) because a larger denominator with the same numerator will be less, not greater. From here, either use the Bowtie method from Chapter 2 to find common denominators or convert the fractions to decimals. is equivalent to 0.75 and is equivalent to 0.8. Choice (B) is the correct answer.

Percents

9.A certain high school has 35 seniors, 40 juniors, 55 sophomores, and 70 freshmen. What percent of the students at this high school are juniors?

(A) 20%

(B) 25%

(C) 35%

(D) 40%

Start with some POE. There are 40 juniors, but more than 100 students, so the juniors are less than 40%. Eliminate (A). Now add all the students together to find the total, which will become the denominator of the fraction: 35 + 40 + 55 + 70 = 200. To find the percent of the juniors, or the part out of the whole, place the juniors in the numerator: . Simplify to , which is equal to 20%. The correct answer is (A).

Percent means out of 100, and the word of in a word problem tells you to multiply.

Exponents—Middle/Upper Levels Only

10.Which of the following is NOT equal to 2⁶?

(A) 2³ × 2³

(B) 4³

(C) 6³

(D) 8²

Note the NOT. Three of your answers will be equivalent to 2⁶ and to each other. This question is testing MADSPM. When multiplying the same base, add the exponents. A power raised to a power will result in multiplication. Choice (A) requires addition of the exponents since the bases are the same: 3 + 3 = 6, so 2³ × 2³ = 2⁶, so eliminate (A). Choice (B) does not have the same base as 2, but it is related. 2² = 4, and the expression can be rewritten (2²)³ = 2⁶. Eliminate (B). Since 6 is not a perfect square or cube of 2, leave (C). Since 8 can be rewritten as 2³, (D) can read (2³)² = 2⁶. Eliminate (D). The correct answer is (C).

Square Roots—Middle/Upper Levels Only

11.The square root of 40 falls between what two integers?

(A) Between 4 and 5

(B) Between 5 and 6

(C) Between 6 and 7

(D) Between 7 and 8

Think of perfect squares near 40. and = 7. Since 40 is between 36 and 49, must be between 6 and 7. If you have trouble with this one, try working backwards. As we discussed in Chapter 2, a square root is just the opposite of squaring a number. Using the numbers in the answer choices, 4² is 16, 5² is 25, 6² is 36, and 7² is 49. You can stop there because 40 is between 36 and 49. The correct answer is (C).

Basic Algebraic Equations—Middle/Upper Levels Only

12.—104 = 8n. What is the value of n ?

(A) —18

(B) —13

(C) 13

(D) 18

Divide both sides by 8. Since a negative divided by a positive is negative, eliminate (C) and (D). — = —13. If you find the equation confusing or you're better at multiplying than dividing, use the answer choices and work backwards. Each of the answer choices is a possible value for x. 8 × —13 = —104. The correct answer is (B).

Solve for Variable—Upper Level Only

13.5x — 14 = 8x + 4. Find the value of x.

(A) —6

(B) —3

(C) 4

(D) 6

To isolate x, manipulate the expressions by always doing the same thing to both sides. Use the opposite operation to “undo” portions of the equation to get x by itself. Subtract 5x from both sides to yield —14 = 3x + 4. To get x by itself, subtract 4 from both sides: —18 = 3x. Now, divide by 3 on both sides to get —6 = x. The correct answer is (A). Just as in question 12, if you get stuck, use the answer choices to work backwards.

Percent Algebra—Upper Level Only

14.15% of 50% of what is equal to 24 ?

(A) 36

(B) 80

(C) 160

(D) 320

Use translation to turn English into math here. Need to review? Go back to Chapter 2! Percent translates to ÷100, of translates into multiplication, a what is an unknown value, so . Simplify to find that . Continue to simplify to find that . Multiply both sides by 200 to find that 15x = 4,800. Now, divide by 15 on both sides to get x by itself to find that x = 320. Remember that you can also use the choices and work backwards. You might notice that (A) is too small, because 50% of 36 is 18, which is smaller than 24. So start with choice (C) in the middle. 50% of 160 is 80, and 15% of 80 is 12. Since that's still too small, cross out (B) and (C), leaving you with correct answer (D).

Geometry

15.BCDE is a rectangle with an area of 45. If the length of BC is 15, what is the perimeter of BCDE ?

(A) 120

(B) 90

(C) 60

(D) 36

If the area of the rectangle is 45 and one of its sides is 15, find the other side by using the area formula l × w = A. Therefore, 15 × w = 45, which means that w equals 3. Find the perimeter by adding all the sides: 15 + 15 + 3 + 3 = 36. The correct answer is (D).

16.If the perimeter of the polygon is 42, what is the value of x + y ?

(A) 6

(B) 13

(C) 14

(D) 15

Use the figure to guesstimate that x and y aren't dramatically smaller than the other sides, so cross out (A). Add all the sides to find the perimeter: 6 + 8 + 7 + x + 6 + y = 42. Simplify to find that 27 + x + y = 42. Subtract 27 from both sides to find that x + y = 15. The correct answer is (D).

Word Problems

17.Peyton is reading a book at a rate of 5 pages every 12 minutes. If he continues reading the book at this rate and finishes the book in 288 minutes, how many pages long is the book?

(A) 24

(B) 60

(C) 100

(D) 120

Start with some POE. Since Peyton reads 5 pages in 12 minutes, and 288 minutes is more than 10 times 12, it must take him longer than 50 minutes. Cross out (A). Now recognize that this is a proportion question because we have two sets of data we are comparing. Set up your fractions.

Remember that the units match on the top and on the bottom.

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 12 × 24 = 288, we must multiply 5 × 24 = 120.

So Peyton's book 5 × 24 = 120 pages long. The correct answer is (D).

18.Half of the crayons in a box of 40 have been used. Of these used crayons, of them are missing their paper wrapping labels. The rest of the used crayons still have their labels. How many used crayons still have a label?

(A) 5

(B) 10

(C) 15

(D) 20

Work through tedious word problems one piece of information at a time. First, find half of the crayons: half of 40 is 20. Of the 20, of them are missing their wrappings, so 20 × = 15. Therefore, 5 of the used crayons still have a label. The correct answer is (A).

19.Maddie has a bag of 24 marbles. She divides the marbles into 2 piles. Maddie keeps 1 pile and gives the other pile to her sister. If her sister’s pile has half as many marbles as Maddie’s pile, how many marbles does Maddie have?

(A) 6

(B) 8

(C) 12

(D) 16

Let the answer choices help! If Maddie has the bigger pile, she has more than half of the 24 marbles. (24) = 12, so Maddie must have more than 12. Cross out answer choices (A), (B), and (C). That leaves answer choice (D). If Maddie has 16 marbles, her sister would have 8, and 16 + 8 = 24. The correct answer is (D).

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 distinct factors does the number 16 have?

(A) 2

(B) 4

(C) 5

(D) 6

2.Which of the following contains all the common factors of 12 and 48 ?

(A) 1, 2, and 6

(B) 12 and 24

(C) 1, 2, 3, 4, and 6

(D) 1, 2, 3, 4, 6, and 12

3.Which of the following is a multiple of 7 ?

(A) 71

(B) 87

(C) 91

(D) 104

4.Which of the following is NOT a multiple of 8 ?

(A) 4

(B) 16

(C) 32

(D) 56

5.Which of the following is a multiple of both 4 and 6 ?

(A) 18

(B) 32

(C) 48

(D) 54

6.The number 1,026 is NOT divisible by which of the following?

(A) 2

(B) 4

(C) 6

(D) 9

7.The sum of three consecutive, odd, positive integers is 21. What is the square of the smallest of the three integers?

(A) 9

(B) 25

(C) 36

(D) 49

8.A company’s profit was $95,000 in 2008. In 2018, its profit was $570,000. The profit in 2018 was how many times as great as the profit in 2008 ?

(A) 4

(B) 6

(C) 8

(D) 10

9.Wenhao is playing a video game with 45 levels. If he has completed two-fifths of the levels, how many levels has Wenhao completed?

(A) 3

(B) 9

(C) 18

(D) 27

10.Alex’s fish tank is currently one-fourth full of water only. When at its maximum capacity, the tank holds 112 gallons of water. How many gallons of water are presently in the fish tank?

(A) 28

(B) 37

(C) 56

(D) 84

11.Jake stands five-eighths of every day he works. In a four-day period, Jake stands the equivalent of how many full workdays?

(A) 1.5

(B) 2

(C) 2.5

(D) 3

12.Which of the following has the greatest value?

(A)

(B)

(C)

(D)

13.

(A) 2

(B)

(C) 3

(D)

14.The product of 0.027 and 10,000 is approximately

(A) 2.7

(B) 27

(C) 270

(D) 2,700

15.1.576 =

(A)

(B)

(C)

(D)

When You Are Done

Don’t forget to check your answers in Chapter 17, this page.

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 right all the questions you find easy or sort-of easy, 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 cross off wrong answers in your test booklet, on your scratch paper, or using the strikeout function on the screen 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 numbers)?

Write It Down

Did you work out the practice questions on paper? Did you move too quickly or skip steps on problems you found easier? Did you always double-check 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 double-check 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 test-taking techniques we’ve talked about: guessing, POE, order of difficulty, pacing, choosing a letter-of-the-day 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 10 and 150 inclusive are multiples of both 3 and 5 ?

(A) 9

(B) 10

(C) 15

(D) 20

5.3⁴ × 3⁴ × 3⁴ =

(A) 3⁸

(B) 3¹²

(C) 3⁶⁴

(D) 3(3⁴)

6.For what integer value of x does x⁴ = 4x + 8 ?

(A) 1

(B) 2

(C) 3

(D) 4

7.If 18 = 3(3x — 6), then x + 6 =

(A) 4

(B) 6

(C) 10

(D) 26

8.What is the largest multiple of 6 that is less than 53 ?

(A) 36

(B) 48

(C) 52

(D) 54

9.One-third of the cars available for purchase at a used car dealership are silver. If there are 24 silver cars at the dealership, how many used cars are available for purchase?

(A) 8

(B) 24

(C) 48

(D) 72

10.Marie’s garden contains 30 yellow roses, 50 red roses, and 40 white roses. Of the roses in her garden, what percent are NOT yellow?

(A) 30%

(B) 50%

(C) 75%

(D) 90%

11.A bookstore sells a signed copy of a particular book for $55 and an unsigned copy for $40. By approximately what percent is the signed copy marked up?

(A) 27%

(B) 30%

(C) 38%

(D) 45%

12.Which of the following is closest to 75% of $49.95 ?

(A) $7.50

(B) $12.50

(C) $37.00

(D) $40.00

13.Candace normally scores 25 points in a basketball game. During the last game of her season, she scored only 15 points. What is the percent change in the number of points she scored?

(A) 10%

(B) 40%

(C) 60%

(D) 67%

14.The dues to enter a tournament are $24. A team with three participants will split the tournament dues evenly. If they add a fourth person to the team and still split the dues evenly, how much will each participant then pay?

(A) $2

(B) $6

(C) $8

(D) $32

15.If a regular pentagon has a perimeter of 65, what is the length of each side?

(A) 5

(B) 11

(C) 13

(D) 16

16.What is the perimeter of a right triangle with legs that measure 3 cm and 4 cm?

(A) 12 cm

(B) 10 cm

(C) 5 cm

(D) 4 cm

When You Are Done

Don’t forget to check your answers in Chapter 17, this page.

More Practice—Upper Level Only

17.If b = 60, then v =

(A) 8

(B) 5

(C)

(D) 4

18.What is one-fourth of the difference between the number of degrees in a rectangle and the number of degrees in a triangle?

(A) 45

(B) 90

(C) 120

(D) 180

19.If one-half the perimeter of a square is equal to its area, what is the length of one side?

(A) 1

(B) 2

(C) 4

(D) 8

20.The area of a circle with a radius of 3 is equal to the circumference of a circle with a diameter of

(A) 2

(B) 4

(C) 6

(D) 9

21.Two right circular cylinders have equal volumes. The formula for the volume of a cylinder is V = πr²h, where r is the radius of the cylinder and h is its height. The cylinder has a radius of 3 and a height of 4. If the other cylinder has a radius of 6, what is its height?

(A) 1

(B) 2

(C) 4

(D) 8

22.If the perimeter of a square is 36n², what is the length of one side?

(A) 6n

(B) 9n

(C) 6n²

(D) 9n²

23.If AB = 12 and AC = 20, what is the perimeter of the figure above?

(A) 32

(B) 44

(C) 52

(D) 64

24.If ABCD is a rectangle, and x = 70, what is the value of y° + z° — w° ?

(A) 20°

(B) 70°

(C) 90°

(D) 110°

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 fencing are needed to surround a yard that measures 32 meters wide by 28 meters long?

(A) 60 meters

(B) 120 meters

(C) 448 meters

(D) 896 meters

27.What is the slope of a line that is perpendicular to line segment AB ?

(A)

(B)

(C)

(D)

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) 117°

(B) 100°

(C) 95°

(D) 46°

30.What is the perimeter of this figure if ABC is an equilateral triangle?

(A) 6 + 3π

(B) 6 + 6π

(C) 12 + 3π

(D) 12 + 6π

31.If MNPQ is a square, what is the area of the trapezoid?

(A) 48

(B) 64

(C) 88

(D) 112

When You Are Done

Don’t forget to check your answers in Chapter 17, this page.

Now we're going to transition into concepts that go beyond what was covered in the Fundamentals (Chapter 2).

RATIOS—LOWER, MIDDLE, AND UPPER LEVELS

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 people wearing purple shirts to people wearing yellow shirts 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 Anna’s Ice Cream Shop, ice cream cones are sold as single, double, or triple scoops. The ratio of single to double to triple scoops sold on a particular Saturday was 3:5:1. If the total number of cones sold was 45, how many double scoops were sold?

(A) 5

(B) 15

(C) 20

(D) 25

2.In Jade’s class, 14 of the 26 students have brown eyes. What is the ratio of the students in Jade’s class with brown eyes to students with another eye color?

(A) 7:13

(B) 6:7

(C) 7:6

(D) 13:12

3.Casper runs 3 miles every day and swims 2 miles every day. When Casper has completed a total of 25 miles, how many miles did he swim?

(A) 5

(B) 10

(C) 12.5

(D) 15

Stop. Check your time for this drill:

When You Are Done

Don’t forget to check your answers in Chapter 17, this page.

PLUGGING IN—LOWER, MIDDLE, AND UPPER LEVELS

The ISEE will often ask you questions about real-life 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.Draw 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

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

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.

If 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!

Don’t forget to check every answer choice when Plugging In!

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.

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.

PRACTICE DRILL 4—PLUGGING IN

1.At a charity fund-raiser, 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

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

When You Are Done

Don’t forget to check your answers in Chapter 17, this page.

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

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) 6^y

(D) 6 + 6y

5.If p pieces of candy cost a total of 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. Middle and Upper level students should keep on drilling!

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, Sarolta ate one-half of a fruit tart. On Tuesday, Sarolta then ate one-fourth of what was left of the tart. What fraction of the tart did Sarolta 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) p² + 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) m³ is even.

12.Antonia has twice as many baseball cards as Krissi, who has one-third as many baseball cards as Ian. If Krissi has k baseball cards, how many baseball cards do Antonia and Ian have together?

(A)

(B)

(C)

(D)

13.The product of b and a² can be written as

(A) (ab)²

(B)

(C)

(D)

14.x^a = (x³)³

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 3x²y⁵?

(A) xy

(B) 3x²y⁵

(C) 3x²y³

(D) 27x³y³

When You Are Done

Don’t forget to check your answers in Chapter 17, this page.

PLUGGING IN THE ANSWERS (PITA)—LOWER, MIDDLE, AND UPPER LEVELS

Plugging In the Answers (or PITA) 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 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.

Jayden baked a batch of cookies. He gave half to his friend Imani and six to his mother. If he now has eight cookies left, how many did Jayden 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 Jayden baked originally must be either 8, 12, 20, or 28 (the four choices). So pick one—start in the middle with either (B) or (C)—and then work backwards to determine whether you have the right choice.

Let’s start with (C): Jayden baked 20 cookies. Now work through the events listed in the question. He had 20 cookies and he gave half to Imani. That leaves Jayden with 10 cookies. Then, he gave 6 to his mom. Now he’s got 4 left.

Keep going. The problem says that Jayden now has 8 cookies left. But if he started with 20—(C)—he would have only 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), Jayden ended up with fewer cookies than we wanted him to have, didn’t he? 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.

Jayden started out with 28 cookies. The first thing he did was give half, or 14, to Imani. That left Jayden with 14 cookies. Then he gave 6 cookies to his mother. 14 — 6 = 8. Jayden has eight cookies left over. Keep going with the question. It says, “If he now has eight cookies left….” He has 8 cookies left and, voilà—he’s supposed to have 8 cookies left.

What does this mean? It means you’ve got the right answer!

PRACTICE DRILL 5—PLUGGING IN THE ANSWERS

Remember to time yourself during this drill!

1.Tina can read 60 pages per hour. Nick can read 45 pages per hour. If both Tina and Nick 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—Abigail, Emiko, and Diego—want to put their money together to buy a $90 radio. If Emiko agrees to pay twice as much as Diego, and Abigail agrees to pay three times as much as Emiko, how much must Emiko pay?

(A) $10

(B) $20

(C) $30

(D) $45

3.Four less than a certain number is two-thirds of that number. What is the number?

(A) 1

(B) 6

(C) 8

(D) 12

More Practice: Lower Level

4.There are 12 more cat-lovers than dog-lovers in a club. If there are 30 total club members, and every club member is either a cat-lover or a dog-lover (but not both), how many cat-lovers are there in the club?

(A) 9

(B) 12

(C) 20

(D) 21

5.Victoria, Jonathan, and Russell buy a home theater system. Victoria pays twice as much as Jonathan, and Victoria pays half as much as 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.Anshuman is half as old as Brinda and three times as old as Cindy. If the sum of their ages is 40, what is Brinda’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 buying a video game. 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

When You Are Done

Don’t forget to check your answers in Chapter 17, this page.

GEOMETRY—LOWER, MIDDLE, AND UPPER LEVELS

Guesstimating: A Second Look

Guesstimating worked well back in the introduction when we were just using it to estimate 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

Lower 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!

Not necessarily. Look at the picture. What fraction of the square looks like it is shaded? Half? Three-quarters? Less than half? It looks like about one-quarter of the area of the square is shaded. You’ve just done most of the work necessary to solve this problem.

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

Try these values when guesstimating:

π ≈ 3+

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

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.

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 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

Start by labeling your figure with the information given if you're working in a test booklet, or by drawing the figure on your scratch paper. The next 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 need to 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 the midpoints of 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 — 4π

(C) 8 + 4π

(D) 4π

When You Are Done

Don’t forget to check your answers in Chapter 17, this page.

FUNCTIONS—MIDDLE AND UPPER LEVELS ONLY

Remember that functions are just a set of directions. Need to review? Go back to Chapter 2!

PRACTICE DRILL 6—FUNCTIONS

Remember to time yourself during this drill!

1.Let b* = 2b + 7. What is the value of 5* ?

(A) —1

(B) 6

(C) 14

(D) 17

2.If ¿n¿ = 4n — 4 and ¿n¿ = 20, what is the value of n?

(A) 4

(B) 6

(C) 20

(D) 76

3.If a∆b = 4a + 3b, then 3∆b =

(A) 4a + 9

(B) 7 + 3b

(C) 12 + b²

(D) 12 + 3b

4.In the three-digit number, 3H8, H represents a digit. If 3H8 is divisible by 3, which of the following could be H ?

(A) 2

(B) 3

(C) 5

(D) 7

5.For any integer c, let = 2c + c(c + 3)

What is ?

(A) 24

(B) 84

(C) 126

(D) 150

When You Are Done

Don’t forget to check your answers in Chapter 17, this page.

QUANTITATIVE COMPARISON—MIDDLE AND UPPER LEVELS ONLY

Quant Comp: Same Book, Different Cover

Quantitative comparison (or "quant comp") is a type of question—one slightly different from the traditional multiple-choice 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, and only in the Quantitative Reasoning section (Section 2).

Lower Level Test Takers

The ISEE’s Primary and Lower Level tests do 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 in the test booklet, we suggest that you write “A B C D” next to each question. Online test takers will see the answer choices on the screen with each question. Regardless of modality, when you know you can eliminate an answer, cross it off.

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” multiple-choice 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.

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.

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.

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:

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.

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).

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).

When a quant comp question contains no variables and no unknown quantities, the answer cannot be (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 7—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

When You Are Done

Don’t forget to check your answers in Chapter 17, this page.

QUANT COMP PLUGGING IN—MIDDLE AND UPPER LEVELS ONLY

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 quant comp part of the Quantitative Reasoning section.

Follow these three simple steps, and you won’t go wrong.

Step 1: If you're taking a paper-based test, write “A B C D” next to the problem. Online testers, get ready to use the strikeout tool.

Step 2: Plug in an “easy” number for x. By easy number, we mean a nice simple integer, such as 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: Now 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).

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 with variables.

Weird Numbers

For your second Plug In, use ZONE F to remind yourself of "weird" options:

Zero

One

Negative

Extreme

Fraction

PRACTICE DRILL 8—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

When You Are Done

Don’t forget to check your answers in Chapter 17, this page.

PRACTICE DRILL 9—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!

(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

When You Are Done

Don’t forget to check your answers in Chapter 17, this page.

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 ?

12.2³ =

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 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: ________________.

When You Are Done

Don’t forget to check your answers in Chapter 17, this page.