Cracking the SSAT & ISEE - The Princeton Review 2019

Fundamental math skills for the SSAT & ISEE
Part i the basics of both tests

INTRODUCTION

Whether you are taking the Lower Level ISEE or the Upper Level SSAT, there are some basic math skills that are at the heart of many of the questions on your test. If you are taking the Lower or Elementary Level exams, the content in this section may be something you learned recently or are learning right now. You should go through this chapter very carefully and slowly. If you are having trouble understanding any of the content, you should ask your parents or teachers for help by having them explain it more thoroughly to you. If you are taking the Middle and Upper Level tests, this chapter may serve more as a chance to review some things you may have forgotten or that you need to practice a little. Even the most difficult-seeming questions on the Upper Level exams are built on testing your knowledge of these same skills. Make sure you read the explanations and do all of the drills before going on to either the SSAT or ISEE math chapter. Answers to these drills are provided in Chapter 3.

A Note to Lower, Elementary, and Middle Level Students

This chapter has been designed to give all students a comprehensive review of the math found on the tests. There are four sections: “The Building Blocks,” “Algebra,” “Geometry,” and “Word Problems.” At the beginnings and ends of some of these sections, you will notice information about what material you should review and what material is only for Upper Level (UL) students. Be aware that you may not be familiar with all the topics on which you will be working. If you are having difficulty understanding a topic, take this book to your teachers or parents and ask them for additional help.

Lose Your Calculator!

You will not be allowed to use a calculator on the SSAT or 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 take it out again after the test is behind you. Do your math 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 adding just a few numbers together, write them down and do the work on paper. Writing things down not only helps to eliminate careless errors but also gives you something to refer back to if you need to double-check your work.

Write It Down; Get It Right!

You don’t get points for doing the math in your head, so don’t do it!

THE BUILDING BLOCKS

Math Vocabulary

The Rules of Zero

Zero has some funny rules. Make sure you understand and remember these rules.

·  Zero is neither positive nor negative.

·  Zero is even.

·  Zero is an integer.

·  Zero multiplied by any number is zero.

·  Zero divided by any number is zero.

·  You cannot divide by zero (9 ÷ 0 = undefined).

The Times Table

Make sure you are comfortable with your multiplication tables up to 12. If you are having trouble with these, break out the flash cards. On one side of the card write down the multiplication problem, and on the other write down the answer. Now quiz yourself. You may also want to copy the table shown below so you can practice. For handy tips on using flash cards effectively, turn to the vocabulary chapter and read the section on flash cards.

Elementary Level

If you haven’t learned the Times Table yet, this is a great opportunity to get ahead of your classmates!

PRACTICE DRILL 1—MATH VOCABULARY

1.How many integers are there between —1 and 6 ?

2.List three consecutive odd integers.

3.How many odd integers are there between 1 and 9 ?

4.What is the tens digit in the number 182.09 ?

5.The product of any number and the smallest positive integer is

6.What is the product of 5, 6, and 3 ?

7.What is the sum of 3, 11, and 16 ?

8.What is the difference between your answer to question 6 and your answer to question 7 ?

9.List three consecutive negative even integers:

10.Is 11 a prime number?

11.What is the sum of the digits in the number 5,647 ?

12.What is the remainder when 58 is divided by 13 ?

13.55 is divisible by what numbers?

14.The sum of the digits in 589 is how much greater than the sum of the digits in 1,207 ?

15.Is 21 divisible by the remainder of 19 ÷ 5 ?

16.What are the prime factors of 156 ?

17.What is the sum of the odd prime factors of 156 ?

18.12 multiplied by 3 is the same as 4 multiplied by what number?

19.What are the factors of 72 ?

20.How many factors of 72 are even? How many are odd?

When You Are Done

Check your answers in Chapter 3.

Working with Negative Numbers

It is helpful to think of numbers as having two component parts: the number itself and the sign in front of it (to the left of the number). Numbers that don’t have signs immediately to the left of them are positive. So +7 can be, and usually is, written as 7.

Adding

If the signs to the left of the numbers are the same, you add the two numbers and keep the same sign. For example:

2 + 5 = (+2) + (+5) = +7 or just plain 7

(—2) + (—5) = —7

If the signs to the left of the numbers are different, you subtract the numbers and the answer takes the sign of the larger number. For example:

5 + (—2) = 5 — 2 = 3, and because 5 is greater than 2, the answer is +3 or just plain 3.

(—2) + 5 = 5 — 2 = 3, and because 5 is greater than 2, the answer is +3 or just plain 3.

(—5) + 2 = 5 — 2 = 3, and because 5 is greater than 2, you use its sign and the answer is —3.

Subtracting—Middle and Upper Levels only

All subtraction problems can be converted to addition problems. This is because subtracting is the same as adding the opposite. “Huh?” you say—well, let’s test this out on something simple that you already know. We know that 7 — 3 = 4, so let’s turn it into an addition problem and see if we get the same answer.

7 — 3 = (+7) — (+3)

Okay, so now we reverse only the operation sign and the sign of the number we are subtracting (the second number). The first number stays the same because that’s our starting point.

(+7) + (—3)

Now use the rules for addition to solve this problem. Because the signs to the left are different, we subtract the two numbers 7 — 3 = 4, and the sign is positive because 7 is greater than 3.

This is just one way to look at subtraction problems. If you have a way that works better for you, use that!

We have just proven that subtraction problems are really just the opposite of addition problems. Now let’s see how this works in a variety of examples.

3 — 7 = (+3) — (+7) = (+3) + (—7) = 7 — 3 = 4 and, because 7 is greater than 3, the answer is —4.

—9 — 3 = (—9) — (+3) = (—9) + (—3) = —12

13 — (—5) = (+13) — (—5) = (+13) + (+5) = +18

(—5) — (—8) = (—5) + (+8) = +3

PRACTICE DRILL 2—ADDING AND SUBTRACTING NEGATIVE NUMBERS

1.6 + (—14) =

2.13 — 27 =

3.(—17) + 13 =

4.12 — (—15) =

5.16 + 5 =

6.34 — (+30) =

7.(—7) + (—15) =

8.(— 42) + 13 =

9.—13 — (—7) =

10.151 + (—61) =

11.(— 42) — (— 42) =

12.5 — (—24) =

13.14 + 10 =

14.(—5) + (—25) =

15.11 — 25 =

When You Are Done

Check your answers in Chapter 3.

Multiplying and Dividing

The rules for multiplying and dividing positive and negative integers are so much easier to learn and use than the rules for adding and subtracting them. You simply multiply or divide as normal, and then determine the sign using the rules below.

Positive (÷ or ×) Positive = Positive

Negative (÷ or ×) Negative = Positive

Positive (÷ or ×) Negative = Negative

Negative (÷ or ×) Positive = Negative

Here are some examples.

6 ÷ 2 = 3

(—6) ÷ (—2) = 3

6 ÷ (—2) = —3

(—6) ÷ 2 = —3

2 × 6 = 12

(—2) × (—6) = 12

2 × (—6) = —12

(—2) × 6 = —12

Helpful Rule of Thumb

When multiplying numbers, simply count the number of negative signs. An even number of negative signs (—6 × —3) means that the product must be a positive number. An odd number of negative signs (2 × —5) means that the product must be negative.

If you are multiplying more than two numbers, simply work from left to right and deal with the numbers two at a time.

2 × (—5) × (—10) = 2 × (—5) = —10 and now (—10) × (—10) = +100

PRACTICE DRILL 3—MULTIPLYING AND DIVIDING NEGATIVE NUMBERS

1.20 ÷ (—5) =

2.(—12) × 3 =

3.(—13) × (—5) =

4.(— 44) ÷ (— 4) =

5.7 × 9 =

6.(—65) ÷ 5 =

7.(—7) × (—12) =

8.(—10) ÷ 2 =

9.81 ÷ 9 =

10.32 ÷ (—4) =

11.25 × (—3) =

12.(—24) × (—3) =

13.64 ÷ (—16) =

14.(—17) × (—2) =

15.(—55) ÷ 5 =

When You Are Done

Check your answers in Chapter 3.

ORDER OF OPERATIONS

How would you attack this problem?

16 — 45 ÷ (2 + 1)² × 4 + 5 =

To solve a problem like this, you need to know which mathematical operation to do first. The way to remember the order of operations is to use PEMDAS.

You can remember the order of operations by using the phrase below:

“Please Excuse My Dear Aunt Sally”

Now, let’s give it a try.

16 — 45 ÷ (2 + 1)² × 4 + 5 =

1.Parentheses:

2.Exponents:

3.Multiplication and division (from left to right):

First Thing’s First

Make sure you remember PEMDAS whenever you see a question with more than one operation.

4.Addition and subtraction (from left to right):

Just take it one step at a time and you’ll be able to do it in no time at all!

PRACTICE DRILL 4—ORDER OF OPERATIONS

1.10 — 3 + 2 =

2.15 + (7 — 3) — 3 =

3.3 × 2 + 3 ÷ 3 =

4.2 × (4 + 6)² ÷ 4 =

5.420 ÷ (10 + 5 × 12) =

6.20 × 5 ÷ 10 + 20 =

7.3 + 5 × 10 × (7 — 6) ÷ 2 — 4 =

8.10 × (8 + 1) × (3 + 1) ÷ (8 — 2) =

9.12 + (5 × 2)² — 33 ÷ 3 =

10.200 — 150 ÷ 3 × 2³ =

When You Are Done

Check your answers in Chapter 3.

Factors

Factors are all the numbers that divide evenly into your original number. For example, 2 is a factor of 10; it goes in 5 times. However, 3 is not a factor of 10 because 10 divided by 3 does not produce an integer quotient (and therefore does not “go in evenly”). When asked to find the factors of a number, just make a list.

The factors of 16 are

1 and 16 (always start with 1 and the original number)

2 and 8

4 and 4

The factors of 18 are

1 and 18

2 and 9

3 and 6

Knowing some of the rules of divisibility can save you some time.

Larger Factors

There’s a quick way to figure out if a number is divisible by larger numbers. Simply take the two smaller factors and check both. If a number is divisible by both 2 and 3, then it’s divisible by 6. If a number is divisible by both 3 and 4, then it’s divisible by 12.

Factor Trees

To find the prime factors of a number, draw a factor tree.

Start by writing down the number and then drawing two branches from the number. Write down any pair of factors of that number. Now if one (or both) of the factors is not prime, draw another set of branches from that factor and write down a pair of factors for that number. Continue until you have only prime numbers at the end of your branches. Each branch end is a prime factor. Remember, 1 is NOT prime!

What are the distinct prime factors of 56? Well, let’s start with the factor tree.

The prime factors of 56 are 2, 2, 2, and 7. Because the question asked for only the distinct prime factors, we have to eliminate the numbers that repeat, so we cross out two of the twos. The distinct prime factors of 56 are 2 and 7.

Multiples

Multiples are the results when you multiply your number by any integer. The number 15 is a multiple of 5 because 5 times 3 equals 15. On the other hand, 18 is a multiple of 3, but not a multiple of 5. Another way to think about multiples is to consider them “counting by a number.”

The first seven positive multiples of 7 are:

7 (7 × 1)

14 (7 × 2)

21 (7 × 3)

28 (7 × 4)

35 (7 × 5)

42 (7 × 6)

49 (7 × 7)

PRACTICE DRILL 5—FACTORS AND MULTIPLES

1.List the first five multiples of:

2

4

5

11

2.Is 15 divisible by 3 ?

3.Is 81 divisible by 3 ?

4.Is 77 divisible by 3 ?

5.Is 23 prime?

6.Is 123 divisible by 3 ?

7.Is 123 divisible by 9 ?

8.Is 250 divisible by 2 ?

9.Is 250 divisible by 5 ?

10.Is 250 divisible by 10 ?

11.Is 10 a multiple of 2 ?

12.Is 11 a multiple of 3 ?

13.Is 2 a multiple of 8 ?

14.Is 24 a multiple of 4 ?

15.Is 27 a multiple of 6 ?

16.Is 27 a multiple of 9 ?

17.How many numbers between 1 and 50 are multiples of 6 ?

18.How many even multiples of 3 are there between 1 and 50 ?

19.How many numbers between 1 and 100 are multiples of both 3 and 4 ?

20.What is the greatest multiple of 3 less than 50 ?

When You Are Done

Check your answers in Chapter 3.

Fractions

A fraction really just tells you to divide. For instance, actually means five divided by eight (which equals 0.625 as a decimal).

Another way to think of this is to imagine a pie cut into eight pieces. represents five of those eight pieces of pie.

The parts of a fraction are called the numerator and the denominator. The numerator is the number on top of the fraction. It refers to the portion of the pie, while the denominator is on the bottom of the fraction and tells you how many pieces there are in the entire pie.

Reducing Fractions

Imagine a pie cut into two big pieces. You eat one of the pieces. That means that you have eaten of the pie. Now imagine the same pie cut into four pieces; you eat two. That’s this time. But look: The two fractions are equivalent!

Remember!

As the denominator gets smaller, the fraction gets bigger. After all, would you rather have of a cake or of one?

To reduce fractions, simply divide the top number and the bottom number by the same amount. Start out with small numbers like 2, 3, 5, or 10 and reduce again if you need to.

In this example, if you happened to see that both 12 and 24 are divisible by 12, then you could have saved two steps. However, don’t spend very much time looking for the largest number possible by which to reduce a fraction. Start out with a small number; doing one extra reduction doesn’t take very much time and will definitely help prevent careless errors.

PRACTICE DRILL 6—REDUCING FRACTIONS

1. =

2. =

3. =

4. =

5. =

6. =

7. =

8. =

9.What does it mean when the number on top is larger than the one on the bottom?

When You Are Done

Check your answers in Chapter 3.

Improper Fractions and Mixed Numbers

Elementary Level

You may not see this topic.

Changing from Improper Fractions to Mixed Numbers

If you knew the answer to number 9 in the last drill or if you looked it up, you now know that when the number on top is greater than the number on the bottom, the fraction is greater than 1. That makes sense, because we also know that a fraction bar is really just another way of telling us to divide. So, is the same as 10 ÷ 2, which equals 5, which is much greater than 1!

A fraction that has a greater numerator than denominator is called an improper fraction. You may be asked to change an improper fraction to a mixed number. A mixed number is an improper fraction that has been converted into a whole number and a proper fraction. To do this, let’s use as the improper fraction that we are going to convert to a mixed number.

First, divide 10 by 8. This gives us our whole number. 8 goes into 10 once.

Now, take the remainder, 2, and put it over the original fraction’s denominator: .

So the mixed number is 1, or 1.

Put Away That Calculator!

Remember that a remainder is just the number left over after you do long division; it is not the decimal that a calculator gives you.

PRACTICE DRILL 7—CHANGING IMPROPER FRACTIONS TO MIXED NUMBERS

1. =

2.=

3.=

4.=

5.=

6.=

7.=

8.=

9.=

10.=

When You Are Done

Check your answers in Chapter 3.

Changing Mixed Numbers to Improper Fractions

It’s important to know how to change a mixed number into an improper fraction because it may be easier to add, subtract, multiply, or divide a fraction if there is no whole number in the way. To do this, multiply the denominator by the whole number and then add the result to the numerator. Then put this sum on top of the original denominator. For example:

1

Multiply the denominator by the whole number: 2 × 1 = 2

Add this to the numerator: 2 + 1 = 3

Put this result over the original denominator:

1 =

PRACTICE DRILL 8—CHANGING MIXED NUMBERS TO IMPROPER FRACTIONS

1.6 =

2.2 =

3.23 =

4.6 =

5.7 =

6.7 =

7.10 =

8.5 =

9.4 =

10.33 =

When You Are Done

Check your answers in Chapter 3.

Adding and Subtracting Fractions with a Common Denominator

To add or subtract fractions with a common denominator, just add or subtract the top numbers and leave the bottom numbers alone.

Adding and Subtracting Fractions When the Denominators Are Different

In the past, you have probably tried to find common denominators so that you could just add or subtract straight across. There is a different way; it is called the Bowtie.

No More “Least Common Denominators”

Using the Bowtie to add and subtract fractions eliminates the need for the least common denominator, but you may need to reduce the result.

This diagram may make the Bowtie look complicated. It’s not. There are three steps to adding and subtracting fractions.

Step 1:Multiply diagonally going up.

First B × C. Write the product next to C.

Then D × A. Write the product next to A.

Step 2:Multiply straight across the bottom, B × D.

Write the product as the denominator in your answer.

Step 3:To add, add the numbers written next to A and C.

Write the sum as the numerator in your answer.

To subtract, subtract the numbers written next to A and C. Write the difference as the numerator in your answer.

PRACTICE DRILL 9—ADDING AND SUBTRACTING FRACTIONS

1. + =

2. + =

3. + =

4. — =

5. + =

6. — =

7. + =

8. — =

9. + =

(Upper Level)

10. + =

11. + =

12. — =

When You Are Done

Check your answers in Chapter 3.

Multiplying Fractions—Middle and Upper Levels only

Multiplying can be a pretty simple thing to do with fractions. All you need to do is multiply straight across the tops and bottoms.

Dividing Fractions—Middle and Upper Levels only

Dividing fractions is almost as simple as multiplying. You just have to flip the second fraction and then multiply.

Dividing fractions can be easy as pie; just flip the second fraction and multiply.

Remember Reciprocals?

A reciprocal results when you flip a fraction—that is, exchange the numerator and the denominator. So the reciprocal of is what? Yep, that’s right: .

PRACTICE DRILL 10—MULTIPLYING AND DIVIDING FRACTIONS

1. × =

2. ÷ =

3. × =

4. × =

5. ÷ =

When You Are Done

Check your answers in Chapter 3.

Decimals

Remember, decimals and fractions are just two different ways of writing the same thing. To change a fraction into a decimal, you just divide the bottom number into the top number.

Be sure you know the names of all the decimal places. Here’s a quick reminder.

Adding Decimals

To add decimals, just line up the decimal places and add.

Subtracting Decimals

To subtract, do the same thing. Line up the decimal places and subtract.

Multiplying Decimals—Middle and Upper Levels only

To multiply decimals, first count the number of digits to the right of the decimal point in the numbers you are multiplying. Then, multiply and, on the product, count that same number of spaces from right to left—this is where you put the decimal point.

(two digits to the right of the decimal point)

Dividing Decimals—Middle and Upper Levels only

To divide, move the decimal points in both numbers the same number of spaces to the right until you are working only with integers.

Now move both decimals over two places and solve the problem.

And you’re done! Remember: you do not put the decimals back into the problem.

PRACTICE DRILL 11—DECIMALS

1.1.43 + 17.27 =

2.2.49 + 1.7 =

3.7 — 2.038 =

4.4.25 × 2.5 =

5.0.02 × 0.90 =

6.180 ÷ 0.03 =

7.0.10 ÷ 0.02 =

When You Are Done

Check your answers in Chapter 3.

Converting Fractions to Decimals and Back Again

From Fractions to Decimals

As we learned when we introduced fractions a little earlier, a fraction bar is really just a division sign.

is the same as 10 ÷ 2, or 5

In the same sense:

= 1 ÷ 2, or 0.5

In fact, we can convert any fraction to its decimal equivalent by dividing the top number by the bottom number:

= 11 ÷ 2 = 5.5

From Decimals to Fractions

To change a decimal to a fraction, look at the digit furthest to the right. Determine what place that digit is in (e.g., tenths, hundredths, and so on) and then put the decimal (without the decimal point) over that number (e.g., 10, 100, and so on). Let’s change 0.5 into a fraction.

5 is in the tenths place so we put it over 10.

reduces to

PRACTICE DRILL 12—CONVERTING FRACTIONS TO DECIMALS AND BACK AGAIN

Fill in the table below by converting the fractions to decimals and vice versa. The fractions and decimals in this table are those most often tested on the SSAT and ISEE, so memorize them now and save yourself time later.

When You Are Done

Check your answers in Chapter 3.

Percents—Middle and Upper Levels Only

Percentages are really just an extension of fractions. Let’s go back to that pie we were talking about in the section on fractions. Let’s say we had a pie that was cut into four equal pieces. If you ate one piece of the pie, then we could say that the fractional part of the pie that you have eaten is:

Now let’s find out what percentage of the pie you have eaten. Percent literally means “out of 100.” When we find a percent, we are really trying to see how many times out of 100 something happens. To determine the percent, you simply take the fractional part and multiply it by 100.

You’ve probably seen percents as grades on your tests in school. What does it mean to get 100% on a test? It means you got every question correct. Let’s say you got 25 questions right out of a total of 25. So we put the number of questions you got right over the total number of questions and multiply by 100.

Let’s says that your friend didn’t do as well on this same test. He answered 20 questions correctly. Let’s figure out the percentage of questions he got right.

What percentage did he get wrong?

Notice that the percentage of questions he got right (80%) plus the percentage of questions he got wrong (20%) equals 100%.

PRACTICE DRILL 13—PERCENTS

1.A bag of candies contains 15 butterscotches, 20 caramels, 5 peppermints, and 10 toffees.

a.)The butterscotches make up what percentage of the candies?

b.)The caramels?

c.)The peppermints?

d.)The toffees?

2.A student answered 75% of the questions on a test correctly and left 7% of the questions blank. What percentage of the questions did the student answer incorrectly?

3.Stephanie’s closet contains 40 pairs of shoes. She has 8 pairs of sneakers, 12 sets of sandals, 16 pairs of boots, and the rest are high heels.

a.)What percentage of the shoes are sneakers?

b.)Sandals?

c.)Boots?

d.)High heels?

e.)How many high heels does Stephanie own?

4.A recipe for fruit punch calls for 4 cups of apple juice, 2 cups of cranberry juice, 3 cups of grape juice, and 1 cup of seltzer. What percentage of the punch is juice?

5.Five friends are chipping in for a birthday gift for their teacher. David and Jakob each contribute $13. Stephanie, Kate, and Janice each contribute $8.

a.)What percentage of the total did the girls contribute?

b.)The boys?

When You Are Done

Check your answers in Chapter 3.

More Percents—Middle and Upper Levels Only

Another place you may have encountered percents is at the shopping mall. Stores offer special discounts on their merchandise to entice shoppers to buy more stuff. Most of these stores discount their merchandise by a certain percentage. For example, you may see a $16 shirt that is marked 25% off the regular price. What does that mean?

Percents are not “real” numbers. In the above scenario, the shirt was not $25 less than the regular price (then they’d have to pay you money!), but 25% less. So how do we figure out how much that shirt really costs and how much money we are saving?

To find how much a percent is in “real” numbers, you need to first take the percent and change it to a fraction.

Because percent means “out of 100,” to change a percent to a fraction, simply put the percent over 100.

25% = =

Now let’s get back to that shirt. Multiply the regular price of the shirt, $16, by the fraction.

$16 × = $4

This means 25% of 16 is $4. You get $4 off the original price of the shirt. If you subtract that from the original price, you find that the new sale price is $12.

Guess what percentage the sale price is of the regular price? If you said 75 percent, you’d be right!

Tip

Changing a decimal to a percent is the same as changing a fraction to a percent. Multiply the decimal by 100 (move the decimal two spaces to the right). So 0.25 as a percent is 0.25 × 100 = 25%.

PRACTICE DRILL 14—MORE PERCENTS

Fill in the missing information in the table below.

Tip:

The word of in word problems means multiply!

1.25% of 84 =

2.33% of 27 =

3.20% of 75 =

4.17% of 300 =

5.16% of 10% of 500 =

6.A dress is marked down 15% from its regular price. If the regular price is $120, what is the sale price of the dress? The sale price is what percentage of the regular price of the dress?

7.Steve goes to school 80% of the 365 days of the year. How many days does Steve go to school?

8.Jennifer answered all 36 questions on her history test. If she got 25% of the questions wrong, how many questions did she get right?

9.During a special one-day sale, the price of a television was marked down 20% from its original price of $100. Later that day, the television was marked down an additional 10%. What was the final sale price?

When You Are Done

Check your answers in Chapter 3.

Exponents—Middle and Upper Levels Only

When in Doubt, Write It Out!

Don’t try to compute exponents in your head. Write them out and multiply!

Exponents are just another way to indicate multiplication. For instance, 3² simply means to multiply three by itself two times, so 3² = 3 × 3 = 9. If you remember that rule, even higher exponents won’t seem very complicated. For example:

2⁵ = 2 × 2 × 2 × 2 × 2 = 32

The questions on the SSAT don’t generally use exponents higher than four or five, so this is likely to be as complicated as it gets.

The rule for exponents is simple: when in doubt, write it out! Don’t try to figure out two times two times two times two times two in your head (just look at how silly it looks written down using words!). Instead, write it as a math problem and just work through it one step at a time.

What would you do if you were asked to solve this problem?

Q³ × Q² =

Let’s look at this one carefully. Q³ means Q × Q × Q and Q² means Q × Q. Put them together and you’ve got:

(Q × Q × Q) × (Q × Q) =

How many Q’s is that? Count them. Five! The answer is Q⁵. Be careful when multiplying exponents like this so that you don’t get confused and multiply the actual exponents, which would give you Q⁶. If you are ever unsure, don’t spend a second worrying; just write out the exponent and count the number of things you are multiplying.

SQUARE ROOTS

A square root is just the opposite of squaring a number. 2² = 2 × 2 or 4, so the square root of 4 is 2.

You will see square roots written this way on a test: .

PRACTICE DRILL 15—EXPONENTS AND SQUARE ROOTS

When You Are Done

Check your answers in Chapter 3.

1.2³ =

2.2⁴ =

3.3³ =

4.4³ =

5.

6.

7.

8.

9.

More Exponents—Upper Level Only

Multiplying and Dividing Exponents with the Same Base

You can multiply and divide exponents with the same base without having to expand out and calculate the value of each exponent. The bottom number, the one you are multiplying, is called the base. (However, note that to multiply 2³ × 5² you must calculate the value of each exponent separately and then multiply the results. That’s because the bases are different.)

For exponents with the same base, remember MADSPM:

When you Multiply with exponents, Add them. When you Divide with exponents, Subtract. When you see Powers with exponents, Multiply.

The exponent rules do NOT apply when adding or subtracting the bases. Pay attention to the operation that is used in the problem when dealing with exponents!

To multiply, add the exponents.

2³ × 2⁴ = 2³⁺⁴ = 2⁷

To divide, subtract the exponents.

2⁸ ÷ 2⁵ = 2^8—5 = 2³

To take an exponent to another power, multiply the exponents.

(2³)³ = 2^3×3 = 2⁹

Anything raised to the first power is itself:

3¹ = 3 x¹ = x

Anything raised to the 0 power is 1:

3⁰ = 1 x⁰ = 1

Negative exponents mean reciprocal: flip it over and get rid of the negative sign in the exponent.

PRACTICE DRILL 16—MORE EXPONENTS

1.3⁵ × 3³ =

2.7² × 7⁷ =

3.5³ × 5⁴ =

4.15²³ ÷ 15²⁰ =

5.4¹³ ÷ 4⁴ =

6.10¹⁰ ÷ 10⁶ =

7.(5³)⁶ =

8.(8¹²)³ =

9.(9⁵)⁵ =

10.(2²)¹⁴ =

When You Are Done

Check your answers in Chapter 3.

REVIEW DRILL 1—THE BUILDING BLOCKS

1.Is 1 a prime number?

2.How many factors does 100 have?

3.—10 + (—20) =

4.100 + 50 ÷ 5 × 4 =

5. —

6. ÷

7.1.2 × 3.4 =

8. × 30 =6

9.1⁵ =

10. =

11.What are the first 10 perfect squares?

When You Are Done

Check your answers in Chapter 3.

ALGEBRA

An Introduction

If you’re a Lower Level, Elementary Level, or Middle Level student, you may not yet have begun learning about algebra in school, but don’t let that throw you. If you know how to add, subtract, multiply, and divide, you can solve an algebraic equation. Lower and Elementary Level students only need to understand the section below titled “Solving Simple Equations.” Middle Level students should complete all of the “Solving Simple Equations” drills and as much of the Upper Level material as possible. Upper Level students need to go through the entire Algebra section carefully to make sure they can solve each of the question types.

Solving Simple Equations

Algebraic equations involve the same basic operations that we’ve dealt with throughout this chapter, but instead of using only numbers, these equations use a combination of numbers and letters. These letters are called variables. Here are some basic rules about working with variables that you need to understand.

·  A variable (usually x, y, or z) replaces an unknown number in an algebraic equation.

·  It is usually your job to figure out what that unknown number is.

·  If a variable appears more than once in an equation, that variable is always replacing the same number.

·  When a variable is directly to the right of a number, with no sign in between them, the operation that is holding them together is multiplication (e.g., 3y = 3 × y).

·  You can add and subtract like variables (e.g., 2z + 5z = 7z).

·  You cannot add or subtract unlike variables (e.g., 2z and 3y cannot be combined).

To solve simple algebraic equations, you need to think abstractly about the equation. Let’s try one.

2 + x = 7

What does x equal?

Well, what number plus 2 gives you 7? If you said 5, you were right and x = 5.

2y = 16

What does y equal?

In the first equation, we subtracted 2 from both sides. In the second equation, we divided both sides by 2.

Now you need to ask yourself what multiplied by 2 gives you 16. If you said 8, you were right! y = 8.

Tip: You can check to see if you found the right number for the variable by replacing the variable with the number you found in the equation. So in the last problem, if we replace y with 8 and rewrite the problem, we get 2 × 8 = 16. And that’s true, so we got it right!

PRACTICE DRILL 17—SOLVING SIMPLE EQUATIONS

1.If 35 — x = 23, then x =

2.If y + 12 = 27, then y =

3.If z — 7 = 21, then z =

4.If 5x = 25, then x =

5.If 18 ÷ x = 6, then x =

6.If 3x = 33, then x =

7.If 65 ÷ y = 13, then y =

8.If 14 = 17 — z, then z =

9.If y = 24, then y =

10.If 136 + z = 207, then z =

11.If 7x = 84, then x =

12.If y ÷ 2 = 6, then y =

13.If z ÷ 3 = 15, then z =

14.If 14 + x = 32, then x =

15.If 53 — y = 24, then y =

When You Are Done

Check your answers in Chapter 3.

Note:

·  Lower and Elementary Level students should stop here. The next section you will work on is Geometry.

·  All Middle and Upper Level students should continue.

Manipulating An Equation—Middle and Upper Levels Only

To solve an equation, your goal is to isolate the variable, meaning that you want to get the variable on one side of the equation and everything else on the other side.

3x + 5 = 17

To solve this equation, follow these two steps.

Step 1: Move elements around using addition and subtraction.

Get variables on one side and numbers on the other.

Simplify.

Step 2: Divide both sides of the equation by the coefficient, the number in front of the variable. If that number is a fraction, multiply everything by the denominator.

For example:

Remember: Whatever you do to one side, you must also do to the other.

Taking Sides

You can do anything you want to one side of the equation, as long as you make sure to do exactly the same thing to the other side.

PRACTICE DRILL 18—MANIPULATING AN EQUATION

1.If 8 = 11 — x, then x =

2.If 4x = 20, then x =

3.If 5x — 20 = 10, then x =

4.If 4x + 3 = 31, then x =

5.If m + 5 = 3m — 3, then m =

6.If 2.5x = 20, then x =

7.If 0.2x + 2 = 3.6, then x =

8.If 6 = 8x + 4, then x =

9.If 3(x + y) = 21, then x + y =

10.If 3x + 3y = 21, then x + y =

11.If 100 — 5y = 65, then y =

When You Are Done

Check your answers in Chapter 3.

Manipulating Inequalities—Middle and Upper Levels Only

Manipulating an inequality is just like manipulating an equation that has an equals sign, except for one rule: if you multiply or divide by a negative number, flip the inequality sign.

Helpful Trick

Think of the inequality sign as an alligator, and the alligator always eats the bigger meal.

Let’s try an example.

—3x < 6

Divide both sides by —3 and then flip the inequality sign.

x > —2

PRACTICE DRILL 19—MANIPULATING INEQUALITIES

Solve for x.

1.4x > 16

2.13 — x > 15

3.15x — 20x < 25

4.12 + 2x > 24 — x

5.7 < —14 — 3x

When You Are Done

Check your answers in Chapter 3.

Solving Percent Questions with Algebra—Middle and Upper Levels Only

Percentages

Solving percent problems can be easy when you know how to translate them from “percent language” into “math language.” Once you’ve done the translation, you guessed it—just manipulate the equation!

Learn a Foreign Language

You can memorize “percent language” quickly because there are only four words you need to remember!

Whenever you see words from the following table, just translate them into math terms and go to work on the equation!

For example:

“What percent” is represented by

PRACTICE DRILL 20—TRANSLATING AND SOLVING PERCENT QUESTIONS

1.30 is what percent of 250 ?

2.What is 12% of 200 ?

3.What is 25% of 10% of 200 ?

4.75% of 20% of what number is 12 ?

5.16% of what number is 25% of 80 ?

6.What percent is equal to ?

7.30 is what percent of 75 ?

8.What is 11% of 24 ?

9.What percent of 24 is equal to 48 ?

10.60% of what percent of 500 is equal to 6 ?

When You Are Done

Check your answers in Chapter 3.

GEOMETRY

An Introduction

Just as in the previous Algebra section, this Geometry section contains some material that is above the level tested on the Lower/Elementary and Middle Level Exams. These students should not work on sections that are indicated for higher levels.

Perimeter

The perimeter is the distance around the outside of any figure. To find the perimeter of a figure, just add up the lengths of all the sides.

What are the perimeters of these figures?

Perimeter = 6 + 6 + 8 + 8 + 10 = 38

Perimeter = 8 + 8 + 12 = 28

Angles—Middle and Upper Levels Only

Straight Lines

Angles that form a straight line always total 180°.

Triangles

All of the angles in a triangle add up to 180°.

The Rule of 180°

There are 180° in a straight line and in a triangle.

Four-Sided Figures

The angles in a square, rectangle, or any other four-sided figure always add up to 360°.

The Rule of 360°

There are 360° in a four-sided figure and in a circle.

Squares and Rectangles

A rectangle is a four-sided figure with four right (90°) angles. Opposite sides are equal in a rectangle. The perimeter is equal to the sum of the sides.

Perimeter = 3 + 3 + 7 + 7 = 20

Perimeter

P = side + side + side…until you run out of sides.

A square is a special type of rectangle in which all the sides are equal.

Perimeter = 5 + 5 + 5 + 5 = 20

Because all sides of a square are equal, you can find the length of a side by dividing its perimeter by four. If the perimeter of a square is 20, then each side is 5, because 20 ÷ 4 = 5.

Area

Area is the amount of space taken up by a two-dimensional figure. One way to think about area is as the amount of paper that a figure covers. The larger the area, the more paper the figure takes up.

To determine the area of a square or rectangle, multiply the length (l) by the width (w).

Remember the formula:

Area = length × width

Area of a Rectangle

A = lw

What is the area of a rectangle with length 9 and width 4 ?

In this case, the length is 9 and the width is 4, so 9 × 4 = 36. Now look at another example.

Area of rectangle ABCD = 6 × 8 = 48

The area of squares and rectangles is given in square feet, square inches, and so on.

To find the area of a square, you multiply two sides, and because the sides are equal, you’re really finding the square of the sides. You can find the length of a side of a square by taking the square root of the area. So if a square has an area of 25, one side of the square is 5.

Area of a Square

A = s²

Volume

Volume is very similar to area, except it takes into account a third dimension. To compute the volume of a figure, you simply find the area and multiply by a third dimension.

For instance, to find the volume of a rectangular object, you would multiply the length by the width (a.k.a. the area) by the height (the third dimension). Since a rectangular solid (like a box) is the only kind of figure you are likely to see in a volume question, simply use the formula below.

length × width × height volume

For example:

What is the volume of a rectangular fish tank with the following specifications?

length: 6 inches

height: 6 inches

width: 10 inches

Volume of a Rectangular Solid

V = lwh

There isn’t much to it. Just plug the numbers into the formula.

length × width × height = volume

6 × 10 ×6 = 360

PRACTICE DRILL 21—SQUARES, RECTANGLES, AND ANGLES

1.What is the value of x ?

2.What is the value of x ?

3.PQRS is a square. What is its perimeter? Area?

4.ABCD is a rectangle with length 7 and width 3. What is its perimeter? Area?

5.STUV is a square. Its perimeter is 12. What is its area?

6.DEFG is a square. Its area is 81. What is its perimeter?

7.JKLM is a rectangle. If its width is 4, and its perimeter is 20, what is its area?

8.WXYZ is a rectangle. If its length is 6 and its area is 30, what is its perimeter?

9.What is the volume of a rectangular solid with height 3, width 4, and length 2 ?

When You Are Done

Check your answers in Chapter 3.

Triangles

A triangle is a geometric figure with three sides.

Isosceles Triangles

Any triangle with two equal sides is an isosceles triangle.

If two sides of a triangle are equal, the angles opposite those sides are always equal. Said another way, the sides opposite the equal angles are also equal.

This particular isosceles triangle has two equal sides (of length 6) and therefore two equal angles (40° in this case).

If you already know that the above triangle is isosceles, then you also know that y must equal one of the other sides and n must equal one of the other angles. Since n = 65 (65° + 50° + n° = 180°), then y must equal 9, because it is opposite the other 65° angle.

Equilateral Triangles

An equilateral triangle is a triangle with three equal sides. If all the sides are equal, then all the angles must be equal. Each angle in an equilateral triangle is 60°.

Right Triangles

A right triangle is a triangle with one 90° angle.

This is a right triangle.

It is also an isosceles triangle.

What does that tell you?

Area

To find the area of a triangle, multiply by the length of the base by the length of the triangle’s height, or b×h.

What is the area of a triangle with base 6 and height 3 ?

(A)3

(B)6

(C)9

(D)12

(E)18

Elementary and Lower Levels

The test writers may give you the formula for the area of a triangle, but memorizing it will still save you time!

Just put the values you are given into the formula and do the math. That’s all there is to it!

b × h = area

() (6) × 3 = area

3 × 3 = 9

Don’t Forget!

A = bh Remember the base and the height must form a 90° angle.

So, (C) is the correct answer.

The only tricky point you may run into when finding the area of a triangle is when the triangle is not a right triangle. In this case, it becomes slightly more difficult to find the height, which is easiest to think of as the distance to the point of the triangle from the base. Here’s an illustration to help.

First look at triangle BAC, the unshaded right triangle on the left side. Finding its base and height is simple—they are both 3. So using our formula for the area of a triangle, we can figure out that the area of triangle BAC is 4.

Now let’s think about triangle BCD, the shaded triangle on the right. It isn’t a right triangle, so finding the height will involve a little more thought. Remember the question, though: how far up from the base is the point of triangle BCD? Think of the shaded triangle sitting on the floor of your room. How far up would its point stick up from the floor? Yes, 3! The height of triangle BCD is exactly the same as the height of triangle BAC. Don’t worry about drawing lines inside the shaded triangle or anything like that, just figure out how high its point is from the ground.

Okay, so just to finish up, to find the base of triangle BCD (the shaded one), use the same area formula, and just plug in 3 for the base and 3 for the height.

b × h = area

()(3) × 3 = area

And once you do the math, you’ll see that the area of triangle BCD is 4.

Not quite convinced? Let’s look at the question a little differently. The base of the entire figure (triangle DAB) is 6, and the height is 3. Using your trusty area formula, you can determine that the area of triangle DAB is 9. You know the area of the unshaded triangle is 4, so what’s left for the shaded part? You guessed it, 4.

Similar Triangles—Middle and Upper Levels Only

Similar triangles are triangles that have the same angles but sides of different lengths. The ratio of any two corresponding sides will be the same as the ratio of any other two corresponding sides. For example, a triangle with sides 3, 4, and 5 is similar to a triangle with sides of 6, 8, and 10, because the ratio of each of the corresponding sides (3:6, 4:8, and 5:10) can be reduced to 1:2.

One way to approach similar triangles questions that ask you for a missing side is to set up a ratio or proportion. For example, look at the question below:

What is the value of EF ?

(A)4

(B)5

(C)6

(D)7

(E)8

These triangles are similar because they have the same angles. To find side EF, you just need to set up a ratio or proportion.

Cross-multiply to get 15(EF) = 18(5).

Divide both sides by 15 to get EF = 6.

Therefore, the answer is (C), 6.

The Pythagorean Theorem—Upper Level Only

For all right triangles, a² + b² = c², where a, b, and c are the lengths of the triangle’s sides.

Always remember that c represents the hypotenuse, the longest side of the triangle, which is always opposite the right angle.

Try It!

Test your knowledge of triangles with the problems that follow. If the question describes a figure that isn’t shown, make sure you draw the figure yourself!

1.What is the length of side BC ?

A)4

B)5

C)6

D)7

E)8

Just put the values you are given into the formula and do the math, remembering that line BC is the hypotenuse:

a² + b² = c²

3² + 4² = c²

9 + 16 = c²

25 = c²

5 = c

So, (B) is the correct answer.

PRACTICE DRILL 22—TRIANGLES

1.What is the value of x ?

2.Triangle PQR is an isosceles triangle. PQ = QR. What is the value of x ?

3.What is the area of right triangle ABC ?

4.What is the area of the shaded region?

5.What is the area of triangle DEF ?

6.What is the area of triangle WXZ ? Triangle ZXY ? Triangle WXY ?

7.What is the length of line QR ?

8.What is the length of side DE ?

9.What is the value of x ?

10.What is the length of the diagonal of rectangle ABCD ?

11.What is the perimeter of square ABCD ?

12.What is the value of x ?

When You Are Done

Check your answers in Chapter 3.

Note:

·  Lower Level students should stop here. The next section you will work on is Word Problems.

·  Middle and Upper Level students should continue.

Circles—Middle and Upper Levels Only

You are probably already familiar with the parts of a circle, but let’s review them anyway.

Any line drawn from the origin (the center of the circle) to its edge is called a radius (r).

Any line that goes from one side of the circle to the other side and passes through the center of the circle is called the diameter (d). The diameter is two times the length of the radius.

Diameter

d = 2r

Circumference

C = πd

Area

A = πr²

Area and Circumference

Circumference (which is written as C) is really just the perimeter of a circle. To find the circumference of a circle, use the formula 2πr or πd. We can find the circumference of the circle above by taking its radius, 3, and multiplying it by 2π.

C = 2πr

C = 2π3

C = 6π

The area of a circle is found by using the formula πr².

A = πr²

A = π3²

A = 9π

You can find a circle’s radius from its circumference by getting rid of π and dividing the number by 2. Or you can find the radius from a circle’s area by getting rid of π and taking the square root of the number.

So if a circle has an area of 81π, its radius is 9. If a circle has a circumference of 16π, its radius is 8.

What’s up with π?

The Greek letter π is spelled “pi” and pronounced “pie.” It is a symbol used with circles. Written as a number, π is a nonrepeating, nonending decimal (3.1415927…). We use π to determine the true length of circles. However, on the ISEE and SSAT, we simply leave π as the Greek letter. So when figuring out area or circumference, make sure that you include π in your equation at the beginning and include it in every step of your work as you solve. Remember, π represents a number and it must always be included in either the area or circumference formula.

PRACTICE DRILL 23—CIRCLES

1.What is the circumference of the above circle? What is the area?

2.What is the area of a circle with radius 4 ?

3.What is the area of a circle with diameter 8 ?

4.What is the radius of a circle with area 9π ?

5.What is the diameter of a circle with area 9π ?

6.What is the circumference of a circle with area 25π ?

When You Are Done

Check your answers in Chapter 3.

3-D Shapes—Upper Level Only

Both the SSAT and ISEE Upper Level tests include 3-D shape geometry questions. While these question types tend to be few and far between, it is important you are prepared for them, just in case they do come up.

Boxes

A three-dimensional box has three important lines: length, width, and height.

This rectangular box has a length of 6, a width of 2, and a height of 3.

The volume formula of a rectangular box is V = lwh.

V = lwh

V = 6(2)(3)

V = 36

Cubes

Cubes are just like rectangular boxes, except that all the sides are equal.

The volume formula for the cube is still just V = lwh, but since the length, width and height are all equal it can also be written as V = s³, where s = side.

V = s³

V = 3³

V = 27

Cylinders

Cylinders are like circles with height added. For a cylinder with a radius of r and a height of h, the volume formula is V = πr²h.

V = πr²h

V = π2²4

V = π4(4)

V = 16π

PRACTICE DRILL 24—3-D SHAPES

1.What is the volume of this cylinder?

2.What is the volume of this cube?

3.What is the volume of this rectangular box?

4.A cube with a side length of 6 has 54 gallons poured into it. How many more gallons must be poured into the cube for it to be completely filled?

5.The rectangular box pictured is filled by identical cubes with side lengths of 2. How many cubes does it take to fill the rectangular box?

6.The cylinder pictured is full of grain. What is the volume of the grain in the cylinder?

When You Are Done

Check your answers in Chapter 3.

WORD PROBLEMS

Many arithmetic and algebra problems are written in paragraph form with many words. The hard part is usually not the arithmetic or the algebra; the hard part is translating the words into math. So let’s focus on translating.

Key Words and Phrases to Translate

Specific words and phrases show up repeatedly in word problems. You should be familiar with all of those on this page.

Proportions

Proportions show relationships between two sets of information. For example, if you wanted to make cookies and you had a recipe for a dozen cookies but wanted to make two dozen cookies, you would have to double all of the ingredients. That’s a proportion. Here’s how we’d look at it in equation form.

Whenever a question gives you one set of data and part of another set, it will ask you for the missing part of the second set of data. To find the missing information, set up the information in fractions like those shown above. Be careful to put the same information in the same place. In our example, we have flour on top and cookies on the bottom. Make sure both fractions have the flour over the cookies. Once we have our fractions set up, we can see what the relationship is between the two elements (in this case, flour and cookies). Whatever that relationship is, it’s the same as the relationship between the other two things.

PRACTICE DRILL 25—WORD PROBLEMS

1.There are 32 ounces in 1 quart. 128 ounces equals how many quarts? How many ounces are there in 7 quarts?

2.A car travels at a rate of 50 miles per hour. How long will it take to travel 300 miles?

3.Betty is twice as old as her daughter Fiona. Fiona is twice as old as her dog Rufus. If Rufus is 11, how old is Betty?

4.A clothing store sold 1,250 pairs of socks this year. Last year, the store sold 250 pairs of socks. This year’s sales are how many times greater than last year’s sales?

5.There are 500 students at Eisenhower High School. of the total students are freshmen. of all the freshmen are girls. How many freshman girls are there?

When You Are Done

Check your answers in Chapter 3.

REVIEW DRILL 2—THE BUILDING BLOCKS

1.If one-third of b is 15, then what is b ?

2.If 7x — 7 = 49, then what is x ?

3.If 4 (y — 5) = 20, then what is y ?

4.8x + 1 < 65. Solve for x.

5.16 is what percent of 10 ?

6.What percent of 32 is 24 ?

7.What is the area of a triangle with base 7 and height 6 ?

(Middle and Upper Levels)

8.What is the diameter of a circle with an area of 49π ?

9.What is the radius of a circle with a circumference of 12π ?

10.What is the area of a circle with a diameter of 10 ?

When You Are Done

Check your answers in Chapter 3.