SAT Test Prep

CHAPTER 7
ESSENTIAL PRE-ALGEBRA SKILLS

1. Numbers and Operations

2. Laws of Arithmetic

3. Fractions

4. Ratios and Proportions

5. Percents

6. Negatives

7. Divisibility

Lesson 1: Numbers and Operations

Integers and Real Numbers

On the SAT, you only need to deal with two kinds of numbers: integers (the positive and negative whole numbers, …, –3, –2, –1, 0, 1, 2, 3, …) and real numbers (all the numbers on the number line, including integers, but also including all fractions and decimals). You don’t have to know about wacky numbers such as irrationals or imaginaries.


The SAT only uses real numbers. It will never (1) divide a number by 0 or (2) take the square root of a negative number because both these operations fail to produce a real number. Make sure that you understand why both these operations are said to be “undefined.”



Don’t assume that a number in an SAT problem is an integer unless you are specifically told that it is. For instance, if a question mentions the fact that , don’t automatically assume that x is 4 or greater. If the problem doesn’t say that x must be an integer, then x might be 3.01 or 3.6 or the like.


The Operations

The only operations you will have to use on the SAT are the basics: adding, subtracting, multiplying, dividing, raising to powers, and taking roots. Don’t worry about “bad boys” such as sinestangents, or logarithms—they won’t show up. (Yay!)


Don’t confuse the key words for the basic operations: Sum means the result of addition, difference means the result of subtraction, product means the result of multiplication, and quotient means the result of division.


The Inverse Operations

Every operation has an inverse, that is, another operation that “undoes” it. For instance, subtracting 5 is the inverse of adding 5, and dividing by –3.2 is the inverse of multiplying by –3.2. If you perform an operation and then perform its inverse, you are back to where you started. For instance, . No need to calculate!


Using inverse operations helps you to solve equations. For example,


Alternative Ways to Do Operations


Every operation can be done in two ways, and one way is almost always easier than the other. For instance, subtracting a number is the same thing as adding the opposite number. So subtracting 5 is the same as adding 5. Also, dividing by a number is exactly the same thing as multiplying by its reciprocal. So dividing by 2/3 is the same as multiplying by 3/2. When doing arithmetic, always think about your options, and do the operation that is easier! For instance, if you are asked to do , you should realize that it is the same as , which is easier to do in your head.


The Order of Operations


Don’t forget the order of operations: P-E-MD-AS. When evaluating, first do what’s grouped in parentheses (or above or below fraction bars or within radicals), then do exponents (or roots) from left to right, then multiplication or division from left to right, and then do addition or subtractionfrom left to right. What is ? If you said –3, you mistakenly did the multiplication before the division. (Instead, do them left to right). If you said –3 or –1/3, you mistakenly subtracted before taking care of the multiplication and division. If you said –5, pat yourself on the back!

When using your calculator, be careful to use parentheses when raising negatives to powers. For instance, if you want to raise –2 to the 4th power, type “(–2)^4,” and not just “–2^4,” because the calculator will interpret the latter as –1(2)^4, and give an answer of –16, rather than the proper answer of 16.


Concept Review 1: Numbers and Operations

3. When is taking the square root of a number not the inverse of squaring a number? Be specific.

__________________________________________________________________________________________________

__________________________________________________________________________________________________

6. The result of an addition is called a __________.

7. The result of a subtraction is called a __________.

8. The result of a multiplication is called a __________.

What is the alternative way to express each of the following operations?

10.–(6) __________

11.÷(4) __________

What is the inverse of each of the following operations?

14.–(6) __________

15.÷(4) __________

Simplify without a calculator:

22. Circle the real numbers and underline the integers:

23. The real order of operations is _____________________________________________________________________.

24. Which two symbols (besides parentheses) are “grouping” symbols? _____________________________________

26. List the three operations that must be performed on each side of this equation (in order!) to solve for x:

 Step 1 __________ Step 2 __________ Step 3 __________ x = __________

SAT Practice 1: Numbers and Operations

1. Which of the following is NOT equal to  of an integer?

(A) 

(B) 1

(C) 

(D) 

(E) 10

2. Which of the following can be expressed as the product of two consecutive even integers?

(A) 22

(B) 36

(C) 48

(D) 60

(E) 72

(A) –3

(B) –2

(C) –1

(D) 2

(E) 3

4. If , what is the value of k?

(A) 11

(B) 64

(C) 67

(D) 121

(E) 132

5. In the country of Etiquette, if 2 is a company and 3 is a crowd, then how many are 4 crowds and 2½ companies?

(A) 14

(B) 17

(C) 23

(D) 28½

(E) 29

6. For what integer value of x is  and ?

7. For all real numbers x, let {x be defined as the least integer greater than x.

(A) –6

(B) –5.7

(C) –5.5

(D) –5

(E) 1

8. Dividing any positive number by ¾ and then multiplying by –2 is equivalent to

(A) multiplying by 

(B) dividing by 

(C) multiplying by –

(D) dividing by 

(E) multiplying by –

9. When 14 is taken from 6 times a number, 40 is left. What is half the number?

10. If the smallest positive four-digit integer without repeated digits is subtracted from the greatest four-digit integer without repeated digits, the result is

(A) 8,642

(B) 1,111

(C) 8,853

(D) 2,468

(E) 8,888

11. If , the value of which of the following expressions increases as x increases?

(A) II only

(B) III only

(C) I and II only

(D) I and III only

(E) I, II, and III

Answer Key 1: Numbers and Operations

Concept Review 1

1. –10 (Remember: “greatest” means farthest to the right on the number line.)

2. –10/1, –20/2, –30/3, etc. (Fractions can be integers.)

3. If the original number is negative, then taking a square root doesn’t “undo” squaring the number. Imagine that the original number is –3. The square of –3 is 9, but the square root of 9 is 3. This is the absolute value of the original number, but not the original number itself.

4. 22 (. Just divide 76 by 4 to get the “middle” of the set = 19.)

5. Infinitely many (If you said –3, don’t assume that unknowns are integers!)

6. sum

7. difference

8. product

10. +(ndash;6)

11. ×(¼)

12. ÷(–3/5)

13. ×(7/6)

14. +(6)

15. ×(4)

16. ÷(–5/3)

17. ×(6/7)

18. –13 (If you said –5, remember to do multiplication/division from left to right.)

19. 0

20. 50

21. –4

22. Circle all numbers except  and underline only 0, , and .

23. PG-ER-MD-AS (Parentheses/grouping (left to right), exponents/roots (left to right), multiplication/division (left to right), addition/subtraction (left to right))

24. Fraction bars (group the numerator and denominator), and radicals (group what’s inside)

25. 120 (It is the least common multiple of 5, 6, 8, and 12.)

26. Step 1: subtract 7; step 2: divide by 3; step 3: take the square root;  or –3 (not just 3!)

SAT Practice 1

1. C  is not  of an integer because , which is not an integer.

.

7. D –5 is the least (farthest to the left on the number line) of all the integers that are greater than (to the right on the number line of) –5.6.

8. A Dividing by ¾ is equivalent to multiplying by 

(Don’t forget to find half the number!)

Don’t forget that 0 is a digit, but it can’t be the first digit of a four-digit integer.

11. D You might “plug in” increasing values of x to see whether the expressions increase or decrease. 1 and 4 are convenient values to try. Also, if you can graph  and y = 10 – 1/x quickly, you might notice that  and  “go up” as you move to the right of 1 on the x-axis.