Although arithmetic is only one of the three types of math tested on the GMAT, arithmetic problems comprise about half of the total number of math questions.

Here are the specific arithmetic topics tested on the GMAT:

1. Axioms and Fundamentals (properties of integers, positive and negative numbers, even and odd). These were covered in Chapter 7.

2. Arithmetic Operations

3. Fractions

4. Decimals

5. Ratios

6. Percentages

7. Averages

8. Exponents and Radicals

In this chapter, we will first discuss the fundamentals of each topic and then show how the test-writers construct questions based on that topic.

1.公理和基本原理（整數，正負數，偶數和奇數的屬性）。 這些已在第7章中介紹。

2.算術運算

3.分數

4.小數

5.比率

6.百分比

7.平均值

8.指數和根號

ARITHMETIC OPERATIONS

There are six arithmetic operations you will need for the GMAT:

GMAT需要進行六種算術運算：

1. Addition (2 + 2): The result of addition is a sum or total.

2. Subtraction (6 – 2): The result of subtraction is a difference.

3. Multiplication (2 × 2): The result of multiplication is a product.

4. Division (8 ÷ 2): The result of division is a quotient.

5. Raising to a power (x2): In the expression x2, the little 2 is called an exponent.

6. Finding a square root :

1.加法（2 + 2）：加法的結果是總和或總計。

2.減法（6 – 2）：減法的結果是不同的。

3.乘法（2×2）：乘法的結果是乘積。

4.除（8÷2）：除的結果是商。

5.求冪（x2）：在表達式x2中，小數2稱為指數。

6.求平方根：

Which One Do I Do First?

In a problem that involves several different operations, the operations must be performed in a particular order, and occasionally GMAC likes to see whether you know what that order is. Here’s an easy way to remember the order of operations:

The first letters stand for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. Do operations that are enclosed in parentheses first; then take care of exponents; then multiply and divide; finally add and subtract, going from left to right.

DRILL 2

Just to get you started, solve each of the following problems by performing the indicated operations in the proper order. The answers can be found in Part VI.

1. 74 + (27 – 24) =

2. (8 × 9) + 7 =

3. 2(9 – (8 ÷ 2)) =

4. 2(7 – 3) + (–4)(5 – 7) =

1. 74 +（27 – 24）=

2.（8×9）+ 7 =

3. 2（9 –（8÷2））=

4. 2（7 – 3）+（–4）（5 – 7）=

Here’s an easy question that shows how GMAC might test PEMDAS.

5. 4(–3(3 – 5) + 10 – 17) =

–27

–4

–1

32

84

It is not uncommon to see a Data Sufficiency problem like this on the GMAT:

6. What is the value of x ?

(1) x3 = 8

(1) x3 = 8

(2) x2 = 4

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are not sufficient.

There are two operations that can be done in any order, provided they are the only operations involved: When you are adding or multiplying a series of numbers, you can group or regroup the numbers any way you like.

2 + 3 + 4 is the same as 4 + 2 + 3

and

4 × 5 × 6 is the same as 6 × 5 × 4

This is called the Associative Law, but the name will not be tested on the GMAT.

Another law that GMAC likes to test states that

a(b + c) = ab + ac and a(b – c) = ab – ac.

This is called the Distributive Law but, again, you don’t need to know that for the test. Sometimes the Distributive Law can provide you with a shortcut to the solution of a problem. If a problem gives you information in “factored form”—a(b + c)—you should distribute it immediately. If the information is given in distributed form—ab + ac—you should factor it.

5. 4（–3（3 – 5）+ 10 – 17）=

–27

–4

–1

32

84

6. x的值是多少？

（1）x3 = 8

（1）x3 = 8

（2）x2 = 4

2 + 3 + 4與4 + 2 + 3相同

4×5×6與6×5×4相同

GMAC喜歡測試的另一條法律指出：

a（b + c）= ab + ac，a（b – c）= ab – ac。

DRILL 3

If the following problems are in distributed form, factor them; if they are in factored form, distribute them. Then do the indicated operations. Answers are in Part VI.

1. x 2 + x

2. (55 × 12) + (55 × 88)

3. a(b + c – d)

4. abc + xyc

A GMAT problem might look like this:

5. If x = 6, what is the value of ?

–30

6

8

30

It cannot be determined from the information given.

It is not uncommon to see a Data Sufficiency problem like this on the GMAT:

6. If ax + ay + az = 15, what is x + y + z ?

(1) x = 2

(2) a = 5

Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.

Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.

BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.

EACH statement ALONE is sufficient.

Statements (1) and (2) TOGETHER are not sufficient.

1. x 2 + x

2.（55×12）+（55×88）

3. a（b + c – d）

4. abc + xyc

GMAT問題可能看起來像這樣：

5.如果x = 6，則值是多少？

–30

6

8

30

6.如果ax + ay + az = 15，x + y + z是多少？

（1）x = 2

（2）= 5

Challenge Question #2

A device calculates the worth of gemstones based on quality such that a gem with a quality rating of q – 1 is worth 5 times more than a gem with a quality rating of q, and a gem with a quality rating of q – 4 is worth 625 times more than a gem with a quality rating of q. According to this device, the worth of a gem with a quality rating of p – r is how many times greater than that of a gem with a rating of p ?

p5 – r5

r5

(p – r)5

5r

5r

p5 – r5

55

（p – r）5

5r

5r

FRACTIONS

Fractions can be thought of in two ways:

A fraction is just another way of expressing division. The expression  1/2 is exactly the same thing as 1 divided by 2. x/y is nothing more than x divided by y. In the fraction x/y , x is known as the numerator and y is known as the denominator.

The other important way to think of a fraction is as part/whole . The fraction 7/10 can be thought of as 7 parts out of a total of 10 parts.

Adding and Subtracting Fractions with the Same Denominator

To add two or more fractions that have the same denominator, simply add the numerators and put the sum over the common denominator. For example: Subtraction works exactly the same way: Adding and Subtracting Fractions with Different Denominators

Before you can add or subtract two or more fractions with different denominators, you must give all of them the same denominator. To do this, multiply the numerator and denominator of each fraction by a number that will give it a denominator in common with the others. If you multiplied each fraction by any old number, the fractions wouldn’t have their original values, so the number you multiply by has to be equal to 1. For example, if you wanted to change  into sixths, you could do the following: We haven’t actually changed the value of the fraction, because  equals 1.

If we wanted to add: The Bowtie

The Bowtie method has been a staple of The Princeton Review’s materials since the company began in a living room in New York City in 1981. It’s been around so long because it works so simply.

To add 3/5 and 4/7 , for example, follow these three steps:

Step One: Multiply the denominators together to form the new denominator. Step Two: Multiply the first denominator by the second numerator (5 × 4 = 20) and the second denominator by the first numerator (7 × 3 = 21) and place these numbers above the fractions, as shown below. Step Three: Add the products to form the new numerator. Subtraction works the same way. Note that with subtraction, the order of the numerators is important. The new numerator is 21 – 20, or 1. If you somehow get your numbers reversed and use 20 – 21, your answer will be –1/35 , which is incorrect. One way to keep your subtraction straight is to always multiply up from denominator to numerator when you use the Bowtie.

Multiplying Fractions

To multiply fractions, just multiply the numerators and put the product over the product of the denominators. For example: Reducing Fractions

When you add or multiply fractions, you often end up with a big fraction that is hard to work with. You can usually reduce such a fraction. To reduce a fraction, find a factor of the numerator that is also a factor of the denominator. It saves time to find the biggest factor they have in common, but this isn’t critical. You may just have to repeat the process a few times. When you find a common factor, cancel it. For example, let’s take the product we just found when we multiplied the fractions above: Get used to reducing all fractions (if they can be reduced) before you do any work with them. It saves a lot of time and prevents errors in computation.

For example, in that last problem, we had to multiply two fractions together: Before you multiplied 2 × 6 and 3 × 5, you could have reduced Dividing Fractions

To divide one fraction by another, just invert the second fraction and multiply: which is the same thing as… You may see this same operation written like this: Again, just invert and multiply. This next example is handled the same way: When you invert a fraction, the new fraction is called a reciprocal. 2/3 is the reciprocal of 3/2. The product of two reciprocals is always 1.

Converting to Fractions

An integer can be expressed as a fraction by making the integer the numerator and 1 the denominator: 16 = 16/1.

The GMAT sometimes gives you numbers that are mixtures of integers and fractions, for example, 3 1/2. It’s easier to work with these numbers if you convert them into fractions. Simply multiply the denominator by the integer, then add the numerator, and place the resulting number over the original denominator.

GMAT有時會為您提供整數和分數混合的數字，例如3 1/2。 如果將這些數字轉換為分數，則使用起來更容易。 只需將分母乘以整數，然後加上分子，然後將結果數字放在原始分母上。 Comparing Fractions

In the course of a problem, you may have to compare two or more fractions and determine which is greater. This is easy to do as long as you remember that you can compare fractions directly only if they have the same denominator. Suppose you had to decide which of these three fractions is greatest: To compare these fractions directly, you need a common denominator, but finding a common denominator that works for all three fractions would be complicated and time consuming. It makes more sense to compare these fractions two at a time. We showed you the classical way to find common denominators when we talked about adding fractions earlier.

Let’s start with 1/2 and 5/9. An easy common denominator for these two fractions is 18 (9 × 2).

1/2                                             5/9 =9/18                                      =10/18

Because 5/9 is greater, let’s compare it with 7/15. Here the easiest common denominator is 45. But before we do that…

Two Shortcuts

Comparing fractions is another situation in which we can use the Bowtie. The idea is that if all you need to know is which fraction is greater, you just have to compare the new numerators. Again, simply multiply the denominator of the first fraction by the numerator of the second and the denominator of the second by the numerator of the first, as shown here. 10 > 9, therefore  5/9 > 1/2

You could also have saved yourself some time on the last problem by a little fast estimation. Again, which is greater? 1/2 ,5/9 , or 7/15 ?

Let’s think about 5/9 in terms of 1/2 . How many ninths equal a half? To put it another way, what is half of 9 ? 4.5. So 4.5/9=1/2. That means 5/9 is greater than 1/2.

Now let’s think about 7/15. Half of 15 is 7.5. , 7.5/15=1/2 , which means that 7/15 is less than 1/2.

PROPORTIONS

A fraction can be expressed in many ways.  also equals  or , etc. A proportion is just a different way of expressing a fraction. Here’s an example:

If 2 boxes hold a total of 14 shirts, how many shirts are contained in 3 boxes?

Here’s How to Crack It

The number of shirts per box can be expressed as a fraction. What you’re asked to do is express the fraction 2/14 in a different way. To find the answer, all you need to do is find a value for x such that 2/14=3/x. The easiest way to do this is to cross-multiply.

2x = 42, which means that x = 21. There are 21 shirts in 3 boxes.

2x = 42，這意味著x =21。3個盒子中有21件襯衫。

DRILL 4

The answers to these questions can be found in Part VI.

1. 5 3/4 + 3/8 =

2. Reduce 12/60

3. Convert 9 2/3 to a fraction

4. Solve for x in 9/2 = x/4

A relatively easy GMAT fraction problem might look like this:

1. 5 3/4 + 3/8 =

2.降低12/60

3.將9 2/3轉換為分數

4.求解x in 9/2 = x / 4 3/100

3/16

1/3

1

7/16

Now that you’ve been reacquainted with the basics of fractions, let’s go a little further. More complicated fraction problems usually involve all of the rules we’ve just mentioned, with the addition of two concepts: part/whole  , and the rest. Here’s a typical medium fraction problem:

A cement mixture is composed of 3 elements. By weight,  1/3 of the mixture is sand,  3/5 of the mixture is water, and the remaining 12 pounds of the mixture is gravel. What is the weight of the entire mixture in pounds?

4

8

36

60

180

4

8

36

60

180

Easy Eliminations

Before we even start doing serious math, let’s use some common sense. The weight of the gravel alone is 12 pounds. Because we know that sand and water make up the bulk of the mixture—sand 1/3 , water 3/5  (which is a bit more than half)—the entire mixture must weigh a great deal more than 12 pounds. Choices (A) and (B) are out of the question. Eliminate them. Here’s How to Crack It

The difficulty in solving this problem is that sand and water are expressed as fractions, while gravel is expressed in pounds. At first there seems to be no way of knowing what fractional part of the mixture the 12 pounds of gravel represent; nor do we know how many pounds of sand and water there are.

The first step is to add up the fractional parts that we do have: Sand and water make up 14 parts out of the whole of 15. This means that gravel makes up what is left over—the rest: 1 part out of the whole of 15. Now the problem is simple. Set up a proportion between parts and weights. Cross-multiply: x = 180. The answer is (E).

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