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# Page 1-Quantitative Reasoning Study Guide for the GRE®

## General Information

There are four types of Quantitative Reasoning questions:

• Quantitative Comparison questions require you to analyze two quantities and determine if they are equal, which one is larger, or if you cannot determine the answer based on the information given in the question.

• Multiple Choice — Select One questions are just what they sound like. You will be given five possible answers from which to select one correct answer.

• Multiple Choice — Select One or More questions require you to select all of the answer choices that satisfy the conditions of the question. The question may not tell exactly how many correct choices are possible.

• Numerical Entry questions do not have any answers from which to choose. Instead, you have to find the answer and enter it into spaces provided. One space will be given for an integer or decimal answer and two spaces for a fraction answer.

The GRE Quantitative Reasoning test contains lots of word problems. Some of them refer to real-life situations and some are strictly mathematical in scope. For all of them, you need to be able to pull up procedures and skills that will work together for a solution.

Four basic areas of math are covered on this test:

• Arithmetic
• Algebra
• Geometry
• Data Analysis

Here are some ideas for review. You will need to know the definitions of any terms you see, as well as how these items function in math, through operations and manipulation within problems. If you still feel unsure about any of this, please seek extra practice online or through workbooks.

## Arithmetic Ideas

### Integers

Integers are negative and positive whole numbers, including 0. Addition, subtraction, and multiplication of integers always yield another integer.

### Greatest Common Factor and Least Common Multiple

The factors of an integer are all of the numbers that can be multiplied by another number to arrive at the number in question. The greatest common factor, or GCF, is the largest positive integer that evenly divides into a group of numbers. The GCF is the largest positive factor common to a set of numbers.

The least common multiple, or LCM, is the smallest positive integer that is a multiple of a set of integers. The LCM is the smallest positive multiple common to a set of numbers.

### Operations with Positive and Negative Numbers

Consider the following results:

$$2 + 3 = + 5$$
$$5 - 3 = + 2$$
$$3 - 5 = - 2$$
$$2 - (-3) = + 5$$
$$-2 - (-3) = + 1$$
$$-2 - 3 = - 5$$

These results can be summarized with the following:

positive + positive = positive
large positive - small positive = positive
small positive - large positive = negative
small negative - large negative = positive
large negative - small negative = negative
negative + negative = negative

### Remainders

In the case where an integer does not evenly divide into another integer, a remainder is left over. The remainder can be represented as a decimal or a fraction.

Consider $$10 \div 4$$, which is equivalent to $$\frac{10}{4}$$.

$$4$$ goes into $$10$$ twice without going over and has a remainder of $$2$$ because $$10 - 2(4) = 2$$.

### Prime Numbers

A number that is only evenly divisible by two distinct values, one and the number itself, is a prime number. The number $$1$$ is not a prime number because it only has one factor; $$2$$ is the first prime number.

### Fractions: Definitions

A number that is not whole is a fraction. A proper fraction contains a numerator (top number) that is less than the denominator (bottom number). The proper fraction $$\frac{2}{3}$$ has a numerator of $$2$$ and a denominator of $$3$$; it can also be read as $$2$$ divided by $$3$$. An improper fraction has a numerator that is larger than the denominator. A mixed number has a whole number with a proper fraction.

A fraction that contains an integer numerator and an integer denominator (other than $$0$$) is called a rational number.

The reciprocal of a fraction is found by switching the numerator and denominator. For example, the reciprocal of $$\frac{2}{3}$$ is $$\frac{3}{2}$$.

### Fractions: Operations

In order to combine fractions through addition or subtraction, the fractions must first share a common denominator.

Consider $$\frac{1}{2} + \frac{2}{3}$$

We can rewrite both fractions with a common denominator by multiplying top and bottom of each fraction with the denominator of the other fraction:

$\frac{1}{2} + \frac{2}{3}$ $\frac{3}{3} \cdot \frac{1}{2} = \frac{3}{6}$ $\frac{2}{2} \cdot \frac{2}{3} = \frac{4}{6}$ $\frac{3}{6} + \frac{4}{6} = \frac{3 + 4}{6} = \frac{7}{6}$

Simplify to get:

$1\;\frac{1}{6}$

Multiplication of fractions is performed by placing the product of the numerators over the product of the denominators. If necessary, convert any mixed numbers to improper fractions before multiplying. To simplify the process, reduce where possible before multiplying.

Consider:

$\frac{9}{10} \cdot 2\;\frac{2}{18}$ $\frac{9}{10} \cdot \frac{38}{18}$ $\frac{1}{5} \cdot \frac{19}{2}$ $\frac{19}{10} = 1\;\frac{9}{10}$

Division of fractions is performed by first converting any mixed numbers to improper fractions and then rewriting the problem as the multiplication of the first fraction with the reciprocal of the second fraction, reducing where possible, and then multiplying across:

$\frac{9}{10} \div \cdot 2\;\frac{2}{18}$ $\frac{9}{10} \div \frac{38}{18}$ $\frac{9}{10} \cdot \frac{18}{38}$ $\frac{9}{10} \cdot \frac{9}{19}$ $\frac{9 \cdot 9}{10 \cdot 19} = \frac{81}{190}$

### Exponents and Roots

A term multiplied by itself a number of times can be expressed using an exponent:

$$2 \cdot 2 \cdot 2 \cdot 2$$ can be written as $$2^4$$, where $$2$$ is the base, and $$4$$ is the exponent.

A root is the same as a fractional exponent. The square root of a number is the number that, when multiplied by itself twice, equals the original number:

The square root of $$4$$ is $$2$$ because $$2 \cdot 2 = 4$$.

The third root of a number is the number that when multiplied by itself three times equals the original number:

The third root of $$8$$ is $$2$$ because $$2 \cdot 2 \cdot 2 = 8$$.

Symbolically, the $$n$$th root of a number is written with a radical or a fractional exponent:

$$\sqrt{4}=4^{\frac{1}{2}}=2$$.

The fractional exponent of $$x^{\frac{3}{4}}$$ means raise $$x$$ to the third power and find the fourth root of $$x$$. For example:

$$2^{\frac{4}{2}} = 16^{\frac{1}{2}} = 4$$, which is the same as reducing the fraction in the first exponent: $$2^{\frac{4}{2}}=2^2$$.

Evaluating a square root yields a positive solution and a negative solution, this is because $$-x \cdot -x = x^2 = x \cdot x$$. This is written as $$\sqrt{4} = \pm2$$

### Place Value and Decimals (including operations with them)

Place value is the value of a digit according to its place in a number. Consider the following number: $$1$$,$$234$$,$$567.987654$$

To the left of the decimal point:

$$1$$ is in the millions place
$$2$$ is in the hundred thousands place
$$3$$ is in the ten thousands place
$$4$$ is in the thousands place
$$5$$ is in the hundred place
$$6$$ is in the tens place
$$7$$ is in the ones place

To the right of the decimal point:

$$9$$ is in the tenths place
$$8$$ is in the hundredths place
$$7$$ is in the thousandths place
$$6$$ is in the ten thousandths place
$$5$$ is in the hundred thousandths place
$$4$$ is in the millionths place

When performing addition and subtraction with decimal numbers, always align the numbers vertically according to place value and position of the decimal point. This ensures that the correct place values are being combined. Consider: $$1.234 + 20.01$$, which can be rewritten as:

\begin{align} 20.010& \\ \underline{+\quad 1.234}& \\ \end{align}

When performing multiplication with decimal numbers, first count the number of digits to the right of the decimal point in the question. Multiply the numbers as usual. In the answer, place the decimal point so that the same number of digits are to the right of the decimal point as there are total decimal places in the numbers being multiplied.

When dividing decimal numbers, move the decimal point in both the divisor and dividend the same number of places such that the divisor becomes a whole number. Divide, as usual, placing the decimal in the quotient in the same place as the dividend: $$\frac{432.102}{0.032}$$ can be rewritten as: $$\frac{432,102}{32}$$.

Notice in the above that all we’ve done is multiply the numerator and denominator by $$1$$,$$000$$.

### Real Numbers

Real numbers are numbers that exist along the number line. They include rational numbers like $$-2, 3,$$ and $$100$$ as well as irrational numbers like $$\pi$$ and $$\sqrt{2}$$.

### Absolute Value

The absolute value of a quantity, expressed as $$\mid x \mid$$, is the distance of the quantity from zero. Absolute value is always positive: $$\mid-2\mid \;= 2$$.

Equations involving absolute value must always be solved for both the positive and negative case, consider the following:

$\mid \;x + 3\mid\; = 12$

To solve equations of this type, rewrite the problem as two equations with one equalling the positive answer and the other equalling the negative answer and removing the absolute value signs:

$$x+3=12$$ and $$x+3=-12$$

Solving gives:

$$x=9$$ and $$x=-15$$.

Always remember that algebra solutions can be verified by plugging the solved value back into the original expression:

$$\mid \;9 + 3 \mid \;= 12$$
$$\mid \;12 \mid \;= 12$$
$$12 = 12$$

and

$$\mid \;-15 + 3 \mid \;= 12$$
$$\mid \;-12 \mid \;= 12$$
$$12=12$$

### Ratio and Proportion

A ratio expresses a part to part or part to the whole relationship. Consider a classroom that has $$3$$ girls for every $$2$$ boys; the ratio of boys to girls can be written as “$$2$$ to $$3$$” or $$2:3$$ or $$\frac{2}{3}$$. Consequently, ratios can be thought of as fractions.

In the case where two things are similar in size or rate, a proportion can be used to solve for unknown quantities. For example, if a right triangle has a side length of $$6$$ and a side length of $$9$$, and another right triangle has a side length of $$12$$ and a side length of $$x$$, the unknown side length can be found with a proportion:

$$\frac{6}{12} = \frac{9}{x}$$, which can be solved through cross multiplication:

$6x = 12 \cdot 9$ $x = 2 \cdot 9 = 18$

### Percent

A percent is a part per $$\bf{100}$$; $$1\%$$ is the same as $$\frac{1}{100}$$.

A valuable formula to commit to memory is:

$\frac{part}{whole} = \frac{\%}{100}$

or

$\frac{is}{of} = \frac{\%}{100}$

This is very useful for questions like “What is $$xx$$ percent of $$yy$$?”

This can be translated to $$\frac{xx}{yy} = \frac{\%}{100}$$.

To convert a percent to a decimal, remove the number’s $$\%$$ sign and divide the number by $$100$$. This is the same as moving the decimal place in the number $$2$$ places to the left. To convert a decimal to a percent, multiply the number by $$100$$ and add a $$\%$$ sign. This is the same as moving the decimal point in the number $$2$$ places to the right.

A percent increase or decrease indicates a positive or negative change in a value from one period of time to another period of time.

The percent change formula is:

$\frac{\text{new value - old value}}{\text{old value}}$

If the answer is positive, there was an increase; if negative, a decrease.

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