### I. Strategy

Strategy

The GMAT rarely asks questions that are simple and straightforward. Expect problems that require you to convert complex written statements into variables and equations. The most effective strategy for solving a word problem is to express the question as an equation or relationship where x, or some other letter, represents the quantity that you need to find.

#### The 5 Steps

1. Read the question.
3. Define variables and relationships.
4. Choose a technique and apply it.
5. Eliminate choices.

### II. Age

Age

Like all word problems, age problems require writing an equation using the words that define the relationships and actions. Age problems just use more of these words than other types of word problems.

The key is to calmly translate the words into manageable equations. Often each sentence requires a new equation.

Know the common words for mathematical actions and relationships. For example, the words that translate to the equals sign include “is,” “was” and “equals.”

### III. Functions

Functions

A function ƒ(x) describes a relationship between one or more inputs and one output. On the GMAT, you can simply think of a function as an instruction for how to treat a particular variable or expression.

#### Combining Functions

You can combine functions using any operation.

#### Composite Functions

Another way of combining functions is with a composite function. This means the functions are nested, so you apply one function to find a value, then apply a second function to that value.

It is important to follow the order of operations, doing the inside function first, then the outside function.

#### Variety of Symbols

On some questions, the functions won’t use the standard ƒ(x) or g(x) format. Instead, they will use symbols, including #, & and ♣. The symbols can be confusing, but just treat them the same as any other function.

### IV. Sequences

Sequences

sequence is an ordered list of numbers. Each number contained in a sequence is called a term.

A sequence is defined by an equation or rule. The key to solving sequence problems is to determine the relationship between the terms.

#### Writing the Rules

There is a common set of variables used in the equation or rule for a sequence:

• Subscripts are used to give the position-number of a term.
• an represents the value of the nth term.
• d is the difference between terms.
• r is the ratio between terms.

#### Arithmetic Sequence

In an arithmetic sequence the difference between consecutive terms is constant. You add (or subtract) the same amount to go from one term to the next.

• arithmetic sequence    an = a+ (n – 1)d

#### Arithmetic Series

series is the sum of the terms in a sequence. The sum of an arithmetic sequence is the mean of the first and last terms multiplied by the number of terms.

arithmetic series

Sn = n \Big( \dfrac{\textit{a}_{\displaystyle{1}} + \textit{a}_{\displaystyle{\textit{n}}}}{2}\Big)

#### Geometric Sequence

In a geometric sequence the ratio between consecutive terms is constant. You multiply by the same number to go from one term to the next.

• geometric sequence     an = a1 × n – 1

#### Sum of a Sequence

The formula for the sum of a sequence has two pieces:

1. The number of integers
2. The arithmetic mean of the sequence

The sum of a sequence with a constant difference of 1 is the number of terms in the sequence times the mean of the sequence, or: (LF + 1) \dfrac{\,\,(\textit{L}+\textit{F})\,\,}{2}

### V. Percent

Percent

A percent is a fraction whose denominator is 100. A percentage is used to compare values. It can describe the relationship of a part to a whole, or it can describe a magnitude of change.

#### Percent and Conversions

• Decimal to Percent: Multiply by 100
• Percent to Decimal: Divide by 100
• Fraction to Decimal: Divide the numerator by the denominator
• Decimal to Fraction: Move the decimal point 2 places to the right and use 100 as the denominator. Then simplify

#### Multiplying by Percent

Multiplying by a percentage is a way to find a “part” of the original. Multiplying by a percentage is the same as multiplying by the decimal equivalent.

#### Percent Increase and Decrease

Percentages can be used to describe the magnitude of change. To find the percentage of change, you compare the new amount to the original amount.

#### Discounts and Markups

• Discount is the decrease in price of an item when the price is decreased by a certain percentage.
• Markup is the increase in price when the cost of an item is increased by a certain percentage.

For markups and discounts, calculate:
new – original/original = percent/100

If the value is negative, it is a discount. If the value is positive, it is a markup.

### VI. Interest

Interest

#### Simple Interest

The formula for calculating simple interest is:

I = P r t

• The amount of interest I is the dollars earned or paid to the investor.
• The principal P is the amount of money invested.
• The interest rate r is the annual interest rate expressed as a decimal value.
• The time period t is measured in years

#### Compound Interest

The formula for calculating compound interest is:

= P (+ r/n)nt

This formula gives the value of the account A, not the interest earned. The new component is n, which is how often the interest is paid or compounded per year.

On the GMAT, you may encounter a question where it seems like compound interest would take too long to calculate. By looking at the answer choices you can determine if an estimate will work.

### VII. Ratio and Proportion

Ratio and Proportion

#### Ratio

A ratio is used to compare two quantities.

The ratio of numbers a and b can be expressed as:

• the ratio of a to b
• a is to b
• a : b
• a/b

#### Proportion

A proportion states that two ratios are equal.

A proportion comparing two ratios can be expressed as:

• a is to b as c is to d
• a : b = c : d
• a/b = c/d

You can solve for any term in a proportion: abc or d.

#### Inversely Proportional

Two quantities are directly proportional if one equals the other multiplied by a constant, or y = cx where c is a constant. When two quantities are directly proportional, both values increase or both values decrease.

Two quantities are inversely proportional if one decreases when the other increases. One value is equal to a constant divided by the other

### VIII. Distance, Rate & Time

Distance, Rate & Time

The distance d that an object will travel is equal to its rate r times its time t.

d = r t

When setting up an equation, make sure that the units match.

The most common methods for solving motion problems are to use a system of equations or to use a proportion.

### IX. Work

Work

The amount of work W, accomplished in time T, depends on the rate R, at which the work is done. This relationship is described by the equation:

W = RT

• The amount of work is often one completed job, so usually W = 1.
• The time T is how long it takes to complete the entire job.
• The rate R is the job divided by time.

### X. Sets

Sets

A set is a group of distinct objects, called elements or members. Sets are used to look at the descriptions and interactions of the elements.

Elements are typically listed in {brackets}. The best way to visualize a set is with a Venn diagram. Use overlapping circles to organize the elements of a set.

#### Number of Elements

Counting problems are one type of set questions. These questions can ask about the number of elements in the set or subsets.

#### Using Sets

Using sets as the basis for questions about probability, averages, and combinations/permutations is a second type of GMAT question that uses sets.

#### Range of Values

Range questions also refer to the elements in a set. When a question asks for a possible range, be sure to check both the lowest and highest possible values.

### XI. Graphs and Data Interpretation

Graphs and Data Interpretation

Data interpretation involves computing and approximating numerical values based on tables and graphs. GMAT questions go beyond just reading the data, requiring you to calculate averages or compare changes, for example. This type of question is usually in the Problem Solving format, and can appear in sets of 2 to 3 questions.

#### Displaying Data

Data can be displayed in many different ways, including tables and graphs. Both tables and graphs can be used in calculations to evaluate and interpret the data.

#### Tables

Tables give values that are organized but not represented visually. You may graph the data from the table to make comparisons and see trends. You may create a table from a graph to make calculations easier.

#### Circle Graphs

Circle graphs represent values as “slices of pie.” Each part is labeled with a fraction or a percent, with the values adding to 1 or 100%.

#### Bar Graphs

Bar graphs can be used to compare non-number categories by showing the number of times each category occurs.

#### Double Bar Graphs

Bar graphs can show more than one set of values for each category.

#### Histograms

histogram is very similar to a bar graph, but has no spaces between the bars. The main difference is that histograms use continuous grouped data to show frequency trends.

#### Line Graphs

One type of line graph is a broken line graph. Segments are used to join the values but the graph has multiple slopes.

### XII. Mean, Median & Mode

Mean, Median, & Mode

An average is a number used to describe a set of data. Averages give a measure of the “middle” of a set.

The most common averages are the arithmetic mean, the median and the mode. The mean is often thought of as the average, but the terms are not interchangeable. The GMAT uses the term arithmetic mean instead of average to be more precise.

#### Arithmetic Mean and Median

The arithmetic mean is the sum of a set of numbers divided by the number of elements in the set.

mean = \dfrac{sum \,of \,the \,values}{number \,of \,values}

The median is the “middle” number of a list of data. To find the median, arrange the numbers in numerical order. If the number of values is odd, the median is the middle number. If the number of values is even, the median is the mean of the two middle numbers.

#### Mode and Range

The mode is the most frequently occurring number in a set of data. There can be more than one mode.

The range isn’t an average. It describes the spread or dispersion of a data set. The range is the difference between the largest and smallest values.

#### Comparing Data

Every set of numbers has a mean, median and range. These three values can be used to compare data by seeing how much the numbers differ.

## Sample Questions

#### Chapter Review

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13 questions with video explanations

100 seconds per question Before attempting these problems, be sure to review this section on Quantitative Comparison questions.