GRE Exam: A Simple Guide To Solving Word Problems

Students of math at almost every level experience some intimidation when confronted with lengthy word problems. The words in the situation often seem to conceal the math necessary to get to the answer. This whole guide is devoted to ways to get past this difficulty.

Word problems fall into several predictable categories, and this guide exploits this predictability by providing you with a framework to implement each time the GRE throws a certain type of word problem at you.

It is worth mentioning now: your ultimate goal is to convert the words into algebra. This is the most difficult and most important component of these questions.

Be sure to give yourself sufficient time to convert the words into algebra. Once you have done so, it is simply a matter of implementing the algebraic concepts that have been covered on Fatskills.

Word Problems
Word problems tend to be intimidating for many test-takers. One of the most common sentiments students express is concern over where to start. You are given a sentence or several sentences and expected to somehow develop mathematical relationships from these sentences. In the following pages, you will see step-by-step approaches for dealing with these situations.

Keep in mind that your ultimate goal should always be the following:
Create algebraic relationships!
Once you’ve created relationships, you can then use your algebra skills to solve the problem.

Let’s look at a typical word problem and a step-by-step methodology for creating these algebraic relationships.
The sum of the lengths of two pipes is 70 feet. The length of the longer piece is 20 feet more than the length of the shorter piece. What is the length of the shorter piece?

Step 1: Identify your unknowns. An unknown is any quantity with an unspecified value. An unknown can be something like Bob’s age, Jack’s height, the number of people in a room, and so on. In the preceding question, there are two unknowns: the length of the shorter pipe and the length of the longer pipe.

Step 2: Assign variables to the unknowns. Since your ultimate goal is to derive algebraic relationships, you should express your unknowns as variables: Let l = the length of the longer pipe and let s = the length of the shorter pipe. You can use other letters as well, but it is helpful to use letters that help you remember which unknown the variable refers to (in this case, you can use l for “longer” and s for “shorter”).

Step 3: Identify relationships among the unknowns. This is the final step in going from words to algebra.

Once you identify a relationship, you can create an equation or inequality and start solving for your variables. Words such as is, equals, is greater, and is less are helpful indicators of relationships.

In the previous example, there are two relationships among the variables:
 

Relationship 1: The sum of the lengths of two pipes is 70 feet. Since sum means addition, you should interpret this information to mean the length of the shorter pipe + length of the longer pipe = 70

Relationship 2: The length of the longer piece is 20 feet greater than the length of the shorter piece. The word is indicates a relationship between l and s. On your paper, write down:

Now translate the wording “20 feet greater.” Since the longer piece is 20 feet more than the shorter piece, it must be true that s alone is not enough to equal l. Thus to make the two sides of the equation equal, you must add 20 to s:

Step 4: Solve for the unknown. At this point, you have two algebraic relationships:

Since the question asked for the length of the shorter piece, the final step is to use substitution to solve for s. Substitute (s + 20) for l in the first equation: (s + 20) + s = 70. Solve for s:


Price and Quantity Relationships
Some word problems will require you to recognize the following relationship: (price/unit) × (number of units) = total price

Though it might be intimidating, this is a translation that most people use every day.

To illustrate this, look at the following example:
If Bob purchased 15 $30 shirts, how much money did he pay for all the shirts?

SOLUTION: Plug the values into the given formula. The price per shirt is $30, and the number of shirts is 15. Thus the total price is 30 × 15 = $450.

Now look at a more GRE-like example:
Bob spends a total of $140 at a certain shop, where he purchases a total of 20 shirts and ties. If the price of each shirt is $10, and the price of each tie is $5, how many shirts does he buy?
Step 1: Assign variables. Since you are not given a value for the number of shirts or the number of ties, let s = the number of shirts and t = the number of ties.
Step 2: Identify relationships. There are two relationships in the question:
Relationship 1: Bob purchases a total of 20 shirts and ties. Thus t + s = 20.
Relationship 2: The total amount Bob pays for the shirts and ties is $140. This amount will be the sum of the amount he paid for the shirts and the amount he paid for the ties. If each shirt costs $10, then he paid 10s for all the shirts. If each tie costs $5, then he paid 5t for all the ties. The algebraic relationship will be 10s + 5t = 140.
Step 3: Combine the equations to solve.

Since you are asked to solve for s, you should use the first equation to write t in terms of s: t = 20 – s.

Next, substitute (20 – s) for t in the second equation:

Solve for s:


Age Questions
A common type of word problem that gives many students difficulty concerns age. The approach toward these questions is very similar to what you have looked at so far, though you will need to keep a couple key facts in mind.

Let’s look at an example:
Bob is 13 years older than Jack. In 3 years Bob will be twice as old as Jack. How old is Jack?
Step 1: Assign variables. Let j = Jack’s current age and let b = Bob’s current age.
Before moving on, it is important to understand the emphasis on current age. In most age questions, you will be given information about the people’s ages at some earlier or later point. By assigning variables for the current age, you will be able to express these new ages in terms of the current ages instead of introducing new variables.
 

Step 2: Identify relationships.
Relationship 1: Bob is 13 years older than Jack

Relationship 2: In 3 years Bob will be twice as old as Jack. Bob’s age in 3 years will be b + 3. Jack’s age in 3 years will be j + 3. Thus you can construct the following equation: b + 3 = 2(j + 3)

Step 3: Combine the equations to solve.

Since the question asks to solve for j, you should substitute (j + 13) for b in the second equation.

Common Translations for Word Problems