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This lesson is part of a series of lessons for the quantitative reasoning section of the GRE revised General Test. In this lesson, we will learn:
 Applications of Algebra
 Translating Words to Algebra
 Examples of using Algebra to solve Word Problems
 Average Word Problems
 Mixture Word Problems
 Distance, Rate, Time Word Problems
 Work Word Problems
 Word Problems that involve Simultaneous Equations
 Word Problems that involve Inequalities
Applications of Algebra
Translating verbal descriptions into algebraic expressions is an essential initial step in solving word problems.
Some examples are given below.
• If the square of the number x is multiplied by 4, and then 15 is added to that product, the result can be represented by 4
x^{2} + 15
• If Peter’s present salary
s is increased by 15 percent, then his new salary is 1.15
s.
• If
y gallons of orange juice are to be distributed among 5 people so that one particular person gets 1 gallon
and the rest of the juice is divided equally among the remaining 4, then each of those 4 people
will get (
y − 1) ÷ 4
This video shows how to translate words and expressions into algebraic expressions involving variables.
Examples of using algebraic techniques to solve word problems
We will now show some examples of using algebraic techniques to solve word problems. More examples of algebra word problems can be found
here.
Average Word Problems
The following are examples of word problems that involve the average or arithmetic mean.
Example 1:
Ellen has received the following scores on 3 exams: 82, 74, and 90. What score will
Ellen need to receive on the next exam so that the average (arithmetic mean) score for the 4 exams will be
85?
Let x represent the score on Ellen’s next exam.
The average
of the 4 exams is
Solving for x we get
Therefore, Ellen will need to attain a score of 94 on the next exam.
Example 2:
On her first four games Jennifer bowled 101, 112, 126, 108. What is the minimum score she must bowl in her fifth game in order to have an average of at least 110?
Mixture Word Problems
The following are some examples of word problems that involve mixture and concentration of mixture.
Example 1:
A mixture of 12 ounces of vinegar and oil is 40 percent vinegar, where all of the
measurements are by weight. How many ounces of oil must be added to the mixture to produce a new
mixture that is only 25 percent vinegar?
Let x represent the number of ounces of oil to be added. Then the total number of ounces of the
new mixture will be 12 + x, and the total number of ounces of vinegar in the new mixture will be (0.40)(12). Since the new mixture must be 25 percent vinegar, we get
7.2 ounces of oil must be added to produce a new mixture that is 25 percent vinegar.
Example 2:
A solution contains 15% hydrochloric acid. How much water should be added to 50 ml of this solution to dilute it to a 2% solution?
Distance, Rate, Time Word Problems
The following are some examples of distance, rate, time word problems.
Example 1:
In a driving competition, Jeff and Dennis drove the same course at average speeds of 51
miles per hour and 54 miles per hour, respectively. If it took Jeff 40 minutes to drive the course, how long
did it take Dennis?
Let x be the time, in minutes, that it took Dennis to drive the course. The distance d, in miles,
is equal to the product of the rate r, in miles per hour, and the time t, in hours; that is, d = rt
Note that since the rates are given in miles per hour, it is necessary to express the times in hours; for
example, 40 minutes equals of an hour.
Using the formula, d = rt, we can get the following table.

Distance 
Rate 
Time 
Jeff 
51 × 
51 

Dennis 
54 × 
54 

Since the distances are equal,
It took Dennis approximately 37.8 minutes to drive the course.
This video gives an introduction to solving word problems on uniform motion (ratetimedistance) using the formula rate x time = distance, or
rt=
d.
Example: Two cyclists start at the same corner and ride in opposite directions. One cyclist rides twice as fast as the other. In 3 hours, they are 81 miles apart. Find the rate of each cyclist. Answer: 9 mph and 18 mph.
This video solves this word problem using uniform motion
rt=
d formula
Example: A jogger started running at an average speed of 6 mph. Half an hour later, another runner started running after him starting from the same place at an average speed of 7 mph. How long will it take for the runner to catch up to the jogger? Answer: 3 hours.
This video solves this word problem using uniform motion
rt=
d formula:
Example: A 555mile, 5hour trip on the Autobahn was driven at two speeds. The average speed of the car was 105 mph on the first part of the trip, and the average speed was 115 mph for the second part. How long did the car drive at each speed? Answer: 105 mph for 2 hours and 115 mph for 3 hours.
This video solves this word problem using uniform motion
rt=
d formula:
Example: Andy and Beth are at opposite ends of a 18mile country road with plans to leave at the same time running toward each other to meet. Andy runs 7 mph while Beth runs 5 mph. How long after they begin will they meet? Answer: 1.5 hours.
This video solves this word problem using uniform motion
rt=
d formula:
Example: A car and a bus set out at 2 pm from the same spot, headed in the same direction. The average speed of the car is twice the average speed of the bus. After 2 hours, the car is 68 miles ahead of the bus. Find the rate of the bus and the car. Answer: Bus speed: 34 mph; Car speed: 68 mph.
This video solves this word problem using uniform motion
rt=
d formula:
Example: A pilot flew from one city to another city averaging 150 mph. Later, it flew back to the first city averaging 100 mph. The total flying time was 5 hours. How far apart are the cities? Answer: 300 miles
Work Word Problems
The following are some examples of word problems that involve work done by individuals or machines.
Word Problems Involving Work
Applications that involve Systems of Equations
The following are examples of applications that involve systems of equations or solving
simultaneous equations.
Example: The perimeter of a rectangle is 160 yd. The width is 4 more than half the length. Find the length and the width.
Example: Assume all Pheonix College courses are either worth 3 or 4 credits. The cross country team is taking a total of 40 courses that are worth 144 credits. How many 3credit courses and how many 4credit courses are being taken?
Applications that involve inequalities
The following are examples of algebra word problems that involve inequalities.
Example: A widget factory has a fixed operating cost of $3,600 per day plus costs $1.40 per widget produced. If a widget sells for $4.20, what is the least number of widgets that must be sold per day to make a profit?
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