An important part of problem solving in mathematics is solving word problems. After all, real-world problems rarely present themselves in a graph, equation, or table. They are common, everyday situations that must be translated into a mathematical representation in order to be solved.
In this lesson, you will focus on ways to use strategies such as equations or tables of values in order to solve meaningful word problems.
Translating verbal situations, or word problems, into equations and tables of values is an important mathematical skill, since real-world problems do not often present themselves in mathematical terms. The mathematics often has to be inferred from the context of the sentence.
A marathon is a foot race that is run at a distance of 26.2 miles. The name marathon comes from the Greek town called Marathon, which is located 26.2 miles from Athens. According to legend, an ancient Greek messenger ran from Athens to Marathon to share the good news that the Greek army had just been victorious in battle. Hence, the distance between Athens and Marathon has become a benchmark distance for a foot race for modern runners.
Source: Marathon de New York: Verrazano Bridge, martineric, www.flickr.com.
You are familiar with the equation that relates distance, rate, and time:
D = rt
In this equation, d represents distance, r represents rate, and t represents time.
You have been given the following information: distance and time.
Use the equation that you just found in order to help solve the following problem:
The finishing times for four friends who ran the Big-D Texas Marathon in Dallas, a distance of 26.2 miles, are shown. Use the relationship between distance, rate, and time to determine their average running speeds.
Practice
Courtney and her family are driving from their home in San Angelo to visit her older sister in College Station, a distance of 290 miles. The drive took them about hours. At this same average speed, how long (in hours and minutes) would it take Courtney and her family to drive from San Angelo to Fort Worth, a distance of 230 miles?
Hai subscribes to an online bookstore. He pays an annual membership of \($\)12.50, and then can download books for either \($\)5.99 or \($\)3.99. Last year, he downloaded 9 books for \($\)5.99 each and 16 books for \($\)3.99 each. How much did Hai pay, in all, to the online bookstore?
Another useful strategy when solving word problems is to make a table of values.
Consider the following problem:
Leslie Massey works as an investment banker. One particular investment that her clients like earns 4.5% interest each year. Copy the table below into your notes and complete the amount of interest earned column for each investment listed, if the investor keeps his or her money in the investment for one year.
| Amount of Investment | Amount of Interest |
| $1,500 | |
| $2,500 | |
| $3,500 | |
| $4,500 |
Practice
Fumiko works at a clothing store that is advertising 20% off all merchandise. Three customers' purchases are shown below.
I. 2 shirts, each with an original price of \($\)25, and one skirt with an original price of \($\)35
II. 2 sweaters, each with an original price of \($\)30, and one pair of pants with an original price of \($\)40
III. 1 pair of shoes with an original price of \($\)45 and 1 pair of sandals with an original price of \($\)25
Which of the customers saved at least \($\)15?
Mara needs to mail a large envelope that weighs 1 pound 3 ounces. The first ounce costs \($\)0.95, and each additional ounce costs \($\)0.20. How much in postage will Mara pay to mail the envelope?
In this lesson, you accomplished two learning goals.
First, you generated an equation or a table of values that can be used to solve a problem that is presented to you in verbal form (a word problem).
A common type of problem you will encounter includes the relationship between distance, rate, and time.
The equation D = rt is particularly useful for solving these types of problems.
The relationship between distance, rate, and time is often expressed using a triangle graphic organizer. To solve for one unknown, cover up the variable representing the unknown, and then either multiply or divide the remaining two. If the remaining two unknowns are horizontally arranged, you will multiply. If the remaining two unknowns are vertically arranged, you will divide.
Review the series of images below to see these relationships.
Second, you used a table of values to help solve a problem and interpret the solutions.
When working with problems involving money, such as interest rates, sales tax, or discounts, it is helpful to use the relationships to make a table of values to help you solve the problem. For example:
Alana works in a bookstore, and her local sales tax rate is 7.5%. She made a table of purchase subtotals to help her estimate the amount of sales tax that a customer will pay if their purchase is close to a certain amount:
|
Amount of |
Amount of |
|---|---|
| $50 | $3.75 |
| $60 | $4.50 |
| $70 | $5.25 |
| $80 | $6.00 |
If a customer purchases \($\)53 worth of books, Alana can use the table to know that the customer will pay close to \($\)3.75 in sales tax. If a customer purchases exactly \($\)70 worth of books, Alana can use the table to know that the customer will pay exactly \($\)5.25 in sales tax.