Modelling Words With Math
In mathematics, when we are working with application problems, it is extremely important to make sure we read and understand the question being asked.
- Often, the words used to describe the situation can be simplified using mathematical operations and equations.
- In this lesson, when we are working through solving application problems involving a linear system, we will be able to take the words given and represent them with two equations.
Before fully working through some examples, let's first take some time to review some common language used in application problems and look at how to develop equations from words.
Translating Words to Math
The table provides some common words used to describe the mathematical operations.
| Operation |
Common Words |
| \(+\) |
sum, added to, more than, increased by |
| \(-\) |
subtracted from, minus, less than, fewer than, decreased by |
| \(\times\) |
times, product, multiplied by, doubled, tripled |
| \(\div\) |
divided by, quotient, per, ratio, half |
In the following examples, we will only look at modelling the words given as equations; we will not work through the calculations to solve and answer the question asked. Later in this lesson, we will work through complete solutions.
Example 1
Cindy thinks of two numbers. Three times the smaller number subtracted from double the larger number is equal to thirty-two. The sum of the two numbers is equal to six. Write two equations to represent the description of Cindy's numbers.
Solution
Before writing equations, it is important to first define variables to represent the unknown quantities.
For this example we will write:
- Let \(l\) represent the larger number that Cindy is thinking of.
- Let \(m\) represent the smaller number that Cindy is thinking of.
Now we can write our equations with these variables.
Let's consider the second sentence and highlight key terms that will help us create the first equation:
"Three times the smaller number subtracted from double the larger number is equal to thirty-two."
Our first equation is
\(2l-3m=32\)
Let's consider the third sentence and highlight key terms that will help us create the second equation:
"The sum of the two numbers is equal to six."
Our second equation is
\(l+m=6\)
Now we have a system of two equations involving the variables \(l\) and \(m\) and could use these equations to solve for the numbers that Cindy is thinking of.
\(\begin{align*} 2l-3m&=32\tag{1}\\ l+m&=6\tag{2}\\ \end{align*}\)
Example 2
Paul orders three hamburgers and four large french fries from a local fast food restaurant. His total bill is \($16.13\).
Sean orders four hamburgers and two large french fries from the same restaurant and his total bill is \($15.54\).
Write two equations to represent the description of orders that would help solve the question: What is the difference in cost between one hamburger and one large french fry order?
Solution
First, we will define variables to represent the unknown quantities.
- Let \(h\) represent the cost of a hamburger in dollars.
- Let \(f\) represent the cost of one large french fry order in dollars.
Based on Paul's order, we can write our first equation as
\(3h+4f=16.13\)
Based on Sean's order, we can write our second equation as
\(4h+2f=15.54\)
Now we have a system of two equations involving the variables \(h\) and \(f\).
\(\begin{align*} 3h+4f&=16.13\tag{1}\\ 4h+2f&=15.54\tag{2}\\ \end{align*}\)
To answer the question asked, we would solve for the values of \(h\) and \(f\) and then find the difference between the two.