Translating Phrases into Mathematics

In this lesson we will explore how to construct mathematical expressions and equations from phrases and sentences.

Mathematics is a language of numbers and symbols that assist us in solving a variety of problems that occur in daily life.  It is challenging, yet important to be able to construct equations and expressions that correspond to such word problems.  The difficulty is that each problem is different from every other problem.  Only through practice can one become adept at constructing these equations.  When you work on translating sentences into expressions and equations, do not attempt to tackle the whole problem all at once.  Instead read the first phrase.  Translate that phrase.  Then read the next phrase. Translate that phrase.  Continue until you have translated all of the phrases.  Finally put the translations together to come up with an expression or equation.  Here are some examples

Example 1

Jose wants to buy several songs for his new MP3 player.  He see an advertisement for $5 off his purchase if he buys in a large volume.  Each song costs $1.20.  Construct a mathematical equation that relates the total cost ,C, to the number ,n, of songs purchased.

Solution

To solve this problem we want to concentrate on the arithmetic: addition, subtraction, multiplication, or division.  Notice that the discount will be calculated after the regular price is calculated, so the discount will be the last piece of arithmetic to do.  Without the $5 discount, the cost would just be 1.20 for each song.  Notice that if 2 songs were purchased, the cost would be $2.40 which is
(1.20)(2).  If 3 songs were purchased, we would multiply (1.20)(3) to get 3.60.  Notice that the arithmetic done here is multiplication.  The factor 1.20 is present regardless of how many songs are purchased.  The second factor is the number of songs purchased.  The problem stated that this is n.  We have

        Cost Without the Discount  =  1.20 n

To see what arithmetic the discount corresponds to notice that the discounted price will be $5 less than the cost without the discount.  In other words, $5 will be taken away from the cost without the discount.  The arithmetic is subtraction.  We have

        Discounted Cost  =  1.20 n - 5

or

        C  =  1.20 n - 5

Now try one by yourself.  If you want to see the answer, put your mouse on the yellow rectangle and the answer will appear.

Exercise 1

Shawndrea went to a restaurant where she ordered several taquitos that cost $1.25 each (tax included).  She tips the waitress $1.50.  Write down an expression that expresses the total cost C for the meal in terms of the number n of taquitos Shawndrea orders.

Answer

Do not be surprised if there are more variables involved in the equation or expression.  Here is another example

Example 2

Two weeks ago Juanita stepped onto the scale and weighed x pounds.  She went on a diet for a week and lost 10 pounds, but then went off her diet and gained 4P pounds this week.  Write an expression that represents Juanita's weight today in pounds.

Solution

Juanita began weighing x pounds.  Losing 10 pounds is the same as subtracting 10 from her weight x.  So,

        Juanita's Weight After the First Week  =  x - 10

Gaining 4P pounds is the same as adding 4P to her weight.  So

        Juanita's Weight Today  =  (x - 10) + 4P

So the solution is

        x - 10 + 4P

Now try one by yourself.  If you want to see the answer, put your mouse on the yellow rectangle and the answer will appear.

Exercise 2

In the morning, the supermarket had M cartons of milk in stock.  During the day customers purchased 3w of these cartons and just before closing the distributor brought 45 cartons of milk to the supermarket.  Write down an expression that gives the number of cartons of milk in the store at closing time.

Answer

Let's try another one.

Example 3

The cost for renting an economy car is x dollars per day.  The car rental company charges $20 more per day to upgrade to a midsized car.  Ishmael rented a midsized car for 4 days.  Write an expression for the number of dollars that Ishmael had to pay for his rental car.

Solution

First let's find an expression that represents the cost of renting the midsized car for 1 day.  The words "$20 more per day" mean add 20 to the cost of the economy car.  So,

        Cost of Midsized Car for 1 Day  =  x + 20

Since Ishmael rented the car for 3 days, we need to multiply the cost for 1 day by 3.  We get

        Cost of Midsized Car for 3 Days  =  3(x + 20)

It is important to notice that this is not the same as 3x + 20, since the 3 multiplies the sum not just the first term x.  The answer is

        3(x + 20)

Now try one by yourself.  If you want to see the answer, put your mouse on the yellow rectangle and the answer will appear.

Exercise 3

The department store is having a sale where they are taking $4 off the regular price, r, of their T-shirts.  Patricia bought 3 T-shirts.  Write down an expression that represents the number of dollars that Patricia spent on T-shirts.

Answer

The CAHSEE often includes questions that describe how a number is manipulated and then asks the student to write down an algebraic expression or equation that represents the given description.  Here is an example of such a question.

Example 4

 Subtract 7 from a number, n,  and divide the result by 10.  The answer is 37.  Write an equation that represents this statement.

Solution

First translate, "Subtract 7 from a number, n".  This is

        n - 7

Notice that when you subtract a first number from a second number, it becomes

        second number - first number

Now "divide the result by 10."  We must divide expression n - 7 by 10, not just the 7.  We get

        n - 7
                   
         10

"The answer is 37" tell us to write what we have and put "= 37" on the right hand side.  Thus, we get

        n - 7
                    =  37
         10

Now try one by yourself.  If you want to see the answer, put your mouse on the yellow rectangle and the answer will appear.

Exercise 4

Multiply a number by 6 and add 5 to the result.  The answer is 15.  Write an equation that represents this statement.

Answer

Another important type of word problem involves inequalities such as less than, greater than, less than or equal to, and greater than or equal to.  Below are the 4 important symbols and their definitions to remember.

        <        Less than

        <        Less than or equal to

        >        Greater than

        >        Greater than or equal to

Here is an example that uses an inequality

Example 5

Write down the inequality that represents the statement,

        "A number, x, increased by 12 is greater than or equal to 15"

Solution

First, "A number, x, increased by 12" involves an addition problem because of the word "increased".  This becomes

        x + 12

Next, "greater than or equal to 15" involves the symbol ">".   We can put together these two ideas to get

       x + 12  >  15

Now try one by yourself.  If you want to see the answer, put your mouse on the yellow rectangle and the answer will appear.

Exercise 5

Write down the inequality that represents the statement,

        "A number, x, decreased by 8 is less than 10"

Answer