Example Problems

Let’s look at some examples:

A cable company charges homeowners a one-time setup fee of $200, plus dollars for each month of service.  If a homeowner paid $980 for 10 months of service, including the setup fee, what is the value of ?

A) 65
B) 98
C) 87
D) 78

We see numbers in the answer choice, so we know we need to turn the words into an equation.  Because there is a variable in the problem, we almost certainly need to generate a linear equation.  Remember the slope-intercept form of linear equations:

$$y=mx+b$$

Our y value is always the total.
Our m is the amount of change in y per one increase in 
Our is our initial starting point or flat fee.

Knowing this, we can generate the equation

$$980=10d+200$$

We then just need to solve for by subtracting 200 and then dividing by 10.  Doing so gives us answer D).

Sometimes as part of the word problem, they will even give you the equation to use.  Solving these is no different – you might just have to do some basic manipulations to the equation to get it into slope-intercept form, and then we plug in our numbers from the word problem.  For example:

$$4x+2y=16$$

Gets transformed into: $$y=-2x+8$$

Another example:

A runner is training for a marathon.  Each week she has to run 40 miles.  She decides to train harder, and add 3 more miles every week she trains.  Which equation represents how many miles she will need to run after x weeks?

A) (y=40x+3)
B) (y=120)
C) (y=3x+40)
D) (3y=40x)

All of the answer choices are equations, so there must be coordinates in the word problem:

The total number of miles is y.

The thing that is changing over time is how many additional miles she is running, 3.

The starting point is 40.  She has to do this every week regardless.

Our correct answer is C).

Another example:

A chemist mixes together two kinds of powders.  Powder A contains 70 percent sulfur by weight and Powder B contains 30 percent sulfur by weight.  Together, the powders have 12 kilograms of sulfur.  Which equation models this relationship, where x is Powder A andy is Powder B?

A) (70x+30y=12)
B) (30x+70y=120)
C) (0.3x+0.7y=12)
D) (0.7x+0.3y=12)

Because we are dealing with percentages, we must convert to decimal to perform mathematical operations.  We can immediately rule out A) and B).  Looking at the words, we need to match up Powder A, 70 percent, and x.  This leads us to answer D).

This leads us to the last type of word problem – the one where we look at the answers and see a bunch more words. Fortunately, we’ve already covered how to answer these.

Our y value is always the total.

Our m is the amount of change in y per one increase in x

Our is our initial starting point or flat fee.

For instance:

Howard wants to buy tickets to a concert.  He goes online and finds out the vendor charges a one-time service fee in addition to the cost of the tickets.  The equation (R=80t+30) represents the amount of dollars, , that he must pay for t tickets.  What does 30 represent in this equation?

A) The price of one ticket, in dollars
B) The total amount of dollars Howard will pay for any tickets
C) The amount of the service fee, in dollars
D) The number of friends he can invite to the concert

We are able to see that 30 represents  b in  (y=mx+b), and we know that b always means a starting point or flat fee.  This leads us to C).

A slightly trickier example:

The function q , defined by f(q)=at+b, where and b are constants, models the height, in centimeters of bamboo plants in a contained environment after t days, during a period in which the growth is approximately linear.  What does a represent?

A) The predicted number of centimeters the bamboo plant grows each day
B) The predicted height, in centimeters, of the bamboo plant at the end of the period
C) The height, in centimeters, of the bamboo plant at the start of the period
D) The predicted total increase in height of the bamboo plant during the period

They are trying to make this look harder by using functional notation and describing values as constants.  If we take a step back, we see that we still have the same (y=mx+b) equation we’ve been dealing with.  They have called as here to try and confuse us, but we know that the thing that is changing over time is the height of the plant, every day it will grow a centimeters, so A) is our answer.