Renting a Car Word Problem

Here is the word problem:

Value Rent-A-Car rents a luxury car at a daily rate of $41.72 plus 10 cents per mile. A business person is allotted $130 for car rental each day. How many miles can the business person travel on the $130?

Here is a demo of me working through the problem.


You might be asking yourself where the four parts are? I was focused on explaining in the demo, but on a test we want to be more careful about our presentation of the solution. Here are the four parts of the answer that I want to make sure are on my paper before I turn it in.

• Part 1: Defining the Variable  x is the number of miles the person travels
• Part 2: Writing the Equation, Inequality, or System of Equations  41.72 + .10x = 130
• Part 3: Writing the Solution to the Equation, Inequality, or System of Equations x = 882.8
• Part 4: Write the Answer to Your Question The person can travel 882.8 miles for $130.

Parking at a Hospital Word Problem

Here is the word problem:

A hospital parking lot charges $2 for the first hour or part thereof, and $1 for each additional hour or part thereof. A weekly pass costs $47 and allows unlimited parking for 7 days. If each visit Jonny makes to the hospital lasts two and a half hours, what is the minimum number of visits for which buying a pass would be less expensive than paying each time?

Here is a demo of me working through the problem.


You might be asking yourself where the four parts are? I was focused on explaining in the demo, but on a test we want to be more careful about our presentation of the solution. Here are the four parts of the answer that I want to make sure are on my paper before I turn it in.

• Part 1: Defining the Variable  n is the number of times he goes
• Part 2: Writing the Equation, Inequality, or System of Equations  4n = 47
• Part 3: Writing the Solution to the Equation, Inequality, or System of Equations x = 11.75
• Part 4: Write the Answer to Your Question If Jonny visits 12 times or more, the weekly pass makes sense. 

Finding Angles of a Triangle

Here is the word problem:

The second angle of a triangle is 55 degrees more than the first angle. The third angle is 15 degrees less than twice the first angle. Find all three angles.

Here is a demo of me working through the problem.


You might be asking yourself where the four parts are? I was focused on explaining in the demo, but on a test we want to be more careful about our presentation of the solution. Here are the four parts of the answer that I want to make sure are on my paper before I turn it in.

• Part 1: Defining the Variable  x is the first angle
• Part 2: Writing the Equation, Inequality, or System of Equations  x + x+55 + 2x-15 = 180
• Part 3: Writing the Solution to the Equation, Inequality, or System of Equations x = 35
• Part 4: Write the Answer to Your Question The first angle is 35 degrees, the second angle is 90 degrees, and the third angle is 55 degrees.

A Board Problem

Here is the word problem:

A 190-inch board is cut into two pieces. One piece is four times the length of the other. Find the lengths of the two pieces. Find the length of the short piece. 

Here is a demo of me working through the problem.





You might be asking yourself where the four parts are? I was focused on explaining in the demo, but on a test we want to be more careful about our presentation of the solution. Here are the four parts of the answer that I want to make sure are on my paper before I turn it in.

• Part 1: Defining the Variable  x is the smaller of the two sub-pieces of the board
• Part 2: Writing the Equation, Inequality, or System of Equations  x + 4x =190
• Part 3: Writing the Solution to the Equation, Inequality, or System of Equations x = 38
• Part 4: Write the Answer to Your Question The small piece is 38 inches long. 

Consecutive Integer Problem

Here is the word problem:

The sum of three consecutive odd integers is 189. What are the integers?

Here is a demo of me working through the problem.





You might be asking yourself where the four parts are? I was focused on explaining in the demo, but on a test we want to be more careful about our presentation of the solution. Here are the four parts of the answer that I want to make sure are on my paper before I turn it in.

• Part 1: Defining the Variable  x is the first integer
• Part 2: Writing the Equation, Inequality, or System of Equations  x + x+2 + x+4 = 189
• Part 3: Writing the Solution to the Equation, Inequality, or System of Equations x = 61
• Part 4: Write the Answer to Your Question The three integers are 61, 63, 65.

Problems with Writing Answer (Part 4)

Here are a few common mistakes that are made on Part 4 of the word problems. Check them out and avoid them :)

• Not writing the answer out in a complete sentence: There are lots of variations on this mistake. To keep yourself straight, just remember that a word problem needs a word answer. Sometimes because of time, I see people simply copy there answer from Part 3. E.g. x=3, or x is 3. Remember, part 4 shouldn't have any variables in it, just words. Also I get some single number answers like 3. There is no explanation what the number is supposed to mean. Give me a nice complete sentence.

• Not answering the whole question: A lot of times the word problem asks for more than one quantity, but in our equation we only have one variable. Sometimes only the quantity that the variable represents makes it into the student's answer in part 4. It looks something like this.
• Question: Anna, Lea, and Simon have ages that add up to be 78. If they were all born in consecutive years, how old is each one?
• Part 1-- x: the youngest child
• Part 2--  x + (x+1)  + (x+2) = 78
• Part 3--  x=25
• Part 4-- The youngest one is 25.

Everything is great except for Part 4. Make sure that you include all the info the question is asking for. A full credit answer would be Anna, Lea, and Simon are 25, 26, and 27. 

• Not interpreting negative solutions from Part 3 correctly: This is a problem in Intermediate Algebra from Ch 5.9 on. Sometimes there are two solutions in Part 3, often a positive and negative one. Sometimes we can interpret the negative solution, sometimes we can't. Here's an example where the student doesn't realize that he/she can interpret it, which would give two solutions in Part 4. The question is What two consecutive even numbers have a product of 440?

The student got x = 20 and x = -22 in part 3, which is right. If 20 is the first integer, 22 is the second integer. If x = -22, then the next integer is -20. Since x is defined as an integer and integers can be negative numbers, it makes sense to interpret x = -22 into our answer for part 4. A nice Part 4 might be...There are two pairs of consecutive even integers that multiply to be 400. First the pair 20 and 22, and second the pair -22 and -20.

• Interpreting inequality answers as equalities. In Ch 2.8 in particular, MyMathLab asks a lot of questions involving inequalities. Key phrases in those kind a word problems are "more than", "at least as many".... A lot of times students set up an equality instead of an inequality. I understand this. Equalities are better understood and Ch 2.8 comes after Ch 2.6, the equality chapter. Bottom line, I won't count off if you use an equality in Part 2, but if the question asks for a range of answers, that should be reflected in the way you word your answer in Part 4. Here is an example were the student has missed that nuance.

 Great problem, except Part 4 should read "The length of the base needs to be at least 32 ft."

Problems with Writing the Solution to Your Equation (Part 3)

If you seem to be having trouble with part 3 of the word problems, here are a few common mistakes of which to be wary.

• Just writing a number: This is being a bit of a stickler, but I definitely take points off for it. Sometimes a student has a great solution going and just needs to be more careful about the presentation of the answer. Take for example the word problem.
• Anna, Lea, and Simon have ages that add up to be 78. If they were all born in consecutive years, how old is each one?
• A student might answer
• Part 1-- x: the youngest child
• Part 2--  x + (x+1)  + (x+2) = 78
• Part 3--  25
• Part 4-- Anna, Lea, and Simon are 25, 26, 27
• This student really knows what they are doing, but got sloppy in part 3. What I'm looking for in part 3 is "x = 25"

• Don't forget to carry down your inequality, also don't forget to check it's direction: Here is a student example of a good looking word problem with a careless mistake.

The problem is that the inequality in part 2 became an equality in part 3. Part 3 should read

• Write all solutions to the equation: This tends to be an issue in Intermediate Algebra after chapter 5.9. Starting in Chapter 5, equations will often have two solutions. Sometimes both solutions will lead to answers in part 4, but often only the positive one will have an interpretation. Regardless, part 3 of the word problem is only asking for the solutions to the equation. It doesn't care wether those solutions can be interpreted or not. That's the difference between the information provided in parts 3 and 4. Part 3 is asking for the solution to the equation. Part 4 is asking for the answer to the word problem. Here is a good example. This student wrote a little more than they had to in parts 1 and 2, but this is a great answer....except for part 3. His/her equation from part 2 is 180=5w(w). He/she needs to provide all w's that solve the equation. The correct answer for part 3 would be w=6,-6. He/she doesn't even need to say L=30.