Hints for Solving Story Problems

When working a story problem, use these four steps as a guideline.

Definition

Write a good definition statement. You should define all variables used in the problem. It is usually better to let the initial variable be the smallest quantity or the one that all others are defined in terms of. The reason the smallest is the best is because everything else will be in terms of addition and multiplication, instead of subtraction and division. Some instructors have been known to give no credit on the problem, even though the answer may be correct, if there is no definition statement. And yes, these instructors are still teaching here, so be careful if you go on to another class. Get in the habit now.

A good definition statement often involves more than writing down "let x be the first number and y be the second". For example, if one line of the problem says "twice the larger is three times the smaller", then you should define the variables in terms of the size: "let x be the smaller number, let y be the larger number". This will help later in the second step.

Another thing to watch out for is consecutive integer problems.

• consecutive integers: let n be the first, let n + 1 be the second, let n + 2 be the third
• consecutive even integers: let n be the first, let n + 2 be the second, let n + 4 be the third
• consecutive odd integers : let n be the first, let n + 2 be the second, let n + 4 be the third

Equation(s)

Write one or more equations describing the problem. Since you defined all of your variables, this should be easier. If the problem says "twice the larger is three times the smaller", you don't need to worry about getting the variables right because you defined them properly in the definition stage. Using the good definition statement above, this becomes "2y = 3x".

Solution

Solve the equation(s) arrived at in the second step. Write down the answer in terms of the original problem. "A story problem deserves a story answer". I have known instructors who have taken off points if the answer isn't a complete sentence. And yes, these instructors are still teaching here, so be careful if you go on to another class. Get in the habit now.

Check

Check your answer! Not necessarily into the equation(s) arrived at in the second step. You may have solved the equation that you have correctly, but had the wrong equation to begin with. By checking into the original problem and asking yourself if your answer makes sense, you can avoid some simple mistakes.

Let's examine a story problem and some common answers and why the answers don't make sense when checked.

Example

Billy can paint a wall in 5 hours and Suzie can paint a wall in 4 hours. How long will it take them, working together, to paint the wall.

• 9 hours: Since Billy takes 5 hours, and Suzie takes 4 hours, then together they take 5 + 4 = 9 hours. This doesn't make sense because Billy could do it in 5 alone. When people work together, things are supposed to go faster.
• 4.5 hours: This person averaged the two together to get 4.5 hours. However, Suzie could do the task in 4 hours alone. The same concept as before applies. When people work together the overall time will be less than the time of the fastest person.
• 1 hour: The reasoning here is Billy can do it in 5 hours, Suzie can do it in 4 hours. 5 - 4 = 1 hour. This logic is just flawed. Let's say it took both of them 5 hours. Then this reasoning would dictate that together they could do it in 5 - 5 = 0 hours, or no time. Sometimes the answer may make sense, that is, 1 hour certain seems reasonable, it is, after all, faster than the fastest single time. However, you also need to consider the method used to arrive at the answer.
• 2 2/9 hours: This is the correct answer. The times were converted into rates: Billy can paint 1/5 of the wall per hour and Suzie can paint 1/4 of the wall per hour. Rates can be added together to get a combined rate: Together, they can paint 1/5 + 1/4 = 9/20 of the wall per hour. If they can paint 9/20 of the wall per hour, then it takes them 20/9 = 2 2/9 hours to paint the entire wall. While rates can be added to get a combined rate, times can not be added together to get a combined time.