# GED Mathematical Reasoning: Solving Word Problems

Word problems – with all those words – can be very intimidating. Like many things in life, having a plan of action to use when you are faced with a challenging situation is often very helpful.

Here is a step-by-step strategy you may use for solving word problems:

Step 1: Read the problem carefully at least one time. Identify what the problem is asking you to find.

Step 2: Make a note of the information given in the problem. It might helpful in some cases to draw a picture to make sense of the situation. Not all of the information may be necessary — some of the information given might be extra information that is not needed. Distinguish between the information needed to solve the problem and the information given that is not necessary.

Step 3: Create a plan for what to do with the necessary information and solve the problem to answer the question. Look for key words – these are words that imply that a certain operation (addition, subtraction, multiplication, or division) should be used to solve the problem.

Here is a directory of commonly used key words for your reference – they are listed under the operation they often (but do not always) imply:

Addition | Subtraction | Multiplication | Division |

plus | less | product | divided by |

combine | subtract | double | divided into |

more than | difference | triple | quotient |

add/added to | less/more/farther than | times | goes into |

increased by | fewer | of | divided equally |

sum | decreased by | twice | per |

total | loss | multiply | each |

gain | minus | average | |

altogether/in all | take away | split | |

change |

So if you see the word “total” in a problem, the operation needed will probably be addition. Or, if the problem includes the word “twice,” you will likely need to multiply by 2.

Here are some other rules of thumb:

- Use addition when combining amounts.
- Use subtraction when comparing one amount to another.
- Use multiplication when finding several of a given amount or when given the part and finding the whole.
- Use division when sharing, cutting, or splitting. Also, when given an amount and finding one part.

And here’s one more thing to keep in mind. as you complete step 3: A word problem may require just one step or operation. But be aware that a problem may require more than one step or operation.

Step 4: Solve the problem based on the information given and the key word (or words) and clearly state your answer, using proper units. It is a good idea to form an estimate of the final answer before doing the math. That way when you solve the problem exactly, you have a “ball park” idea of what your answer should be.

Step 5: Ask yourself the following questions: Does my answer make sense? Is it close to my estimation? If not, you may need to go back and recheck your work and/or redo the math.

**Example 1**

To prepare for the first day of a new school year, Bo purchased a calculator for $14.95, a backpack for $29.99, three pairs of jeans for $23.50 each, and sneakers for $42.30. Which expression can be used to calculate the amount Bo paid for the calculator and the jeans?

- $84.96
- $14.96 + $23.50
- $14.96 + $23.50 – $23.50 – $23.50
- $14.96 + $23.50 + $23.50 + $23.50

Our tendency might be to dive right into calculating the cost, but that’s not what the question is asking.

For step 1, notice that the problem is not asking us to find the actual sum – it is simply asking us to identify the expression that represents the math we would perform were we to find the amount spent on the calculator and jeans.

Moving on to step 2, let’s list the information given.

- The calculator was: $14.95
- The backpack was: $29.99
- The cost of the jeans was: $23.50 for each pair and he bought three pairs total
- The sneakers were: $42.50

Of this, the necessary information we’ll need is the cost of the calculator and the cost of the jeans.

- The calculator was: $14.95
- The cost of the jeans was: $23.50 for each pair and he bought three pairs total

Now for step 3. In this problem we are combining amounts which implies we need to use addition.

To complete step 4, we are asked to find the expression representing the combined cost of the calculator and the three pairs of jeans and we have established that we will use addition to do so. We would use the expression: $14.96 + $23.50 + $23.50 + $23.50 – answer (4).

Finally, step 5. Does our answer make sense? Yes – the expression involves the addition of the cost of the calculator costing $14.96 as well as three pairs of jeans, each costing $23.50.

**Example 2**

Mark has been analyzing the cost per mile to drive his car each month. In November, costs related to his vehicle were $879 and he drove 324 miles. Estimate Mark’s cost per mile for November.

- $2 per mile
- $3 per mile
- $4 per mile
- $5 per mile

First note, we are being asked to estimate the cost per mile for November. We are not being asked to find the exact cost per mile.

Second, we are given that vehicle costs were $879 and miles driven were 324 and both pieces of information will be necessary to solve this problem.

To create our plan for step 3, notice that we are being asked to find cost per mile. The word “per” implies division, therefore we will divide.

To estimate the answer for step 4, we will round 879 to the hundreds place, which is 900. And we will round 324 to the hundreds place, which is 300. Then, since we’re trying to find cost per mile, we will divide the estimated cost of 900 by the estimated miles of 300. 900 divided by 300 is 3. Therefore, the cost per mile is $3 per mile – answer (2).

For our final step, we ask ourselves: Does our answer make sense? Yes – Mark’s vehicle costs are approximately three times the number of miles he drove.

**Example 3**

To rent a car, LeAnn paid an initial one-time, non-refundable deposit of $30. She also paid $25 a day for each of the three days she rented the car. What was LeAnn’s total cost for renting the car?

- $105
- $55
- $80
- $115

For step 1, we are being asked to find the total cost.

The information given for step 2 is that: LeAnn paid an initial deposit of $30 and she also paid $25 a day for three days. Both pieces of information are necessary to solve this problem.

For step 3, the word “total” implies we will add. In addition, we need to find “several for a given amount” to determine the cost of 3 days at $25 each day. This implies we will also multiply.

In our fourth step, we estimate the total cost to be around $100 since LeAnn essentially has 4 cost components (for the deposit as well as day 1, day 2, and day 3) each costing exactly or about $25. Now let’s calculate the actual cost. The daily cost can be calculated by multiplying 3 times 25, which equals $75. Then, we will add the cost of the one-time deposit. So $75 + $30 = $105 – answer (1).

For our final step, let’s assess whether or not our answer seems reasonable. Yes – it does seem reasonable since our actual cost of $105 is very close to our estimate of $100.

As a final note, there are no magic tricks for mastering word problems. Practice makes perfect! If you’re faced with a word problem and are unsure of how to proceed, find a similar example to model in a textbook or online. Follow along with each step of the example – taking notes and asking questions as needed. Then, use that same process to complete the given problem. With time and practice, you will learn to connect key words in the problem to the operation or process needed to find the solution.