Understanding Word Problems
Key Terms
- Arithmetic Reasoning (AR): Tests the ability to solve word problems, primarily involving basic arithmetic operations, such as addition, subtraction, multiplication and/or division. AR tests the ability to quickly analyze information and determine the appropriate formula to apply to the problem.
- Math Operation Keywords: Certain words indicate whether you should perform addition, subtraction, multiplication or division in order to arrive at a solution to a word problem.
Math Operation Keywords
Addition Keywords Sum, total, combined, altogether, increased by, more than, plus, added to, in all
Subtraction Keywords Difference, remaining, left over, less than, fewer, decreased by, minus, how many more, how much more
Multiplication Keywords Product, times, of, each, per, every, doubled, tripled, twice
Division Keywords Quotient, per, average, split, shared equally, divided, ratio, out of
[VISUAL-MANUAL: Math operation keywords reference chart for ASVAB Arithmetic Reasoning study guide, “Understanding Word Problems” page.
Layout: Four-quadrant grid or four horizontal rows, one for each operation. Each section has the operation symbol, name, and associated keywords.
Content: Section 1 – Addition (+) Keywords: sum, total, combined, altogether, increased by, more than, plus, in all
Section 2 – Subtraction (−) Keywords: difference, remaining, left over, less than, fewer, decreased by, how much more
Section 3 – Multiplication (×) Keywords: product, times, of, each, per, every, doubled, tripled, twice
Section 4 – Division (÷) Keywords: quotient, per, average, split, shared equally, divided, ratio
Style: Clean, professional study guide aesthetic. Use distinct colors for each operation section (suggest: blue for addition, orange for subtraction, green for multiplication, purple for division). Include the mathematical symbol prominently in each section. Make it suitable for printing as a quick-reference card.]
Translation Process
In order to turn any word problem into a solvable equation, follow the following steps:
Step 1: Read the entire problem once. Do not begin to solve the problem until you have read the entire problem through to completion.
Step 2: Identify the question. Locate the final sentence of the problem. What exactly is being asked? Mentally circle or underline the question.
Step 3: List the relevant data. Write down the numbers that pertain to the question. Disregard any extraneous information.
Step 4: Determine the relationship. How do the numbers in step 3 relate to each other and to the question?
Step 5: Write the equation and solve. Translate your understanding into math, and then calculate the solution.
Step 6: Verify the answer. Does the answer make logical sense? Did the answer address the actual question being asked?
Here is a sample application of the above steps to a problem:
“The carpenter has a 96 inch board. He wants to cut it into six equal lengths, 8 inches each. How many inches of wood will remain?”
- Question: How many inches of wood will remain?
- Data: 96 inches, 8 inches each length
- Relationship: Cutting equal lengths = division, but we need the remainder
- Equation: 96 ÷ 8 = 12 pieces with no inches of wood remaining.
- Solution: 0 inches
Note that the question is asking for how many inches of wood will remain, and not how many pieces of wood there will be. Therefore, if you arrived at 12 as the solution, you correctly solved the problem, but incorrectly addressed the question being asked.
Mistakes to Avoid
Solving the incorrect problem. This is the most common mistake. It may seem logical, but you may have calculated perfectly and provided the correct solution, but solved the wrong problem. Example: In a word problem, you may be able to determine the total cost of an item, but the question being asked may be the change that is due back to the customer. Re-read the question prior to providing a solution.
Ignoring non-relevant information. Word problems include unnecessary information. As a test taker, you need to evaluate the information given and determine what information is relevant and what is not. If a problem states three different quantities, but only two relate to the question being asked, disregard the third quantity.
Interpreting “more than” and “less than” incorrectly. When reading “5 more than 12”, 12 + 5 = 17. However, 5 + 12 = 17. Likewise, “8 less than 20” equates to 20 – 8 = 12. Conversely, 8 – 20 is 12. When reading subtraction problems, the number after “than” always comes first.
Not accounting for units. When working with units of measurement (i.e. hours and minutes, feet and inches), ensure that your units are consistent throughout the problem. Many times, converting the units of measure at the conclusion of the problem results in an inaccurate solution.
Not evaluating whether the solution is reasonable. If a problem is asking for how many hours someone worked, and your solution is 300 hours in a week, something went amiss. A quick evaluation of reasonableness can help identify the error.
Practice Questions
Question 1 A bakery sold 145 muffins on Friday and 238 muffins on Saturday. How many more muffins were sold on Saturday than on Friday?
- A) 93 B) 103 C) 383 D) 193
Answer: A) 93
The keyword “how many more” indicates a subtraction operation. Thus, we need to determine how many more muffins were sold on Saturday than on Friday by performing the operation 238 – 145 = 93. Choice C represents the correct solution if we had performed an addition operation instead of a subtraction operation.
Question 2 A factory produces 1250 units of merchandise daily. How many units will it produce during a 5-day workweek?
- A) 250 B) 1255 C) 6000 D) 6250
Answer: D) 6250
Since we need to determine the total number of units produced during a 5-day workweek and we know that the factory produces 1250 units per day, we will multiply 1250 * 5 = 6250. Choices A and B represent solutions obtained if we had performed division instead of multiplication.
Question 3 Elena purchased 3 notebooks for $4.50 each and 2 pens for $1.25 each. How much did she spend in total?
- A) $5.75 B) $11.75 C) $13.50 D) $16.00
Answer: D) $16.00
First, we will calculate the total amount spent for the 3 notebooks: 3 * $4.50 = $13.50. Next, we will calculate the total amount spent for the 2 pens: 2 * $1.25 = $2.50. Finally, we will add the total amount spent for the notebooks and pens together: $13.50 + $2.50 = $16.00. Choice C would represent only the amount spent on the 3 notebooks.
Question 4 A runner ran 15 miles in 2.5 hours. What was his average speed in miles/hour?
- A) 5 mph B) 6 mph C) 12.5 mph D) 37.5 mph
Answer: B) 6 mph
To calculate the average speed, we will divide the distance traveled by the time elapsed. Therefore, we will compute 15 / 2.5 = 6. Choices D represents the result of multiplying 15 * 2.5 = 37.5 mph.
Question 5 Michael has $75. He purchases a jacket costing $48.50 and a hat costing $12.75. How much money does Michael have left?
- A) $13.75 B) $14.75 C) $26.25 D) $61.25
Answer: A) $13.75
We will first calculate the total amount spent on the jacket and hat: $48.50 + $12.75 = $61.25. Then we will subtract the total amount spent ($61.25) from Michael’s original amount of money ($75): $75 – $61.25 = $13.75. Choice D is the total amount spent by Michael, and not the amount of money he has left. Therefore, choice D is the answer to a different question.
Quick Tips
- Read the question before doing anything else. Knowing the specific question that is being asked allows you to focus on the relevant data and avoid getting bogged down in the extraneous details contained within the question.
- Identify and circle or underline keywords. Circles or underlines around key words such as “total”, “remainder”, and “each” will force you to pay attention to them.
- Write down your equation. While you can perform calculations mentally, writing down the equation will provide evidence that you have translated the problem into an equation.
- Estimate before calculating. Performing a quick estimate of the answer will allow you to identify if there is a large error in the calculation, and also help you to eliminate obviously incorrect answers.
- Verify your answer against the original question. Once you have determined the answer to the question, verify that the answer actually addresses the original question, and not some other question that you may have interpreted the original question to mean.
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