Find practice problems and solutions for these concepts at Word Problem Pitfalls Practice Problems.

** We've seen how** to translate keywords into operations and how to decide which operation to use based on the context of a word problem. However, sometimes keywords and phrases can be misleading. In this chapter, we'll look at how word problems try to trick you—and how you can avoid being tricked.

### Backward Phrases

A **backward phrase** is a group of words and numbers that describes an operation in which the numbers are given in the opposite order that they will appear in a number sentence. For instance, 3 *less than* 4 is a backward phrase because when the numbers 3 and 4 are placed in a number sentence, 4 appears before 3: 4 – 3.

The phrase *add* 4 *and* 6 can be converted into the number sentence 4 + 6. The numbers fit right into the number sentence as they appear in the phrase. What about the phrase *subtract* 4 *from* 6? At first glance, you may want to write 4 – 6. Read the phrase again carefully: subtract 4 *from* 6. We are starting with 6 and taking away 4. This phrase, written as a number sentence, is 6 – 4. This kind of phrase is a backward phrase.

#### Example

- If nine pencils are added to a box that holds eight pencils, how many pencils are in the box?

The number 9 is added to 8, the original quantity. The phrase *added* to is a backward phrase. There were already eight pencils in the box, so we are adding 8 and 9: 8 + 9 = 17 pencils. As we've already learned, addition is commutative, so even if we had added 8 to 9, we still would have found the same answer: 9 + 8 = 17. However, not all operations are commutative.

#### Example

- North has 14 fewer photos than Jackie, who has 63 photos. How many photos does North have?

The words *fewer* and *than* may be separated, but these words signal that the order of the numbers given in the word problem will need to be switched. *Fewer* is a keyword that signals subtraction, so to find how many photos North has, begin with 63 and subtract 14: 63 – 14 = 49 photos.

## Inside TrackSubtraction word problems that involve real-life situations will often be written as a larger number minus a smaller number. In the last example, even if you thought that the problem should be solved by subtracting 63 from 14, that would have given you an answer of –49. It would be impossible for North to have –49 photos, since it is impossible to have a negative quantity of a real-life object. However, a word problem that simply involves numbers could have a negative answer. As always, refer to the question being asked to find the final clue of what the correct answer should be. |

#### Example

- What is fourteen less than nine?

In this example, *less than* is a backward phrase. It's also a keyword phrase that signals subtraction. To find 14 less than 9, we must start with 9 and subtract 14: 9 – 14 = –5.

Multiplication, like addition, is commutative, so you don't have to worry about backward phrases for multiplication. Division, like subtraction, is not commutative, so we must be wary of backward phrases for division.

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