# Word Problems

### Topics

Any English phrase in which one quantity is taken away from another translates into a mathematical expression with a minus sign. You've seen those self-help books about dealing with loss? Those are probably *riddled* with minus sign implications.

Here are some phrases that can be abbreviated with the – sign.

- Five
**minus**three means 5 – 3.

- Ten
**take away**four means 10 – 4. If you have ten brownies and you take away (eat) four, you have 10 – 4 = 6 left. Good luck explaining this to your nine brothers and sisters.

- Five
**subtract**two means 5 – 2.

- Sixty-four
**decreased by**(or**diminished by**or**made smaller by**or**reduced by**) two means 64 – 2. On the number line, we start at 64 and move two places to the left. In time with the music, if you can.

- The
**difference between**8 and 6 means 8 – 6.

When talking about subtraction in English, we can play with the order in which we talk about the quantities involved. We can first say how many things we start out with ("start with five pencils, take away two''), or we can first say how many things we'll be taking away ("take away two of the five pencils''). In English, either order is fine as long as we're consistent with what we're saying. English is flexible like that. You should see it do a center split.

In mathematics, however, the rules are stricter. English is the lenient substitute teacher who wants to make it through the day alive, while math is a regular teacher who won't stand for you flying paper airplanes across the room. In every mathematical expression, we must begin by saying how many things we start out with, and then how many things we'll take away. We write

(amount we start out with) – (amount to take away)

Except with more numbers and fewer words. You get the picture.

Below are some English phrases that mean "subtraction." Not ones you'll hear as often as "Have a nice day" or "Do you want fries with that?" but English phrases nevertheless. Each starts by saying how many things we'll be taking away. The number line can be helpful for figuring out which number should be written first in the mathematical translation. Oh, number line. You're always there for us when we need you.

- Four
**less than**seven means 7 – 4. On the number line, we start at 7 and move 4 spaces to the left:

- Four
**fewer then**seven also means 7 – 4.

- Two
**smaller than**five means 5 – 2. However, if someone ever phrases something that way, you have our permission to smack them. With your eyes. We start at 5 on the number line and move two spaces to the left:

- Two
**subtracted from**five also means 5 – 2.

Whenever we're translating English into mathematics and subtraction is involved, we need to identify

- which number is the amount we start out with, and

- which number is the amount we'll take away.

There's a big difference between starting with 10 cookies and having someone take 9 of them away, and starting with 9 cookies and having someone take 10 of them away. We *hate* it when someone leaves us with negative cookies. How will we ever put on our winter weight?