Candace's school is doing a Jump Rope for Heart event to raise money for the American Heart Association. Technically, it is about raising money for a worthy cause, but, unofficially, the students all keep track of their jump rope high scores and try to beat each other. Last year, Candace's friend, Lauren, got the most jumps without missing: 487. Candace only made 324 jumps without missing, but she's been practicing all year. Assuming that Lauren does about the same this year as last, how many more jumps will Candace need to do to beat her?

In this concept, you will learn to take the difference between two integers with the same sign.

### Subtracting Integers with the Same Sign

A **positive number **is a number greater than zero. It can be written with or without a + symbol in front of it. A gain in something is written with a positive number.

A **negative number **is a number that is less than zero. It is always written with a - symbol in front of it. A loss is written with a negative number.

An **integer **is any positive whole number or its opposite. Here, opposite means sign. So a positive integer has a negative opposite and vice versa.

The rules are different when subtracting two positive numbers or two negative numbers. When subtracting two positive numbers, first take the smaller number out of the larger number. Then, look at which number was bigger in the original problem. If the bigger number was first, then the answer is positive, as in this example:

\begin{align*}9 – 4 = 5\end{align*}

If the bigger number was second, then the answer is negative, as in this example:

The process is longer when subtracting negative numbers because you end up subtracting a negative number, which is the same as adding a positive number, as in this example:

The subtraction became addition. This is always the case when subtracting a negative number.

So, when subtracting two negative numbers, the first step is always to change the subtraction to an addition and re-write the problem. Then, you have an addition problem between a positive number and a negative number. Next, re-write the problem so that it looks like the difference between two positive numbers.

\begin{align*}-8+5=5-8 \end{align*}

Finally, solve it as you did the difference between two positive numbers.

In this example, the difference between 8 and 5 is 3, and because the bigger integer was second, the answer is negative.

\begin{align*}5-8=-3\end{align*}

Here is another example.

Next, re-write as the difference of two positive numbers.

\begin{align*}6-3=\end{align*}

Then, take the difference.

The difference between 6 and 3 is 3.

Finally, notice whether the bigger integer was first or second.

In this case, it was first, so the answer is positive.

The answer is 3.

### Examples

#### Example 1

Earlier, you were given a problem about Candace and her jump rope vendetta.

She wants to beat her friend's jump rope high score of 487. Her own high score is 324. How many more jumps will she have to do this year without missing?

To figure this out, first she sets up a subtraction problem between two integers of the same sign.

\begin{align*}487-324\end{align*}

Next, she takes the smaller number from the bigger.

\begin{align*}487-324=153\end{align*}

Then, she decides which number was bigger.

In this case, Lauren had more jumps, so Candace knows that she has to make at least 153 more jumps herself to beat Lauren.

#### Example 2

Joaquin made fifteen sugar cookies. He wanted to give one to everyone in his class of 25. How many more does he need to make?

First, write the situation as a mathematical expression.

He starts with 15 cookies and wants to give away 25.

\begin{align*}15-25\end{align*}

Next, take the smaller number from the larger number.

25 less 15 is 10.

Then, check to see which number is bigger.

In this case, the second is bigger, so the answer is negative.

The answer is -10. This means that Juaquin is 10 cookies shy of his goal. He will need to make ten more to give everyone in his class one.

#### Example 3

Find the difference.

5 – 10 = ____

First, take the smaller number from the bigger.

10 less 5 is 5.

Next, see which number in the original problem was bigger.

In this case, it was the second so the answer is negative.

The answer is -5.

#### Example 4

Find the difference.

14 – 7 = ____

First, take the smaller number from the bigger.

14 less 7 is 7.

Next, see which number in the original problem was bigger.

In this case, it was the first so the answer is positive.

The answer is 7.

#### Example 5

Find the difference.

-4 – -8 = ____

First, re-write the problem as an addition.

-4 + 8

Next, re-write the problem as the difference between two positive numbers.

8 - 4

Then, take the smaller number from the larger.

8 less 4 is 4.

Finally, decide which number was bigger.

In this case, the 8 was bigger, which was the first. So the answer is positive.

The answer is 4.

### Review

In the following problems, subtract each pair of integers.

- \begin{align*}14 - 19\end{align*}
- \begin{align*}24 - 19\end{align*}
- \begin{align*}-1 - 7\end{align*}
- \begin{align*}-4 - 12\end{align*}
- \begin{align*}-14 - 29\end{align*}
- \begin{align*}-24 - (-19)\end{align*}
- \begin{align*}9 - 11\end{align*}
- \begin{align*}13 - (-1)\end{align*}
- \begin{align*}23 - 19\end{align*}
- \begin{align*}-31 - 15\end{align*}
- \begin{align*}-18 - (-19)\end{align*}
- \begin{align*}-14 - 6\end{align*}
- \begin{align*}-74 - 39\end{align*}
- \begin{align*}54 - (-29)\end{align*}
- \begin{align*}64 - 99\end{align*}

### Review (Answers)

To see the Review answers, open this PDF file and look for section 11.10.