Have you ever taken a loan? Take a look at this dilemma.

The student theater club borrowed a total of $354 to put on the school play. The loan will be divided among six students. Write an integer that represents the amount each student owes.

**This Concept will show you how to write an equation and solve an real-world problem that contains integers.**

### Guidance

We already mentioned money as a real-life application for working with integers. In fact, losses and gains of any kind make the most sense when we apply integers to the operations. It can help us to keep track of what actually occurred. We will also be able to see whether or not we have a gain or a loss in our sum or in our difference.

Let’s look at a real-life problem using integers.

**Caitlin borrowed $950 to buy a computer. So far, she has paid back $175. How much does Caitlin still owe?**

Think of a debt like a negative number. For example, if you have $10, you have +10. If you give that away, you have 0. If you then owe another $10, you have -10.

To find the amount Caitlin still owes, write a simple equation to represent the problem. Let \begin{align*}x\end{align*} represent the amount Caitlin still owes.

\begin{align*}x &= -950+175\\ &=175-950\\ &=-775\end{align*}

**Caitlin still owes $775.**

Here is another one.

**The population of a certain town in 2002 was 312,980. In 2006, the population increased to 391,740. To the nearest thousand, what was the population increase from 2002 to 2006?**

This is an estimation problem. Round each number to the nearest thousand, and then find the difference.

Round the first number. Since there is a 9 in the hundreds place, 312,980 rounds up to 313,000.

Round the second number. Since there is a 7 in the hundreds place, 391,740 rounds up to 392,000.

Now find the difference.

\begin{align*}392,000-313,000=79,000\end{align*}

**The population increased by 79,000 people.**

Here are a few for you to try.

#### Example A

John earned three bonus points on his test. If he started with a 78, what was his final score?

**Solution: \begin{align*}81\end{align*}**

#### Example B

A football team lost fifteen yards at the 20 yard line. Since football moves backwards, what yard line did the team start the next play on?

**Solution: The 35 yard line**

#### Example C

A gain of 16 can be represented by which integer?

**Solution: \begin{align*}+16\end{align*}**

Now let's go back to the dilemma from the beginning of the Concept.

Write the total loan as an integer. Since it is a debt, it can be represented with a negative integer: -354.

Now divide to find the amount each student owes.

\begin{align*}-354 \div 6=-59\end{align*}

**Each student owes $59.**

### Vocabulary

- Integer
- the set of whole numbers and their opposites, this includes both positive and negative whole numbers.

### Guided Practice

Here is one for you to try on your own.

Yuri saved $215 each month for six months. About how much has Yuri saved? What is the exact amount that Yuri saved?

**Solution**

First, we need to find an estimate. Round the first number to a number that is easy to multiply. Then find the product.

215 rounds down to 200.

\begin{align*}200 \times 6=1,200\end{align*}

Yuri saved about $1,200.

To find the exact amount Yuri saved, write a simple equation to represent the problem. Let \begin{align*}x\end{align*} represent the total amount Yuri saved.

\begin{align*}x &= 215 \times 6\\ &= 1,290\end{align*}

**Yuri saved exactly $1,290.**

### Video Review

Problem Solving Using Integers

### Practice

Directions: Use simple integer equations to solve each real-world problem.

1. Karen saved fifteen dollars a week for eight weeks. How much money did she have at the end of this time?

2. Jocelyn spent as much as Karen had saved. Write an integer to show the amount that Jocelyn spent.

3. If a car backs up fifteen feet and then goes forward forty feet. How many feet did the car advance?

4. Tasha owes her brother fifty dollars. She paid five dollars towards the debt. How much does she still owe her brother?

5. The temperature on Monday began at 5 degrees, then went up to 20 degrees and then decreased to 7 degrees. Show the temperature change in an equation.

6. Represent this change in temperature by writing an integer.

7. Joshua spent fifteen dollars, then he spent five more dollars and then he spent three dollars and fifty cents. Write an equation to show his spending.

8. How much money did Joshua spend in all?

9. If Joshua had started shopping with $30.00, would he have gotten any change?

10. How much change would he have gotten?

11. Jessica is shortening her dress length three inches. Write an integer to show this change.

12. If the length of the dress is 40 inches, how long will the dress be after the alteration?

13. If Jessica is five feet tall, how far will the hem be from the floor?

14. Carly is scuba diving. She descends to fifteen feet and then proceeds to descend another 35 feet. Show this change using an equation.

15. What is her final depth?