A train is hauling 150 freight cars full of grain and corn. Each train car can hold either 90.25 tons of corn or 114 tons of grain. If there are 90 cars full of grain, how much weight is the train hauling?

In this concept, you will write equations and solve real-world problems using simple integer equation

### Applying Integer Equations to Solve Real World Problems

Integers are used in everyday life when you solve real-world problems. Let’s look at a real-life problem using integers.

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

First, write a simple equation to represent the problem. Let \begin{align*}x\end{align*} be the amount that Kaitlyn still owes.

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

Next, rearrange the equation to isolate \begin{align*}x\end{align*}.

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

Then, solve for \begin{align*}x\end{align*}.

\begin{align*}x=775\end{align*}

The answer is 775.

Kaitlyn still owes $775.

Here is another real-world example.

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?

First, since you are estimating, round the populations in 2002 and 2006 to the nearest thousand.

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.

Next, find the difference between the two populations.

\begin{align*}392000-313000=79000\end{align*}

The answer is 79000.

The population increased by 79,000 people.

### Examples

#### Example 1

Remember the freight train hauling the grain and corn? There are 150 cars in total, 90 carrying grain and 60 carrying corn.

First, write an expression for the weight of each type of load for all of the freight cars. Let \begin{align*}x\end{align*} equal the total weight being hauled by the train.

\begin{align*}x = 90.25 \frac{\ \text{tons}}{\ \text{car}} \times 60 \ \text{cars} + 114 \ \frac{\text{tons}}{\ \text{car}} \times 90 \ \text{cars}\end{align*}

Next, simplify the equation.

\begin{align*}x=5415 \ \text{tons} + 10260 \ \text{tons}\end{align*} Then, solve for \begin{align*}x\end{align*}.

\begin{align*}x=15675\end{align*}The answer is 15675.

The amount of grain and corn is 15675 tons.

#### Example 2

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

First, you need to find an estimate the amount Yuri saved.

Round the first number to a number that is easy to multiply. The number 215 rounds down to 200.

Next, multiply 200 by the 6 months.

\begin{align*}200 \times 6 = 1200\end{align*}

The answer is 1200.

Yuri saved about $1,200.

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

\begin{align*}\begin{array}{rcl} x &=& 215 \times 6 \\ x &=& 1290 \end{array}\end{align*}

The answer is 1290.

Yuri saved exactly $1,290.

Here are a few for you to try.

#### Example 3

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

First, write an equation to represent this problem.

\begin{align*}x= 78+3\end{align*}

Next, solve for \begin{align*}x\end{align*}.

\begin{align*}x=81\end{align*}The answer is 81.

John’s final score on his test was 81.

#### Example 4

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?

First, write an equation to represent this problem.

\begin{align*}x =20 - (-15)\end{align*}

Next, solve for \begin{align*}x\end{align*}.

\begin{align*}\begin{array}{rcl} x &=& 20 + 15 \\ x &=& 35 \end{array}\end{align*}

The answer is 35.

The next play is from the 35 yard line.

#### Example 5

You are helping to make potato salad for a family picnic. You can peel 2 potatoes every minute. How long will it take you to peel the 50 potatoes you need?

First, write an equation to represent this problem.

\begin{align*}x = \frac{50 \ \text{potatoes}}{2 \ \text{potatoes/min}}\end{align*}

Next, solve for \begin{align*}x\end{align*}.

\begin{align*}x=25\end{align*}

The answer is 25.

It will take you 25 minutes.

### Review

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

- Karen saved fifteen dollars a week for eight weeks. How much money did she have at the end of this time?
- Jocelyn spent as much as Karen had saved. Write an integer to show the amount that Jocelyn spent.
- If a car backs up fifteen feet and then goes forward forty feet. How many feet did the car advance?
- Tasha owes her brother fifty dollars. She paid five dollars towards the debt. How much does she still owe her brother?
- 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.
- Represent this change in temperature by writing an integer.
- 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.
- How much money did Joshua spend in all?
- If Joshua had started shopping with $30.00, would he have gotten any change?
- How much change would he have gotten?
- Jessica is shortening her dress length three inches. Write an integer to show this change.
- If the length of the dress is 40 inches, how long will the dress be after the alteration?
- If Jessica is five feet tall, how far will the hem be from the floor?
- Carly is scuba diving. She descends to fifteen feet and then proceeds to descend another 35 feet.
- Show this change using an equation.
- What is her final depth?

### Review (Answers)

To see the Review answers, open this PDF file and look for section 2.14.