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# Solving Real-World Problems with Two-Step Equations

## Equations that require multiple steps solve real world problems. This involves translating words into algebraic equations and solving them.

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Solve Real-World Problems by Using Rational Numbers and Simple Equations

Have you ever done any woodwork? Take a look at this dilemma.

Jill had a piece of wood 4 feet long. She cut of a piece that was 258\begin{align*}2 \frac{5}{8}\end{align*} feet long. How long was the piece of wood she had left?

To figure this out, you will need to use rational numbers and an equation. This Concept will show you how to apply rational numbers and equations when problem solving. By the end of the Concept, you will know how to figure out the length of the piece of wood.

### Guidance

You can use fractions, decimals, and integers to solve real-world problems. You can use rational numbers to write and solve equations to represent real-world problems too. Let's look at how to do that successfully.

Take a look at this situation.

Candy is 5 years older than one-third Liam’s age. If Candy is 16, how old is Liam?

Choose a variable to represent Liam’s age. Let l\begin{align*}l\end{align*} represent Liam’s age.

Write an equation using the information in the problem.

The phrase “5 years older” translates to “+5.” The phrase “one-third Liam’s age” translates to “13l\begin{align*}\frac{1}{3} \cdot l\end{align*}.” Since the problem tells you that Candy is 16, set the expression that shows Candy’s age equal to 16.

Put the parts together to form an equation.

13l+5=16\begin{align*}\frac{1}{3}l+5=16\end{align*}

Now solve the equation for l\begin{align*}l\end{align*}. Remember that any operation you perform on one side of the equation must also be performed on the other side.

First, subtract 5 from both sides to isolate the variable term on one side of the equation.

13l+5513l=165=11\begin{align*}\frac{1}{3}l+5-5 &= 16-5\\ \frac{1}{3}l &= 11\end{align*}

Now you need to remove the 13\begin{align*}\frac{1}{3}\end{align*} from the variable term. In order to eliminate a fractional coefficient, multiply by the reciprocal.

3113ll=1131=33\begin{align*}\frac{3}{1} \cdot \frac{1}{3} l &= 11 \cdot \frac{3}{1}\\ l &= 33\end{align*}

Liam is 33 years old.

#### Example A

1112+212\begin{align*}-\frac{11}{12} + \frac{2}{12}\end{align*}

Solution:  34\begin{align*}-\frac{3}{4}\end{align*}

#### Example B

.89+.987\begin{align*}-.89+.987\end{align*}

Solution:  .097\begin{align*}.097\end{align*}

#### Example C

78+28\begin{align*}-\frac{7}{8} + -\frac{2}{8}\end{align*}

Solution:  98=118\begin{align*}-\frac{9}{8} = -1 \frac{1}{8}\end{align*}

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

Jill had a piece of wood 4 feet long. She cut of a piece that was 258\begin{align*}2 \frac{5}{8}\end{align*} feet long. How long was the piece of wood she had left?

First convert the mixed number to an improper fraction: 258=16+58=218\begin{align*}2 \frac{5}{8}=\frac{16+5}{8}=\frac{21}{8}\end{align*}

Then convert the first number to a fraction with a denominator of 8: 4=41×88=328\begin{align*}4 = \frac{4}{1} \times \frac{8}{8}=\frac{32}{8}\end{align*}

Subtract to find the length of the piece of wood.

328218=118=138\begin{align*}\frac{32}{8}-\frac{21}{8}=\frac{11}{8}=1 \frac{3}{8}\end{align*}

The piece of wood is 138\begin{align*}1 \frac{3}{8}\end{align*} feet long.

### Vocabulary

Rational Number
number that can be written in fraction form.
Integer
the set of whole numbers and their opposites.
Percent
number representing a part out of 100.

### Guided Practice

Here is one for you to try on your own.

Ben bought one share of stock for $40. He watched the price of his stock every day for a week. On Monday, the stock moved 2.5 points up. On Tuesday, it moved 1.75 points down. On Wednesday, it moved p points. On Thursday and Friday, the stock moved 0.75 points up each day. If the price of the stock was$45 at the end of the week, what is the value of p\begin{align*}p\end{align*}?

Solution

Use the information in the problem to write an equation to represent the price of the stock.

Use the order of operations to simplify the equation. First, multiply. Then add and subtract in order from left to right.

40+2.51.75+p+2(0.75)=45\begin{align*}40+2.5-1.75+p+2(0.75)=45\end{align*}

Now apply the commutative property in order to combine the decimals.

40+2.51.75+p+1.542.51.75+p+1.540.75+p+1.5=45=45=45\begin{align*}40+2.5-1.75+p+1.5 &= 45\\ 42.5-1.75+p+1.5 &= 45\\ 40.75+p+1.5 &= 45\end{align*}

Finally, isolate the variable by subtracting 42.25 from both sides. You can then apply the commutative and additive identity properties to the left side of the equation to solve for p\begin{align*}p\end{align*}.

42.25+p42.25p+42.2542.25p+0p=4542.25=4542.25=2.75=2.75\begin{align*}42.25+p-42.25 &= 45-42.25\\ p+42.25-42.25 &= 45-42.25\\ p+0 &= 2.75\\ p &= 2.75\end{align*}

The stock moved 2.75 points up on Wednesday.

### Practice

Directions: Use what you have learned to work with each problem concerning rational numbers.

1. The expression 2p34m\begin{align*}2p^3-4m\end{align*} can be used to find the sales profit of a company where p\begin{align*}p\end{align*} is the number of products they sell and m\begin{align*}m\end{align*} is the number of miles they travel. If they sold 5 products and traveled 10 miles, what was their profit?
2. If they sold 6 products and traveled 15 miles, what was their profit?
3. If they sold 10 products and traveled 20 miles, what was their profit?
4. The expression 11s2+7t\begin{align*}\frac{11s}{2}+7t\end{align*} can be used to find the group admission price, where s\begin{align*}s\end{align*} is the number of students and t\begin{align*}t\end{align*} is the number of teachers. If there are 20 students and 4 teachers, what is the group admission price?
5. If there are 25 students and 4 teachers, what is the group price?
6. If there are 20 students and 5 teachers, what is the group price?
7. Brooke needs to save $146 for a trip. She has$35 in her savings account. She saves $15.75 each week. She also has to spend$15 to buy a present for a friend. How many weeks will Brooke need to save to have enough for her trip?
8. Vinnie is 12\begin{align*}\frac{1}{2}\end{align*} as old as Julie. Julie is 24. How old is Vinnie?
9. Manuel starts with $30. He earns$8.00 per hour plus an additional bonus of $12 each day. He spends$8.00 for lunch. If he has \$94 at the end of the day, for how many hours did he work?
10. A formula for the perimeter of a rectangle is P=2(l+w)\begin{align*}P=2(l+w)\end{align*}, where P\begin{align*}P\end{align*} is the perimeter, l\begin{align*}l\end{align*} is the length, and w\begin{align*}w\end{align*} is the width. If the perimeter of a rectangle is 312 centimeters and the width is 67.3 centimeters, what is the length?
11. If the length of a rectangle is 32 centimeters and the width is 65.5 centimeters, what is the perimeter?
12. If the length of a rectangle is 64 centimeters and the width is 22 centimeters, what is the perimeter?
13. If the length of a rectangle is 32 centimeters and the width is 65.5 centimeters, what is the area?
14. If the length of a rectangle is 30 centimeters and the width is 16 centimeters, what is the area?
15. If the perimeter of a rectangle is 32 centimeters and the width is 8 centimeters, what is the length?

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Color Highlighted Text Notes

### Vocabulary Language: English

Integer

The integers consist of all natural numbers, their opposites, and zero. Integers are numbers in the list ..., -3, -2, -1, 0, 1, 2, 3...

Percent

Percent means out of 100. It is a quantity written with a % sign.

rational number

A rational number is a number that can be expressed as the quotient of two integers, with the denominator not equal to zero.