Example

Pete bought a shirt on sale for $[latex]18[/latex], which is one-half the original price. What was the original price of the shirt? Solution: Step 1. Read the problem. Make sure you understand all the words and ideas. You may need to read the problem two or more times. If there are words you don't understand, look them up in a dictionary or on the Internet.

• In this problem, do you understand what is being discussed? Do you understand every word?

Step 2. Identify what you are looking for. It's hard to find something if you are not sure what it is! Read the problem again and look for words that tell you what you are looking for!

• In this problem, the words "what was the original price of the shirt" tell you what you are looking for: the original price of the shirt.

Step 3. Name what you are looking for. Choose a variable to represent that quantity. You can use any letter for the variable, but it may help to choose one that helps you remember what it represents.

• Let [latex]p=[/latex] the original price of the shirt

Step 4. Translate into an equation. It may help to first restate the problem in one sentence, with all the important information. Then translate the sentence into an equation. Step 5. Solve the equation using good algebra techniques. Even if you know the answer right away, using algebra will better prepare you to solve problems that do not have obvious answers.

Write the equation.	[latex]18=\Large\frac{1}{2}p[/latex]
Multiply both sides by 2.	[latex]\color{red}{2}\cdot18=\color{red}{2}\cdot\Large\frac{1}{2}\normalsize p[/latex]
Simplify.	[latex]36=p[/latex]

Step 6. Check the answer in the problem and make sure it makes sense.

• We found that [latex]p=36[/latex], which means the original price was [latex]\text{\$36}[/latex]. Does [latex]\text{\$36}[/latex] make sense in the problem? Yes, because [latex]18[/latex] is one-half of [latex]36[/latex], and the shirt was on sale at half the original price.

Step 7. Answer the question with a complete sentence.

• The problem asked "What was the original price of the shirt?" The answer to the question is: "The original price of the shirt was [latex]\text{\$36}[/latex]."

If this were a homework exercise, our work might look like this:

Example

Translate the table into algebraic expressions:

some number	the sum of the number and [latex]3[/latex]	twice the sum of the number and [latex]3[/latex]
a length	double the length	double the length, decreased by [latex]6[/latex]
a cost	the difference of the cost and [latex]20[/latex]	[latex]2[/latex] times the difference of the cost and [latex]20[/latex]
some quantity	the difference of [latex]5[/latex] and the quantity	the difference of [latex]5[/latex] and the quantity, divided by [latex]2[/latex]
an amount of time	triple the amount of time	triple the amount of time, increased by [latex]5[/latex]
a distance	the sum of [latex]-4[/latex] and the distance	the sum of [latex]-4[/latex] and the twice the distance

Answer:

[latex]a[/latex]	[latex]a+3[/latex]	[latex]2\left(a+3\right)[/latex]
[latex]l[/latex]	[latex]2l[/latex]	[latex]2l-6[/latex]
[latex]c[/latex]	[latex]c-20[/latex]	[latex]2\left(c-20\right)[/latex]
[latex]q[/latex]	[latex]5-q[/latex]	[latex]\frac{5-q}{2}[/latex]
[latex]t[/latex]	[latex]3t[/latex]	[latex]3t+5[/latex]
[latex]d[/latex]	[latex]-4+d[/latex]	[latex]-4+2d[/latex]

Example

Yash brought apples and bananas to a picnic. The number of apples was three more than twice the number of bananas. Yash brought [latex]11[/latex] apples to the picnic. How many bananas did he bring?

Answer: Solution:

Step 1. Read the problem.
Step 2. Identify what you are looking for.	How many bananas did he bring?
Step 3. Name what you are looking for.Choose a variable to represent the number of bananas.	Let [latex]b=\text{number of bananas}[/latex]
Step 4. Translate. Restate the problem in one sentence with all the important information.Translate into an equation.	[latex]11\enspace\Rightarrow[/latex] The number of apples[latex]=\enspace\Rightarrow[/latex] was [latex]3\enspace\Rightarrow[/latex] three [latex]+\enspace\Rightarrow[/latex] more than [latex]2b\enspace\Rightarrow[/latex] twice the number of bananas
Step 5. Solve the equation.	[latex]11=2b+3[/latex]
Subtract 3 from each side.	[latex]11\color{red}{-3}=2b+3\color{red}{-3}[/latex]
Simplify.	[latex]8=2b[/latex]
Divide each side by 2.	[latex]\Large\frac{8}{\color{red}{2}}\normalsize =\Large\frac{2b}{\color{red}{2}}[/latex]
Simplify.	[latex]4=b[/latex]
Step 6. Check: First, is our answer reasonable? Yes, bringing four bananas to a picnic seems reasonable. The problem says the number of apples was three more than twice the number of bananas. If there are four bananas, does that make eleven apples? Twice 4 bananas is 8. Three more than 8 is 11.
Step 7. Answer the question.	Yash brought 4 bananas to the picnic.

Example

Twenty-eight less than five times a certain number is [latex]232[/latex]. What is the number?

Answer: Following the steps provided:

1. Read and understand: we are looking for a number.
2. Constants and variables: [latex]28[/latex] and [latex]232[/latex] are constants, "a certain number" is our variable, because we don't know its value, and we are asked to find it. We will call it [latex]x[/latex].
3. Translate: five times a certain number translates to [latex]5x[/latex] Twenty-eight less than five times a certain number translates to [latex]5x-28[/latex], because subtraction is built backward. "is 232" translates to "[latex]=232"[/latex] since "is" is associated with equals.
4. Write an equation: [latex]5x-28=232[/latex]
5. Solve the equation using what you know about solving linear equations:

   [latex]\begin{array}{r}5x-28=232\\5x=260\\x=52\,\,\,\end{array}[/latex]

6. Check and interpret: We can substitute [latex]52[/latex] for [latex]x[/latex].

   [latex]\begin{array}{r}5\left(52\right)-28=232\\5\left(52\right)=260\\260=260\end{array}[/latex]

   TRUE!

example

Nga's car insurance premium increased by [latex]\text{\$60}[/latex], which was [latex]\text{8%}[/latex] of the original cost. What was the original cost of the premium?

Answer: Solution:

Step 1. Read the problem. Remember, if there are words you don't understand, look them up.
Step 2. Identify what you are looking for.	the original cost of the premium
Step 3. Name. Choose a variable to represent the original cost of premium.	Let [latex]c=\text{the original cost}[/latex]
Step 4. Translate. Restate as one sentence. Translate into an equation.
Step 5. Solve the equation.	[latex]60=0.08c[/latex]
Divide both sides by [latex]0.08[/latex].	[latex]\Large\frac{60}{\color{red}{0.08}}\normalsize =\Large\frac{0.08c}{\color{red}{0.08}}[/latex]
Simplify.	[latex]c=750[/latex]
Step 6. Check: Is our answer reasonable? Yes, a [latex]\text{\$750}[/latex] premium on auto insurance is reasonable. Now let's check our algebra. Is 8% of 750 equal to [latex]60[/latex]?[latex]750=c[/latex] [latex-display]0.08(750)=60[/latex-display] [latex]60=60\quad\checkmark[/latex]
Step 7. Answer the question.	The original cost of Nga's premium was [latex]\text{\$750}[/latex].