2.2 Use a Problem Solving Strategy

Step 1. Read the problem.
Step 2. Identify what you are looking for.	the number
Step 3. Name what you are looking for and choose a variable to represent it.	Let n = the number.
Step 4. Translate: Restate the problem as one sentence. Translate into an equation.
Step 5. Solve the equation. Subtract eight from each side and simplify. Divide each side by seven and simplify.
Step 6. Check. Is the sum of seven times four plus eight equal to 36? $\begin{array}{ccc}\hfill 7·4+8& \stackrel{?}{=}\hfill & 36\hfill \\ \hfill 28+8& \stackrel{?}{=}\hfill & 36\hfill \\ \hfill 36& =\hfill & 36✓\hfill \end{array}$
Step 7. Answer the question.	The number is 4.

Did you notice that we left out some of the steps as we solved this equation? If you’re not yet ready to leave out these steps, write down as many as you need.

$\begin{array}{cc}\mathbf{\text{Step 1. Read}}\text{the problem.}\hfill & \\ \mathbf{\text{Step 2. Identify}}\text{what you are looking for.}\hfill & \text{three consecutive even integers}\hfill \\ \mathbf{\text{Step 3. Name.}}\hfill & \text{Let}n=1^{\text{st}}\text{even integer.}\hfill \\ & n+2=2^{\text{nd}}\text{consecutive even integer}\hfill \\ & n+4=3^{\text{rd}}\text{consecutive even integer}\hfill \\ \mathbf{\text{Step 4. Translate.}}\hfill & \\ \text{Restate as one sentence.}\hfill & \text{The sum of the three even integers is}120.\hfill \\ \text{Translate into an equation.}\hfill & n+n+2+n+4=120\hfill \\ \mathbf{\text{Step 5. Solve}}\text{the equation.}\hfill & n+n+2+n+4=120\hfill \\ \text{Combine like terms.}\hfill & 3n+6=120\hfill \\ \text{Subtract 6 from each side.}\hfill & 3n=114\hfill \\ \text{Divide each side by 3.}\hfill & n=381^{\text{st}}\text{integer}\hfill \\ \\ & n+22^{\text{nd}}\text{integer}\hfill \\ & 38+2\hfill \\ & 40\hfill \\ \\ & n+43^{\text{rd}}\text{integer}\hfill \\ & 38+4\hfill \\ & 42\hfill \\ \mathbf{\text{Step 6. Check.}}\hfill & \\ \begin{array}{cc}\hfill 38+40+42& \stackrel{?}{=}120\hfill \\ \hfill 120& =120✓\hfill \end{array}\hfill \\ \mathbf{\text{Step 7. Answer}}\text{the question.}\hfill & \text{The three consecutive integers are 38, 40, and 42.}\hfill \end{array}$

$\begin{array}{cccccc}\mathbf{\text{Step 1. Read}}\text{the problem.}\hfill & & & & & \\ \mathbf{\text{Step 2. Identify}}\text{what you are looking for.}\hfill & & & & & \text{How much does the husband earn?}\hfill \\ \mathbf{\text{Step 3. Name.}}\hfill & & & & & \\ \text{Choose a variable to represent}\hfill & & & & & \text{Let}h=\text{the amount the husband earns.}\hfill \\ \text{the amount the husband earns.}\hfill & & & & & \\ \text{The wife earns \$16,000 less than twice that.}\hfill & & & & & \\ \mathbf{\text{Step 4. Translate.}}\hfill & & & & & \\ \begin{array}{c}\text{Restate the problem in one sentence}\hfill \\ \text{with all the important information.}\hfill \\ \text{Translate into an equation.}\hfill \end{array}\hfill & & & & & \begin{array}{c}2h-\mathrm{16,000}=\text{the amount the wife earns}\hfill \\ \text{Together the husband and wife earn \$110,000.}\hfill \end{array}\hfill \\ & & & & & \begin{array}{ccc}\hfill h+2h-\mathrm{16,000}& =\hfill & \mathrm{110,000}\hfill \end{array}\hfill \\ \begin{array}{c}\mathbf{\text{Step 5. Solve}}\text{the equation.}\hfill \\ \text{Combine like terms.}\hfill \\ \text{Add 16,000 to both sides and simplify.}\hfill \\ \text{Divide each side by three.}\hfill \end{array}\hfill & & & & & \begin{array}{ccc}\hfill h+2h-\mathrm{16,000}& =\hfill & \mathrm{110,000}\hfill \\ \hfill 3h-\mathrm{16,000}& =\hfill & \mathrm{110,000}\hfill \\ \hfill 3h& =\hfill & \mathrm{126,000}\hfill \\ \hfill h& =\hfill & \mathrm{42,000}\hfill \end{array}\hfill \\ & & & & & \hfill \begin{array}{c}\hfill \text{\$42,000 amount husband earns}\hfill \\ \\ \\ \hfill 2h-\mathrm{16,000}\text{amount wife earns}\hfill \\ \hfill 2(\mathrm{42,000})-\mathrm{16,000}\hfill \\ \hfill \mathrm{84,000}-\mathrm{16,000}\hfill \\ \hfill \mathrm{68,000}\hfill \end{array}\hfill \\ \mathbf{\text{Step 6. Check:}}\hfill & & & & & \\ \text{If the wife earns \$68,000 and the husband}\hfill & & & & & \\ \text{earns \$42,000, is that \$110,000? Yes!}\hfill & & & & & \\ \mathbf{\text{Step 7. Answer}}\text{the question.}\hfill & & & & & \text{The husband earns \$42,000 a year.}\hfill \end{array}$

ⓐ

Translate into algebra. Let n = the number.
Multiply.
			37.8 is 45% of 84.

ⓑ

Translate. Let n = the amount.
Multiply.
Divide both sides by 0.085 and simplify.
			8.5% of $56 is $4.76

ⓒ

We are asked to find percent, so we must have our result in percent form.
Translate into algebra. Let p = the percent.
Multiply.
Divide both sides by 112 and simplify.
Convert to percent.
			168 is 150% of 112.

ⓐ

Identify what you are asked to find, and choose a variable to represent it.	What is the amount of mark-up? Let $m=$ the amount of mark-up.
Write a sentence that gives the information to find it.
Translate into an equation.
Solve the equation.
Write a complete sentence.	The mark-up on the painting was $300.

ⓑ

Identify what you are asked to find, and choose a variable to represent it.	What is the list price? Let $p=$ the list price.
Write a sentence that gives the information to find it.
Translate into an equation.
Solve the equation.
Check.	Is the list price more than the original cost? Is $1,050 more than $750? Yes.
Write a complete sentence.	The list price of the painting was $1,050.

$\begin{array}{ccc}\hfill I& =\hfill & ?\hfill \\ \hfill P& =\hfill & \text{\$}950\hfill \\ \hfill r& =\hfill & 3\%\hfill \\ \hfill t& =\hfill & 5\text{years}\hfill \end{array}$

$\begin{array}{cccc}\text{Identify what you are asked to find, and choose a}\hfill & & & \text{What is the simple interest?}\hfill \\ \text{variable to represent it.}\hfill & & & \text{Let}I=\text{interest.}\hfill \\ \text{Write the formula.}\hfill & & & I=Prt\hfill \\ \text{Substitute in the given information.}\hfill & & & I=(950)(0.03)(5)\hfill \\ \text{Simplify.}\hfill & & & I=142.5\hfill \\ \text{Check.}\hfill & & & \\ \text{Is \$142.50 a reasonable amount of interest on \$950?}\hfill & & & \\ \text{Yes.}\hfill & \\ \text{Write a complete sentence.}\hfill & & & \text{The interest is \$142.50.}\hfill \end{array}$

$\begin{array}{ccc}\hfill I& =\hfill & \text{\$}1500\hfill \\ \hfill P& =\hfill & \text{\$}7500\hfill \\ \hfill r& =\hfill & ?\hfill \\ \hfill t& =\hfill & 5\text{years}\hfill \end{array}$

$\begin{array}{cccc}\text{Identify what you are asked to find, and choose}\hfill & & & \text{What is the rate of simple interest?}\hfill \\ \begin{array}{c}\text{a variable to represent it.}\hfill \\ \text{Write the formula.}\hfill \\ \text{Substitute in the given information.}\hfill \\ \text{Multiply.}\hfill \\ \text{Divide.}\hfill \\ \text{Change to percent form.}\hfill \end{array}\hfill & & & \begin{array}{ccc}\hfill \text{Let}r& =\hfill & \text{rate of interest.}\hfill \\ \hfill I& =\hfill & Prt\hfill \\ \hfill \mathrm{1,500}& =\hfill & (\mathrm{7,500})r(5)\hfill \\ \hfill \mathrm{1,500}& =\hfill & \mathrm{37,500}r\hfill \\ \hfill 0.04& =\hfill & r\hfill \\ \hfill 4\%& =\hfill & r\hfill \end{array}\hfill \\ \text{Check.}\hfill & & & \\ \begin{array}{ccc}\hfill I& =\hfill & Prt\hfill \\ \hfill \mathrm{1,500}& \stackrel{?}{=}\hfill & (\mathrm{7,500})(0.04)(5)\hfill \\ \hfill \mathrm{1,500}& =\hfill & \mathrm{1,500}✓\hfill \end{array}\hfill & & & \\ \text{Write a complete sentence.}\hfill & & & \text{The rate of interest was 4\%.}\hfill & & & \end{array}$

$\begin{array}{ccc}\hfill I& =\hfill & \mathrm{4,866.25}\hfill \\ \hfill P& =\hfill & ?\hfill \\ \hfill r& =\hfill & 8.5\%\hfill \\ \hfill t& =\hfill & 5\text{years}\hfill \end{array}$

$\begin{array}{cccc}\text{Identify what you are asked to find,}\hfill & & & \text{What is the amount borrowed (the principal)?}\hfill \\ \begin{array}{c}\text{and choose a variable to represent it.}\hfill \\ \text{Write the formula.}\hfill \\ \text{Substitute in the given information.}\hfill \\ \text{Multiply.}\hfill \\ \text{Divide.}\hfill \end{array}\hfill & & & \begin{array}{ccc}\hfill \text{Let}P& =\hfill & \text{principal borrowed.}\hfill \\ \hfill I& =\hfill & Prt\hfill \\ \hfill \mathrm{4,866.25}& =\hfill & P(0.085)(5)\hfill \\ \hfill \mathrm{4,866.25}& =\hfill & 0.425P\hfill \\ \hfill \mathrm{11,450}& =\hfill & P\hfill \end{array}\hfill \\ \text{Check.}\hfill & & & \\ \begin{array}{ccc}\hfill I& =\hfill & Prt\hfill \\ \hfill \mathrm{4,866.25}& \stackrel{?}{=}\hfill & (\mathrm{11,450})(0.085)(5)\hfill \\ \hfill \mathrm{4,866.25}& =\hfill & \mathrm{4,866.25}✓\hfill \end{array}\hfill & & & \\ \\ \\ \text{Write a complete sentence.}\hfill & & & \text{The principal was \$11,450.}\hfill \end{array}$