3.1 Use a problem-solving strategy  (Page 5/8)

Solution

Step 1. Read the problem.
Step 2. Identify what we are looking for.		three consecutive integers
Step 3. Name each of the three numbers.		Let $n=1^{\mathrm{st}}$ integer.
		$n+1=$ 2 nd consecutive integer
		$n+2=$ 3 rd consecutive integer
Step 4. Translate.
Restate as one sentence.		The sum of the three integers is −42.
Translate into an equation.
Step 5. Solve the equation.
Combine like terms.
Subtract 3 from each side.
Divide each side by 3.
Step 6. Check.
$\begin{array}{ccc}\hfill \mathrm{-13}+\left(\mathrm{-14}\right)+\left(\mathrm{-15}\right)& \stackrel{?}{=}\hfill & \mathrm{-42}\hfill \\ \hfill \mathrm{-42}& =\hfill & \mathrm{-42}✓\hfill \end{array}$
Step 7. Answer the question.		The three consecutive integers are −13, −14, and −15.

Solution

$\begin{array}{cccc}\mathbf{\text{Step 1. Read}}\text{the problem.}\hfill & & & \\ \mathbf{\text{Step 2. Identify}}\text{what we are looking for.}\hfill & & & \text{three consecutive even integers}\hfill \\ \mathbf{\text{Step 3. Name}}\text{the integers.}\hfill & & & \text{Let}n=1^{\text{st}}\text{even integer.}\hfill \\ & & & n+2=2^{\text{nd}}\text{consecutive even integer}\hfill \\ & & & n+4=3^{\text{rd}}\text{consecutive even integer}\hfill \\ \mathbf{\text{Step 4. Translate.}}\hfill & & & \\ \text{Restate as one sentence.}\hfill & & & \text{The sum of the three even integers is}84.\hfill \\ \text{Translate into an equation.}\hfill & & & n+n+2+n+4=84\hfill \\ \mathbf{\text{Step 5. Solve}}\text{the equation.}\hfill & & & \\ \text{Combine like terms.}\hfill & & & n+n+2+n+4=84\hfill \\ \text{Subtract 6 from each side.}\hfill & & & 3n+6=84\hfill \\ \text{Divide each side by 3.}\hfill & & & 3n=78\hfill \\ & & & n=261^{\text{st}}\text{integer}\hfill \\ \\ \\ & & & \begin{array}{cccc}\hfill n& +\hfill & 2\hfill & 2^{\text{nd}}\text{integer}\hfill \\ \hfill 26& +\hfill & 2\hfill & \\ & 28\hfill & & \\ \\ \\ \hfill n& +\hfill & 4\hfill & 3^{\text{rd}}\text{integer}\hfill \\ \hfill 26& +\hfill & 4\hfill & \\ & 30\hfill & & \end{array}\hfill \\ \mathbf{\text{Step 6. Check.}}\hfill & & & \\ \\ \\ \begin{array}{ccc}\hfill 26+28+30& \stackrel{?}{=}\hfill & 84\hfill \\ \hfill 84& =\hfill & 84✓\hfill \end{array}\hfill & & & \\ \mathbf{\text{Step 7. Answer}}\text{the question.}\hfill & & & \text{The three consecutive integers are 26, 28, and 30.}\hfill \end{array}$

Solution

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