Solving Number Problems

Now that we have a problem solving strategy, we will use it on several different types of word problems. The first type we will work on is “number problems.” Number problems give some clues about one or more numbers. We use these clues to write an equation. Number problems don’t usually arise on an everyday basis, but they provide a good introduction to practicing the problem solving strategy outlined above.

Example

The difference of a number and six is 13. Find the number.

Solution

Step 1. Read the problem. Are all the words familiar?
Step 2. Identify what we are looking for.	the number
Step 3. Name. Choose a variable to represent the number.	Let $n=$ the number.
Step 4. Translate. Remember to look for clue words like "difference... of... and..."
Restate the problem as one sentence.
Translate into an equation.
Step 5. Solve the equation.
Simplify.
Step 6. Check.
The difference of 19 and 6 is 13. It checks!
Step 7. Answer the question.	The number is 19.

Example

The sum of twice a number and seven is 15. Find the number.

Solution

Step 1. Read the problem.
Step 2. Identify what we are looking for.	the number
Step 3. Name. Choose a variable to represent the number.	Let $n=$ the number.
Step 4. Translate.
Restate the problem as one sentence.
Translate into an equation.
Step 5. Solve the equation.
Subtract 7 from each side and simplify.
Divide each side by 2 and simplify.
Step 6. Check.
Is the sum of twice 4 and 7 equal to 15?
$\begin{array}{ccc}\hfill 2\cdot 4+7& \stackrel{?}{=}\hfill & 15\hfill \\ \hfill 15& =\hfill & 15✓\hfill \end{array}$
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.

Some number word problems ask us to find two or more numbers. It may be tempting to name them all with different variables, but so far we have only solved equations with one variable. In order to avoid using more than one variable, we will define the numbers in terms of the same variable. Be sure to read the problem carefully to discover how all the numbers relate to each other.

Example

One number is five more than another. The sum of the numbers is 21. Find the numbers.

Solution

Step 1. Read the problem.
Step 2. Identify what we are looking for.		We are looking for two numbers.
Step 3. Name. We have two numbers to name and need a name for each.
Choose a variable to represent the first number.		Let $n={1}^{\mathrm{st}}$ number.
What do we know about the second number?		One number is five more than another.
		$n+5={2}^{\mathrm{nd}}$ number
Step 4. Translate. Restate the problem as one sentence with all the important information.		The sum of the 1^st number and the 2^nd number is 21.
Translate into an equation.
Substitute the variable expressions.
Step 5. Solve the equation.
Combine like terms.
Subtract 5 from both sides and simplify.
Divide by 2 and simplify.
Find the second number, too.


Step 6. Check.
Do these numbers check in the problem?
Is one number 5 more than the other?	$\phantom{\rule{1.6em}{0ex}}13\stackrel{?}{=}8+5$
Is thirteen 5 more than 8? Yes.	$\phantom{\rule{1.6em}{0ex}}13=13✓$
Is the sum of the two numbers 21?	$8+13\stackrel{?}{=}21$
	$\phantom{\rule{1.6em}{0ex}}21=21✓$
Step 7. Answer the question.		The numbers are 8 and 13.

Example

The sum of two numbers is negative fourteen. One number is four less than the other. Find the numbers.

Solution

Step 1. Read the problem.
Step 2. Identify what we are looking for.		We are looking for two numbers.
Step 3. Name.
Choose a variable.		Let $n={1}^{\mathrm{st}}$ number.
One number is 4 less than the other.		$n-4={2}^{\mathrm{nd}}$ number
Step 4. Translate.
Write as one sentence.		The sum of the 2 numbers is negative 14.
Translate into an equation.
Step 5. Solve the equation.
Combine like terms.
Add 4 to each side and simplify.
Simplify.




Step 6. Check.
Is −9 four less than −5?	$\phantom{\rule{1.1em}{0ex}}-5-4\stackrel{?}{=}-9$
	$\phantom{\rule{2.7em}{0ex}}-9=-9✓$
Is their sum −14?	$-5+\left(-9\right)\stackrel{?}{=}-14$
	$\phantom{\rule{2.3em}{0ex}}-14=-14✓$
Step 7. Answer the question.		The numbers are −5 and −9.

Example

One number is ten more than twice another. Their sum is one. Find the numbers.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.		We are looking for two numbers.
Step 3. Name.
Choose a variable.		Let $x={1}^{\mathrm{st}}$ number.
One number is 10 more than twice another.		$2x+10={2}^{\mathrm{nd}}$ number
Step 4. Translate.
Restate as one sentence.		Their sum is one.
		The sum of the two numbers is 1.
Translate into an equation.
Step 5. Solve the equation.
Combine like terms.
Subtract 10 from each side.
Divide each side by 3.




Step 6. Check.
Is ten more than twice −3 equal to 4?	$2\left(-3\right)+10\stackrel{?}{=}4$
	$\phantom{\rule{1em}{0ex}}-6+10\stackrel{?}{=}4$
	$\phantom{\rule{3.8em}{0ex}}4=4✓$
Is their sum 1?	$\phantom{\rule{1.5em}{0ex}}-3+4\stackrel{?}{=}1$
	$\phantom{\rule{3.8em}{0ex}}1=1✓$
Step 7. Answer the question.		The numbers are −3 and −4.

Some number problems involve consecutive integers. Consecutive integers are integers that immediately follow each other. Examples of consecutive integers are:

$\begin{array}{c}1,2,3,4\hfill \\ -10,-9,-8,-7\hfill \\ 150,151,152,153\hfill \end{array}$

Notice that each number is one more than the number preceding it. So if we define the first integer as n, the next consecutive integer is $n+1.$ The one after that is one more than $n+1,$ so it is $n+1+1,$ which is $n+2.$

$\begin{array}{cccc}\hfill n\hfill & & & {1}^{\text{st}}\phantom{\rule{0.2em}{0ex}}\text{integer}\hfill \\ \hfill n+1\hfill & & & {2}^{\text{nd}}\phantom{\rule{0.2em}{0ex}}\text{consecutive integer}\hfill \\ \hfill n+2\hfill & & & {3}^{\text{rd}}\phantom{\rule{0.2em}{0ex}}\text{consecutive integer . . . etc.}\hfill \end{array}$

Example

The sum of two consecutive integers is 47. Find the numbers.

Solution

Step 1. Read the problem.
Step 2. Identify what you are looking for.		two consecutive integers
Step 3. Name each number.		Let $n={1}^{\mathrm{st}}$ integer.
		$n+1=$ next consecutive integer
Step 4. Translate.
Restate as one sentence.		The sum of the integers is 47.
Translate into an equation.
Step 5. Solve the equation.
Combine like terms.
Subtract 1 from each side.
Divide each side by 2.



Step 6. Check.
$\begin{array}{ccc}\hfill 23+24& \stackrel{?}{=}\hfill & 47\hfill \\ \hfill 47& =\hfill & 47✓\hfill \end{array}$
Step 7. Answer the question.		The two consecutive integers are 23 and 24.

Example

Find three consecutive integers whose sum is $-42.$

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.
$\begin{array}{ccc}\hfill -13+\left(-14\right)+\left(-15\right)& \stackrel{?}{=}\hfill & -42\hfill \\ \hfill -42& =\hfill & -42✓\hfill \end{array}$
Step 7. Answer the question.		The three consecutive integers are −13, −14, and −15.

Now that we have worked with consecutive integers, we will expand our work to include consecutive even integers and consecutive odd integers. Consecutive even integers are even integers that immediately follow one another. Examples of consecutive even integers are:

$\begin{array}{c}18,20,22\hfill \\ 64,66,68\hfill \\ -12,-10,-8\hfill \end{array}$

Notice each integer is 2 more than the number preceding it. If we call the first one n, then the next one is $n+2.$ The next one would be $n+2+2$ or $n+4.$

$\begin{array}{cccc}\hfill n\hfill & & & {1}^{\text{st}}\phantom{\rule{0.2em}{0ex}}\text{even integer}\hfill \\ \hfill n+2\hfill & & & {2}^{\text{nd}}\phantom{\rule{0.2em}{0ex}}\text{consecutive even integer}\hfill \\ \hfill n+4\hfill & & & {3}^{\text{rd}}\phantom{\rule{0.2em}{0ex}}\text{consecutive even integer . . . etc.}\hfill \end{array}$

Consecutive odd integers are odd integers that immediately follow one another. Consider the consecutive odd integers 77, 79, and 81.

$\begin{array}{c}77,79,81\hfill \\ n,n+2,n+4\hfill \end{array}$

$\begin{array}{cccc}\hfill n\hfill & & & {1}^{\text{st}}\phantom{\rule{0.2em}{0ex}}\text{odd integer}\hfill \\ \hfill n+2\hfill & & & {2}^{\text{nd}}\phantom{\rule{0.2em}{0ex}}\text{consecutive odd integer}\hfill \\ \hfill n+4\hfill & & & {3}^{\text{rd}}\phantom{\rule{0.2em}{0ex}}\text{consecutive odd integer . . . etc.}\hfill \end{array}$

Does it seem strange to add 2 (an even number) to get from one odd integer to the next? Do you get an odd number or an even number when we add 2 to 3? to 11? to 47?

Whether the problem asks for consecutive even numbers or odd numbers, you don’t have to do anything different. The pattern is still the same—to get from one odd or one even integer to the next, add 2.

Example

Find three consecutive even integers whose sum is 84.

Solution

\(\begin{array}{cccc}\mathbf{\text{Step 1. Read}}\phantom{\rule{0.2em}{0ex}}\text{the problem.}\hfill & & & \\ \mathbf{\text{Step 2. Identify}}\phantom{\rule{0.2em}{0ex}}\text{what we are looking for.}\hfill & & & \text{three consecutive even integers}\hfill \\ \mathbf{\text{Step 3. Name}}\phantom{\rule{0.2em}{0ex}}\text{the integers.}\hfill & & & \text{Let}\phantom{\rule{0.2em}{0ex}}n={1}^{\text{st}}\phantom{\rule{0.2em}{0ex}}\text{even integer.}\hfill \\ & & & n+2={2}^{\text{nd}}\phantom{\rule{0.2em}{0ex}}\text{consecutive even integer}\hfill \\ & & & n+4={3}^{\text{rd}}\phantom{\rule{0.2em}{0ex}}\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}\phantom{\rule{0.2em}{0ex}}84.\hfill \\ \text{Translate into an equation.}\hfill & & & n+n+2+n+4\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}84\hfill \\ \mathbf{\text{Step 5. Solve}}\phantom{\rule{0.2em}{0ex}}\text{the equation.}\hfill & & & \\ \text{Combine like terms.}\hfill & & & n+n+2+n+4\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}84\hfill \\ \text{Subtract 6 from each side.}\hfill & & & \phantom{\rule{4.45em}{0ex}}3n+6\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}84\hfill \\ \text{Divide each side by 3.}\hfill & & & \phantom{\rule{6.1em}{0ex}}3n\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}78\hfill \\ & & & \phantom{\rule{6.7em}{0ex}}n\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}26\phantom{\rule{0.4em}{0ex}}{1}^{\text{st}}\phantom{\rule{0.2em}{0ex}}\text{integer}\hfill \\ \\ & & & \phantom{\rule{6em}{0ex}}\begin{array}{cccc}\hfill n& +\hfill & 2\hfill & {2}^{\text{nd}}\phantom{\rule{0.2em}{0ex}}\text{integer}\hfill \\ \hfill 26& +\hfill & 2\hfill & \\ & 28\hfill & & \\ \\ \hfill n& +\hfill & 4\hfill & {3}^{\text{rd}}\phantom{\rule{0.2em}{0ex}}\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}}\phantom{\rule{0.2em}{0ex}}\text{the question.}\hfill & & & \text{The three consecutive integers are 26, 28, and 30.}\hfill \end{array}\)

Example

A married couple together earns $110,000 a year. The wife earns $16,000 less than twice what her husband earns. What does the husband earn?

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.		$2h-16,000$ 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.		$\phantom{\rule{1.1em}{0ex}}$3h − 16,000 = 110,000
Add 16,000 to both sides and simplify.		$\phantom{\rule{4.7em}{0ex}}$3h = 126,000
Divide each side by 3.		$\phantom{\rule{5.2em}{0ex}}$h = 42,000
		$\phantom{\rule{6.2em}{0ex}}$$42,000 amount husband earns
		$\phantom{\rule{4.7em}{0ex}}$2h − 16,000 amount wife earns
		$\phantom{\rule{1.88em}{0ex}}$2(42,000) − 16,000
		$\phantom{\rule{3.05em}{0ex}}$84,000 − 16,000
		$\phantom{\rule{6.85em}{0ex}}$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.

Solving Number Problems