Using a Problem-Solving Strategy to Solve Number Problems

Learning Outcomes

    • Apply the general problem-solving strategy to number problems




  • Identify how many numbers you are solving for given a number problem
  • Solve consecutive integer problems


 

Now we will translate and solve number problems. In number problems, you are given some clues about one or more numbers, and you use these clues to build an equation. Number problems don't usually arise on an everyday basis, but they provide a good introduction to practicing the Problem-Solving Strategy. Remember to look for clue words such as difference, of, and and.

Example

The difference of a number and six is
1313
. Find the number.

Solution:

Step 1. Read the problem. Do you understand all the words?
Step 2. Identify what you are looking for. the number
Step 3. Name. Choose a variable to represent the number. Let
n=the numbern=\text{the number}
Step 4. Translate. Restate as one sentence.

Translate into an equation.
 
n6n-6\enspace\Rightarrow
The difference of a number and 6

==\enspace\Rightarrow
is

1313\enspace\Rightarrow
thirteen
Step 5. Solve the equation.

Add 6 to both sides.

Simplify.
n6=13n-6=13


n6+6=13+6n-6\color{red}{+6}=13\color{red}{+6}


n=19n=19
Step 6. Check:

The difference of
1919
and
66
is
1313
. It checks.
Step 7. Answer the question. The number is
1919
.
 

 

example

The sum of twice a number and seven is
1515
. Find the number.

 

Watch the following video to see another example of how to solve a number problem.



Solving for Two or More Numbers

Some number word problems ask you 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. 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 twenty-one. Find the numbers.

 

Watch the following video to see another example of how to find two numbers given the relationship between the two.



example

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

 

example

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

 

Solving for Consecutive Integers

Consecutive integers are integers that immediately follow each other. Some examples of consecutive integers are:

\begin{array}{c}\phantom{\rule{0.2}{0ex}}\\ \phantom{\rule{0.2}{0ex}}\\ \phantom{\rule{0.2}{0ex}}\\ \phantom{\rule{0.2}{0ex}}\\ \qquad ...1,2,3,4\text{,...}\qquad \end{array}


...10,9,8,7,...\text{...}-10,-9,-8,-7\text{,...}


...150,151,152,153,...\text{...}150,151,152,153\text{,...}


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

n1st integern+12nd consecutive integern+23rd consecutive integer\begin{array}{cccc}n\qquad & & & \text{1st integer}\qquad \\ n+1\qquad & & & \text{2nd consecutive integer}\qquad \\ n+2\qquad & & & \text{3rd consecutive integer}\qquad \end{array}


example

The sum of two consecutive integers is
4747
. Find the numbers.

Solution:

Step 1. Read the problem.
Step 2. Identify what you are looking for. two consecutive integers
Step 3. Name. Let
n=1st integern=\text{1st integer}


n+1=next consecutive integern+1=\text{next consecutive integer}
Step 4. Translate.

Restate as one sentence.

Translate into an equation.
n+n+1n+n+1\enspace\Rightarrow
The sum of the integers

==\enspace\Rightarrow
is

4747\enspace\Rightarrow
 47
Step 5. Solve the equation.
n+n+1=47n+n+1=47
Combine like terms.
2n+1=472n+1=47
Subtract 1 from each side.
2n=462n=46
Divide each side by 2.
n=23n=23
     1st integer
Substitute to get the second number.
n+1n+1
    2nd integer
23+1\color{red}{23}+1
2424
Step 6. Check:
23+24=?4723+24\stackrel{\text{?}}{=}47


47=4747=47\quad\checkmark
Step 7. Answer the question. The two consecutive integers are
2323
and
2424
.
 

try it



 

example

Find three consecutive integers whose sum is
4242
.

 

try it



Watch this video for another example of how to find three consecutive integers given their sum.



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