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 [latex]13[/latex]. 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 [latex]n=\text{the number}[/latex]
Step 4. Translate. Restate as one sentence.
Translate into an equation.
[latex]n-6\enspace\Rightarrow[/latex] The difference of a number and 6
The difference of [latex]19[/latex] and [latex]6[/latex] is [latex]13[/latex]. It checks.
Step 7. Answer the question.
The number is [latex]19[/latex].
try it
example
The sum of twice a number and seven is [latex]15[/latex]. Find the number.
Show Solution
Solution:
Step 1. Read the problem.
Step 2. Identify what you are looking for.
the number
Step 3. Name. Choose a variable to represent the number.
Let [latex]n=\text{the number}[/latex]
Step 4. Translate. Restate the problem as one sentence.
Translate into an equation.
[latex]2n\enspace\Rightarrow[/latex] The sum of twice a number
[latex]+\enspace\Rightarrow[/latex] and
[latex]7\enspace\Rightarrow[/latex] seven
[latex]=\enspace\Rightarrow[/latex] is
[latex]15\enspace\Rightarrow[/latex] fifteen
Step 5. Solve the equation.
[latex]2n+7=15[/latex]
Subtract 7 from each side and simplify.
[latex]2n=8[/latex]
Divide each side by 2 and simplify.
[latex]n=4[/latex]
Step 6. Check: is the sum of twice [latex]4[/latex] and [latex]7[/latex] equal to [latex]15[/latex]?
[latex]2\cdot{4}+7=15[/latex]
[latex]8+7=15[/latex]
[latex]15=15\quad\checkmark[/latex]
Step 7. Answer the question.
The number is [latex]4[/latex].
try it
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.
Show Solution
Solution:
Step 1. Read the problem.
Step 2. Identify what you are looking for.
You are looking for two numbers.
Step 3. Name.
Choose a variable to represent the first number.
What do you know about the second number?
Translate.
Let [latex]n=\text{1st number}[/latex]
One number is five more than another.
[latex]x+5={2}^{\text{nd}}\text{number}[/latex]
Step 4. Translate.
Restate the problem as one sentence with all the important information.
Translate into an equation.
Substitute the variable expressions.
The sum of the numbers is [latex]21[/latex].
The sum of the 1st number and the 2nd number is [latex]21[/latex].
[latex]n\enspace\Rightarrow[/latex] First number
[latex]+\enspace\Rightarrow[/latex] +
[latex]n+5\enspace\Rightarrow[/latex] Second number
[latex]=\enspace\Rightarrow[/latex] =
[latex]21\enspace\Rightarrow[/latex] 21
Step 5. Solve the equation.
[latex]n+n+5=21[/latex]
Combine like terms.
[latex]2n+5=21[/latex]
Subtract five from both sides and simplify.
[latex]2n=16[/latex]
Divide by two and simplify.
[latex]n=8[/latex] 1st number
Find the second number too.
[latex]n+5[/latex] 2nd number
Substitute [latex]n = 8[/latex]
[latex]\color{red}{8}+5[/latex]
[latex]13[/latex]
Step 6. Check:
Do these numbers check in the problem?
Is one number 5 more than the other?
Is thirteen, 5 more than 8? Yes.
Is the sum of the two numbers 21?
[latex]13\stackrel{\text{?}}{=}8+5[/latex]
[latex]13=13\quad\checkmark[/latex]
[latex]8+13\stackrel{\text{?}}{=}21[/latex]
[latex]21=21\quad\checkmark[/latex]
Step 7. Answer the question.
The numbers are [latex]8[/latex] and [latex]13[/latex].
try it
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.
Show Solution
Solution:
Step 1. Read the problem.
Step 2. Identify what you are looking for.
two numbers
Step 3. Name. Choose a variable.
What do you know about the second number?
Translate.
Let [latex]n=\text{1st number}[/latex]
One number is [latex]4[/latex] less than the other.
[latex]n-4={2}^{\text{nd}}\text{number}[/latex]
Step 4. Translate.
Write as one sentence.
Translate into an equation.
Substitute the variable expressions.
The sum of two numbers is negative fourteen.
[latex]n\enspace\Rightarrow[/latex] First number
[latex]+\enspace\Rightarrow[/latex] +
[latex]n-4\enspace\Rightarrow[/latex] Second number
[latex]=\enspace\Rightarrow[/latex] =
[latex]-14\enspace\Rightarrow[/latex] -14
Step 5. Solve the equation.
[latex]n+n-4=-14[/latex]
Combine like terms.
[latex]2n-4=-14[/latex]
Add 4 to each side and simplify.
[latex]2n=-10[/latex]
Divide by 2.
[latex]n=-5[/latex] 1st number
Substitute [latex]n=-5[/latex] to find the 2^{nd} number.
[latex]n-4[/latex] 2nd number
[latex]\color{red}{-5}-4[/latex]
[latex]-9[/latex]
Step 6. Check:
Is −9 four less than −5?
Is their sum −14?
[latex]-5-4\stackrel{\text{?}}{=}-9[/latex]
[latex]-9=-9\quad\checkmark[/latex]
[latex]-5+(-9)\stackrel{\text{?}}{=}-14[/latex]
[latex]-14=-14\quad\checkmark[/latex]
Step 7. Answer the question.
The numbers are [latex]−5[/latex] and [latex]−9[/latex].
try it
example
One number is ten more than twice another. Their sum is one. Find the numbers.
Show Solution
Solution:
Step 1. Read the problem.
Step 2. Identify what you are looking for.
two numbers
Step 3. Name. Choose a variable.
One number is ten more than twice another.
Let [latex]x=\text{1st number}[/latex]
[latex]2x+10={2}^{\text{nd}}\text{number}[/latex]
Step 4. Translate. Restate as one sentence.
Their sum is one.
Translate into an equation
[latex]x+(2x+10)\enspace\Rightarrow[/latex] The sum of the two numbers
[latex]=\enspace\Rightarrow[/latex] is
[latex]1\enspace\Rightarrow[/latex] 1
Step 5. Solve the equation.
[latex]x+2x+10=1[/latex]
Combine like terms.
[latex]3x+10=1[/latex]
Subtract 10 from each side.
[latex]3x=-9[/latex]
Divide each side by 3 to get the first number.
[latex]x=-3[/latex]
Substitute to get the second number.
[latex]2x+10[/latex]
[latex]2(\color{red}{-3})+10[/latex]
[latex]4[/latex]
Step 6. Check.
Is 4 ten more than twice −3?
Is their sum 1?
[latex]2(-3)+10\stackrel{\text{?}}{=}4[/latex]
[latex]-6+10=4[/latex]
[latex]4=4\quad\checkmark[/latex]
[latex]-3+4\stackrel{\text{?}}{=}1[/latex]
[latex]1=1\quad\checkmark[/latex]
Step 7. Answer the question.
The numbers are [latex]−3[/latex] and [latex]4[/latex].
try it
Solving for Consecutive Integers
Consecutive integers are integers that immediately follow each other. Some examples of consecutive integers are:
Notice that each number is one more than the number preceding it. So if we define the first integer as [latex]n[/latex], the next consecutive integer is [latex]n+1[/latex]. The one after that is one more than [latex]n+1[/latex], so it is [latex]n+1+1[/latex], or [latex]n+2[/latex].
Ex: Write and Solve an Equation for Consecutive Natural Numbers with a Given Sum. Authored by: James Sousa (Mathispower4u.com). Located at: https://youtu.be/Bo67B0L9hGs. License: CC BY: Attribution
Write and Solve a Linear Equations to Solve a Number Problem (1) Mathispower4u . Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. Located at: https://youtu.be/izIIqOztUyI. License: CC BY: Attribution
Question ID 142763, 142770, 142775, 142806, 142811, 142816, 142817. Authored by: Lumen Learning. License: CC BY: Attribution. License Terms: IMathAS Community License, CC-BY + GPL
CC licensed content, Specific attribution
Prealgebra. Provided by: OpenStax. License: CC BY: Attribution. License Terms: Download for free at http://cnx.org/contents/caa57dab-41c7-455e-bd6f-f443cda5519c@9.757