Using a Problem-Solving Strategy to Solve Number Problems
Learning Outcomes
Solve number problems
Solve consecutive integer problems
Solving Number 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 thirteen. 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
[latex]=\enspace\Rightarrow[/latex] is
[latex]13\enspace\Rightarrow[/latex] thirteen
Step 5. Solve the equation. Add 6 to both sides.
Simplify.
The sum of twice a number and seven is fifteen. 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]?
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]n+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] twenty-one
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
Now find the second number.
[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?
Another type of number problem involves consecutive numbers. Consecutive numbers are numbers that come one after the other. Some examples of consecutive integers are:
If we are looking for several consecutive numbers, it is important to first identify what they look like with variables before we set up the equation. 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].
For example, let’s say I want to know the next consecutive integer after [latex]4[/latex]. In mathematical terms, we would add [latex]1[/latex] to [latex]4[/latex] to get [latex]5[/latex]. We can generalize this idea as follows: the consecutive integer of any number, [latex]x[/latex], is [latex]x+1[/latex]. If we continue this pattern, we can define any number of consecutive integers from any starting point. The following table shows how to describe four consecutive integers using algebraic notation.
First
[latex]x[/latex]
Second
[latex]x+1[/latex]
Third
[latex]x+2[/latex]
Fourth
[latex]x+3[/latex]
We apply the idea of consecutive integers to solving a word problem in the following example.
example
The sum of two consecutive integers is [latex]47[/latex]. Find the numbers.
Solution:
Step 1. Read the problem.
Step 2. Identify what you are looking for.
two consecutive integers
Step 3. Name.
Let [latex]n=\text{1st integer}[/latex]
[latex]n+1=\text{next consecutive integer}[/latex]
Step 4. Translate. Restate as one sentence.
Translate into an equation.
[latex]n+n+1\enspace\Rightarrow[/latex] The sum of the integers
[latex]=\enspace\Rightarrow[/latex] is
[latex]47\enspace\Rightarrow[/latex] 47
The two consecutive integers are [latex]23[/latex] and [latex]24[/latex].
try it
Example
The sum of three consecutive integers is [latex]93[/latex]. What are the integers?
Show Solution
Following the steps provided:
Read and understand: We are looking for three numbers, and we know they are consecutive integers.
Constants and Variables: [latex]93[/latex] is a constant.
The first integer we will call [latex]x[/latex].
Second integer: [latex]x+1[/latex]
Third integer: [latex]x+2[/latex]
Translate: The sum of three consecutive integers translates to [latex]x+\left(x+1\right)+\left(x+2\right)[/latex], based on how we defined the first, second, and third integers. Notice how we placed parentheses around the second and third integers. This is just to make each integer more distinct. “is 93” translates to “[latex]=93[/latex]” since “is” is associated with equals.
Write an equation: [latex]x+\left(x+1\right)+\left(x+2\right)=93[/latex]
Solve the equation using what you know about solving linear equations: We can’t simplify within each set of parentheses, and we don’t need to use the distributive property so we can rewrite the equation without parentheses.
Check and Interpret: Okay, we have found a value for [latex]x[/latex]. We were asked to find the value of three consecutive integers, so we need to do a couple more steps. Remember how we defined our variables:
The first integer we will call [latex]x[/latex], [latex]x=30[/latex]
Second integer: [latex]x+1[/latex] so [latex]30+1=31[/latex]
Third integer: [latex]x+2[/latex] so [latex]30+2=32[/latex]
The three consecutive integers whose sum is [latex]93[/latex] are [latex]30\text{, }31\text{, and }32[/latex]
example
Find three consecutive integers whose sum is [latex]42[/latex].
Show Solution
Solution:
Step 1. Read the problem.
Step 2. Identify what you are looking for.
three consecutive integers
Step 3. Name.
Let [latex]n=\text{1st integer}[/latex][latex]n+1=\text{2nd consecutive integer}[/latex]
[latex]n+2=\text{3rd consecutive integer}[/latex]
Step 4. Translate. Restate as one sentence.
Translate into an equation.
[latex]n\enspace +\enspace n+1\enspace +\enspace n+2\enspace\Rightarrow[/latex] The sum of the three integers
[latex]=\enspace\Rightarrow[/latex] is
[latex]42\enspace\Rightarrow[/latex] 42
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