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Study Guides > ALGEBRA / TRIG I

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 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.
 
 

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example

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

Answer:

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 n=the numbern=\text{the number}
Step 4. Translate. Restate the problem as one sentence. Translate into an equation. 2n2n\enspace\Rightarrow The sum of twice a number ++\enspace\Rightarrow and 77\enspace\Rightarrow seven ==\enspace\Rightarrow is 1515\enspace\Rightarrow fifteen
Step 5. Solve the equation. 2n+7=152n+7=15
Subtract 7 from each side and simplify. 2n=82n=8
Divide each side by 2 and simplify. n=4n=4
Step 6. Check: is the sum of twice 44 and 77 equal to 1515? 24+7=152\cdot{4}+7=15 8+7=158+7=15 15=1515=15\quad\checkmark
Step 7. Answer the question. The number is 44.

 

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Watch the following video to see another example of how to solve a number problem. https://youtu.be/izIIqOztUyI

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.

Answer:

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 n=1st numbern=\text{1st number}One number is five more than another. n+5=2ndnumbern+5={2}^{\text{nd}}\text{number}
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 2121.The sum of the 1st number and the 2nd number is 2121. nn\enspace\Rightarrow First number ++\enspace\Rightarrow + n+5n+5\enspace\Rightarrow Second number ==\enspace\Rightarrow = 2121\enspace\Rightarrow twenty-one
Step 5. Solve the equation. n+n+5=21n+n+5=21
Combine like terms. 2n+5=212n+5=21
Subtract five from both sides and simplify. 2n=162n=16
Divide by two and simplify. n=8n=8     1st number
Now find the second number. n+5n+5     2nd number
Substitute n=8n = 8 8+5\color{red}{8}+5
1313
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? 13=?8+513\stackrel{\text{?}}{=}8+513=1313=13\quad\checkmark 8+13=?218+13\stackrel{\text{?}}{=}21 21=2121=21\quad\checkmark
Step 7. Answer the question. The numbers are 88 and 1313.

 

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Watch the following video to see another example of how to find two numbers given the relationship between the two. https://youtu.be/juslHscrh8s

example

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

Answer:

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 n=1st numbern=\text{1st number}One number is 44 less than the other. n4=2ndnumbern-4={2}^{\text{nd}}\text{number}
Step 4. Translate.Write as one sentence. Translate into an equation. Substitute the variable expressions. The sum of two numbers is negative fourteen.nn\enspace\Rightarrow First number ++\enspace\Rightarrow + n4n-4\enspace\Rightarrow Second number ==\enspace\Rightarrow = 14-14\enspace\Rightarrow negative fourteen
Step 5. Solve the equation. n+n4=14n+n-4=-14
Combine like terms. 2n4=142n-4=-14
Add 4 to each side and simplify. 2n=102n=-10
Divide by 2. n=5n=-5     1st number
Substitute n=5n=-5 to find the 2nd number. n4n-4     2nd number
54\color{red}{-5}-4
9-9
Step 6. Check:
Is −9 four less than −5?Is their sum −14? 54=?9-5-4\stackrel{\text{?}}{=}-99=9-9=-9\quad\checkmark 5+(9)=?14-5+(-9)\stackrel{\text{?}}{=}-14 14=14-14=-14\quad\checkmark
Step 7. Answer the question. The numbers are 5−5 and 9−9.

 

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example

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

Answer:

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 x=1st numberx=\text{1st number}2x+10=2ndnumber2x+10={2}^{\text{nd}}\text{number}
Step 4. Translate. Restate as one sentence. Their sum is one.
Translate into an equation x+(2x+10)x+(2x+10)\enspace\Rightarrow The sum of the two numbers==\enspace\Rightarrow is 11\enspace\Rightarrow one
Step 5. Solve the equation. x+2x+10=1x+2x+10=1
Combine like terms. 3x+10=13x+10=1
Subtract 10 from each side. 3x=93x=-9
Divide each side by 3 to get the first number. x=3x=-3
Substitute to get the second number. 2x+102x+10
2(3)+102(\color{red}{-3})+10
44
Step 6. Check.
Is 4 ten more than twice −3?Is their sum 1? 2(3)+10=?42(-3)+10\stackrel{\text{?}}{=}46+10=4-6+10=4 4=44=4\quad\checkmark 3+4=?1-3+4\stackrel{\text{?}}{=}1 1=11=1\quad\checkmark
Step 7. Answer the question. The numbers are 3−3 and 44.

 

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Solving for Consecutive Integers

Another type of number problem involves consecutive numbers. Consecutive numbers are numbers that come one after the other.  Some examples of consecutive integers are:

...1,2,3,4,5,6,...\begin{array}{c} \hfill \text{...}1, 2, 3, 4, 5, 6\text{,...}\hfill \end{array} ...10,9,8,7,...\text{...}-10,-9,-8,-7\text{,...} ...150,151,152,153,...\text{...}150,151,152,153\text{,...}

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 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\hfill & & & \text{1st integer}\hfill \\ n+1\hfill & & & \text{2nd consecutive integer}\hfill \\ n+2\hfill & & & \text{3rd consecutive integer}\hfill \end{array}

For example, let's say I want to know the next consecutive integer after 44. In mathematical terms, we would add 11 to 44 to get 55. We can generalize this idea as follows: the consecutive integer of any number, xx, is x+1x+1. 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 xx
Second x+1x+1
Third x+2x+2
Fourth  x+3x+3
We apply the idea of consecutive integers to solving a word problem in the following example.

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

[ohm_question]142817[/ohm_question]

Example

The sum of three consecutive integers is 9393. What are the integers?

Answer: Following the steps provided:

  1. Read and understand: We are looking for three numbers, and we know they are consecutive integers.
  2. Constants and Variables: 9393 is a constant. The first integer we will call xx. Second integer: x+1x+1 Third integer: x+2x+2
  3. Translate: The sum of three consecutive integers translates to x+(x+1)+(x+2)x+\left(x+1\right)+\left(x+2\right), 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 "=93=93" since "is" is associated with equals.
  4. Write an equation: x+(x+1)+(x+2)=93x+\left(x+1\right)+\left(x+2\right)=93
  5. 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.

    x+x+1+x+2=93x+x+1+x+2=93

    Combine like terms, simplify, and solve.

    x+x+1+x+2=933x+3=933     33x=903x3=903x=30\begin{array}{r}x+x+1+x+2=93\\3x+3 = 93\\\underline{-3\,\,\,\,\,-3}\\3x=90\\\frac{3x}{3}=\frac{90}{3}\\x=30\end{array}

  6. Check and Interpret: Okay, we have found a value for xx. 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 xx, x=30x=30 Second integer: x+1x+1 so 30+1=3130+1=31 Third integer: x+2x+2 so 30+2=3230+2=32 The three consecutive integers whose sum is 9393 are 3031, and 3230\text{, }31\text{, and }32

 

example

Find three consecutive integers whose sum is 4242.

Answer:

Solution:
Step 1. Read the problem.
Step 2. Identify what you are looking for. three consecutive integers
Step 3. Name. Let n=1st integern=\text{1st integer}n+1=2nd consecutive integern+1=\text{2nd consecutive integer} n+2=3rd consecutive integern+2=\text{3rd consecutive integer}  
Step 4. Translate. Restate as one sentence. Translate into an equation. n+n+1+n+2n\enspace +\enspace n+1\enspace +\enspace n+2\enspace\Rightarrow The sum of the three integers ==\enspace\Rightarrow is 4242\enspace\Rightarrow 42
Step 5. Solve the equation. n+n+1+n+2=42n+n+1+n+2=42
Combine like terms. 3n+3=423n+3=42
Subtract 3 from each side. 3n=393n=39
Divide each side by 3. n=13n=13      1st integer
Substitute to get the second number. n+1n+1     2nd integer
13+1\color{red}{13}+1
2424
Substitute to get the third number. n+2n+2     3rd integer
13+2\color{red}{13}+2
1515
Step 6. Check: 13+14+15=?4213+14+15\stackrel{\text{?}}{=}4242=4242=42\quad\checkmark
Step 7. Answer the question. The three consecutive integers are 1313, 1414, and 1515.

try it

[ohm_question]142816[/ohm_question]
Watch this video for another example of how to find three consecutive integers given their sum. https://youtu.be/Bo67B0L9hGs In the following video we show an example of a consecutive integer problem in which you are given the sum of four consecutive integers. https://youtu.be/S5HZy3jKodg

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