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? |
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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. |
[latex]n-6=13[/latex]
[latex-display]n-6\color{red}{+6}=13\color{red}{+6}[/latex-display]
[latex]n=19[/latex] |
Step 6. Check:
The difference of [latex]19[/latex] and [latex]6[/latex] is [latex]13[/latex]. It checks. |
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Step 7. Answer the question. |
The number is [latex]19[/latex]. |
example
The sum of twice a number and seven is [latex]15[/latex]. Find the number.
Answer:
Solution:
Step 1. Read the problem. |
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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-display]2\cdot{4}+7=15[/latex-display]
[latex-display]8+7=15[/latex-display]
[latex]15=15\quad\checkmark[/latex] |
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Step 7. Answer the question. |
The number is [latex]4[/latex]. |
Watch the following video to see another example of how to solve a number problem.
https://youtu.be/izIIqOztUyI
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. |
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Step 2. Identify what you are looking for. |
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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. |
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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. |
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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. |
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[latex]n+n+5=21[/latex] |
Combine like terms. |
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[latex]2n+5=21[/latex] |
Subtract five from both sides and simplify. |
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[latex]2n=16[/latex] |
Divide by two and simplify. |
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[latex]n=8[/latex] 1st number |
Find the second number too. |
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[latex]n+5[/latex] 2nd number |
Substitute [latex]n = 8[/latex] |
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[latex]\color{red}{8}+5[/latex] |
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[latex]13[/latex] |
Step 6. Check: |
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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-display]13=13\quad\checkmark[/latex-display]
[latex-display]8+13\stackrel{\text{?}}{=}21[/latex-display]
[latex]21=21\quad\checkmark[/latex] |
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Step 7. Answer the question. |
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The numbers are [latex]8[/latex] and [latex]13[/latex]. |
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. |
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Step 2. Identify what you are looking for. |
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two numbers |
Step 3. Name. Choose a variable.
What do you know about the second number?
Translate. |
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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. |
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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. |
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[latex]n+n-4=-14[/latex] |
Combine like terms. |
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[latex]2n-4=-14[/latex] |
Add 4 to each side and simplify. |
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[latex]2n=-10[/latex] |
Divide by 2. |
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[latex]n=-5[/latex] 1st number |
Substitute [latex]n=-5[/latex] to find the 2nd number. |
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[latex]n-4[/latex] 2nd number |
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[latex]\color{red}{-5}-4[/latex] |
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[latex]-9[/latex] |
Step 6. Check: |
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Is −9 four less than −5?
Is their sum −14? |
[latex]-5-4\stackrel{\text{?}}{=}-9[/latex]
[latex-display]-9=-9\quad\checkmark[/latex-display]
[latex-display]-5+(-9)\stackrel{\text{?}}{=}-14[/latex-display]
[latex]-14=-14\quad\checkmark[/latex] |
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Step 7. Answer the question. |
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The numbers are [latex]−5[/latex] and [latex]−9[/latex]. |
example
One number is ten more than twice another. Their sum is one. Find the numbers.
Answer:
Solution:
Step 1. Read the problem. |
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Step 2. Identify what you are looking for. |
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two numbers |
Step 3. Name. Choose a variable.
One number is ten more than twice another. |
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Let [latex]x=\text{1st number}[/latex]
[latex]2x+10={2}^{\text{nd}}\text{number}[/latex] |
Step 4. Translate. Restate as one sentence. |
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Their sum is one. |
Translate into an equation |
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[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. |
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[latex]x+2x+10=1[/latex] |
Combine like terms. |
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[latex]3x+10=1[/latex] |
Subtract 10 from each side. |
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[latex]3x=-9[/latex] |
Divide each side by 3 to get the first number. |
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[latex]x=-3[/latex] |
Substitute to get the second number. |
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[latex]2x+10[/latex] |
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[latex]2(\color{red}{-3})+10[/latex] |
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[latex]4[/latex] |
Step 6. Check. |
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Is 4 ten more than twice −3?
Is their sum 1? |
[latex]2(-3)+10\stackrel{\text{?}}{=}4[/latex]
[latex-display]-6+10=4[/latex-display]
[latex-display]4=4\quad\checkmark[/latex-display]
[latex-display]-3+4\stackrel{\text{?}}{=}1[/latex-display]
[latex]1=1\quad\checkmark[/latex] |
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Step 7. Answer the question. |
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The numbers are [latex]−3[/latex] and [latex]4[/latex]. |
example
The sum of two consecutive integers is [latex]47[/latex]. Find the numbers.
Solution:
Step 1. Read the problem. |
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Step 2. Identify what you are looking for. |
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two consecutive integers |
Step 3. Name. |
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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. |
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[latex]n+n+1\enspace\Rightarrow[/latex] The sum of the integers
[latex]=\enspace\Rightarrow[/latex] is
[latex]47\enspace\Rightarrow[/latex] 47 |
Step 5. Solve the equation. |
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[latex]n+n+1=47[/latex] |
Combine like terms. |
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[latex]2n+1=47[/latex] |
Subtract 1 from each side. |
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[latex]2n=46[/latex] |
Divide each side by 2. |
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[latex]n=23[/latex] 1st integer |
Substitute to get the second number. |
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[latex]n+1[/latex] 2nd integer |
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[latex]\color{red}{23}+1[/latex] |
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[latex]24[/latex] |
Step 6. Check: |
[latex]23+24\stackrel{\text{?}}{=}47[/latex]
[latex]47=47\quad\checkmark[/latex] |
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Step 7. Answer the question. |
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The two consecutive integers are [latex]23[/latex] and [latex]24[/latex]. |
example
Find three consecutive integers whose sum is [latex]42[/latex].
Answer:
Solution:
Step 1. Read the problem. |
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Step 2. Identify what you are looking for. |
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three consecutive integers |
Step 3. Name. |
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Let [latex]n=\text{1st integer}[/latex]
[latex-display]n+1=\text{2nd consecutive integer}[/latex-display]
[latex-display]n+2=\text{3rd consecutive integer}[/latex-display]
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Step 4. Translate.
Restate as one sentence.
Translate into an equation. |
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[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 |
Step 5. Solve the equation. |
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[latex]n+n+1+n+2=42[/latex] |
Combine like terms. |
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[latex]3n+3=42[/latex] |
Subtract 3 from each side. |
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[latex]3n=39[/latex] |
Divide each side by 3. |
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[latex]n=13[/latex] 1st integer |
Substitute to get the second number. |
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[latex]n+1[/latex] 2nd integer |
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[latex]\color{red}{13}+1[/latex] |
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[latex]24[/latex] |
Substitute to get the third number. |
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[latex]n+2[/latex] 3rd integer |
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[latex]\color{red}{13}+2[/latex] |
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[latex]15[/latex] |
Step 6. Check: |
[latex]13+14+15\stackrel{\text{?}}{=}42[/latex]
[latex]42=42\quad\checkmark[/latex] |
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Step 7. Answer the question. |
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The three consecutive integers are [latex]13[/latex], [latex]14[/latex], and [latex]15[/latex]. |