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Guillermo bought textbooks and notebooks at the bookstore. The number of textbooks was 3 more than twice the number of notebooks. He bought 7 textbooks. How many notebooks did he buy?
2
Gerry worked Sudoku puzzles and crossword puzzles this week. The number of Sudoku puzzles he completed is eight more than twice the number of crossword puzzles. He completed 22 Sudoku puzzles. How many crossword puzzles did he do?
7
Now that we have a problem solving strategy, we will use it on several different types of word problems. The first type we will work on is “number problems.” Number problems give some clues about one or more numbers. We use these clues to write an equation. Number problems don’t usually arise on an everyday basis, but they provide a good introduction to practicing the problem solving strategy outlined above.
The difference of a number and six is 13. Find the number.
| Step 1. Read the problem. Are all the words familiar? | |
| Step 2. Identify what we are looking for. | the number |
| Step 3. Name. Choose a variable to represent the number. | Let the number. |
| Step 4. Translate. Remember to look for clue words like "difference... of... and..." | |
| Restate the problem as one sentence. |
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| Translate into an equation. |
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| Step 5. Solve the equation. |
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| Simplify. |
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| Step 6. Check. | |
| The difference of 19 and 6 is 13. It checks! | |
| Step 7. Answer the question. | The number is 19. |
The difference of a number and eight is 17. Find the number.
25
The sum of twice a number and seven is 15. Find the number.
| Step 1. Read the problem. | |
| Step 2. Identify what we are looking for. | the number |
| Step 3. Name. Choose a variable to represent the number. | Let the number. |
| Step 4. Translate. | |
| Restate the problem as one sentence. |
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| Translate into an equation. |
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| Step 5. Solve the equation. |
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| Subtract 7 from each side and simplify. |
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| Divide each side by 2 and simplify. |
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| Step 6. Check. | |
| Is the sum of twice 4 and 7 equal to 15? | |
| Step 7. Answer the question. | The number is 4. |
Did you notice that we left out some of the steps as we solved this equation? If you’re not yet ready to leave out these steps, write down as many as you need.
The sum of four times a number and two is 14. Find the number.
3
The sum of three times a number and seven is 25. Find the number.
6
Some number word problems ask us 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. In order to avoid using more than 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.
One number is five more than another. The sum of the numbers is 21. Find the numbers.
| Step 1. Read the problem. | ||
| Step 2. Identify what we are looking for. | We are looking for two numbers. | |
| Step 3. Name. We have two numbers to name and need a name for each. | ||
| Choose a variable to represent the first number. | Let number. | |
| What do we know about the second number? | One number is five more than another. | |
| number | ||
| Step 4. Translate. Restate the problem as one sentence with all the important information. | The sum of the 1 st number and the 2 nd number is 21. | |
| Translate into an equation. |
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| Substitute the variable expressions. |
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| Step 5. Solve the equation. |
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| Combine like terms. |
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| Subtract 5 from both sides and simplify. |
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| Divide by 2 and simplify. |
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| Find the second number, too. |
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| 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? | ||
| Step 7. Answer the question. | The numbers are 8 and 13. |
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