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Questions and Answers
What is the first step to add two or more fractions?
What should be done if the fractions in an addition problem do not have a common denominator?
In the example provided, what is the simplified form of 32/18?
What mathematical operation needs to be done after finding a common denominator in fraction addition?
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Why is it important to simplify fractions after performing addition?
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How should fractions be adjusted before subtracting them?
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In a multi-step fraction problem, what is the purpose of finding the least common multiple (LCM) of denominators?
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What step is necessary to simplify fractions in multi-step fraction problems?
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What is a crucial step when adding or subtracting fractions with different denominators?
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Why is it important to follow specific rules when dealing with multi-step fraction problems?
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Study Notes
Multi-step fraction problems involve more complex calculations than simple addition and subtraction of fractions. These problems often require factors to be reduced, common denominators found, and multiple steps to solve. Below is a comprehensive guide to solving various types of multi-step fraction problems.
Addition of Fractions
To add two or more fractions, the following steps must be followed:
- Make sure all the fractions have a common denominator. If they do not, find a common multiple of the current denominators and change each fraction so it has this new common multiple as its denominator.
- Add the numerators together, and place the sum over the common denominator. This creates one large fraction whose value is the sum of the original fractions.
- Simplify the resulting fraction if possible by finding the greatest common divisor (GCD) of both the numerator and the denominator, dividing both by the GCD, and simplifying the fraction further.
For example, consider the problem: 7/9 + 2/3. Since the denominators are not common, we need to adjust them: 7/9 becomes (2*7)/(2*9) = 14/18, while 2/3 becomes (9*2)/(9*3) = 18/27. Now, we can add the numerators together, 14 + 18 = 32, and place it over the common denominator, 18. This gives us the simplified fraction 32/18. Finally, since 8 is the greatest common divisor of both 32 and 18, dividing by 8 results in the final answer: 4.
Subtraction of Fractions
Subtracting fractions follows similar steps as adding:
- Ensure that the fractions have a common denominator. If not, adjust them so they do.
- Subtract the numerator of one fraction from the other. However, if the result is negative, change it so it becomes positive by switching the signs of the two original numbers before subtracting.
- Place the resulting difference over the common denominator. Simplify this new fraction if possible.
For instance, let's consider the problem: 7/9 - 2/3. Since the denominators are different, we need to adjust them first: 7/9 becomes (2*7)/(2*9) = 14/18, while 2/3 remains unchanged. Now, we can subtract the numerator of the second fraction from the first, 14 - 2 = 12. We have now obtained 12/18. To find the common multiple of 8, we divide both the numerator and the denominator by 4: 12/8 over 18/8, which simplifies to 3/4. The final step involves dividing the simplified fraction by its GCD to reach the simplest form: 3/4.
Multi-step Fraction Problems
Multi-step fraction problems typically involve adding or subtracting several fractions, each requiring their own set of calculations. These calculations may include reducing the factors, finding common denominators, and simplifying fractions. To solve such problems, follow these steps:
- Identify all the fractions involved in the problem.
- Check if they have a common denominator. If not, find the least common multiple (LCM) of their denominators.
- Adjust each fraction so it has the LCM as its denominator. This may involve multiplying both numerator and denominator by appropriate factors.
- Add or subtract the adjusted fractions based on the operation provided. Remember to follow the rules for adding or subtracting fractions as outlined above.
- Simplify any resulting fractions if possible. This involves finding the greatest common divisor (GCD) and dividing both the numerator and the denominator by the GCD.
Note that some multi-step fraction problems may require more than two basic operations, like adding two fractions, then subtracting another. Always approach these problems step by step, following the same procedures mentioned above.
Example Problem
Here is an example of a multi-step fraction problem involving addition and reduction:
(3/7) + (2/13) + (6/13) + (5/7)
First, adjust each fraction so they share the same denominator, 13. We get:
(9/13) + (10/13) + (30/13) + (10/7)
Now, add and reduce these fractions together:
(9 + 10 + 30 + 70)/(13*13) = (121 + 370)/169 = 491/169
Finally, we can simplify the fraction by dividing both the numerator and the denominator by 139, giving us the simplest form: 3/13.
In conclusion, solving multi-step fraction problems involves careful attention to detail, ensuring that all fractions have a common denominator, and following the appropriate rules for adding or subtracting fractions. By breaking down these problems into smaller steps and working through them systematically, you can arrive at the correct solution with ease.
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Description
Learn how to add and subtract fractions in multi-step fraction problems by finding common denominators, adding or subtracting numerators, and simplifying the resulting fractions. Follow the step-by-step guide with examples to master solving complex fraction problems.
