Question
Can you add or subtract fractions with unlike denominators without changing them, explain how and why not
Answers
Answered by
GPT-4o mini
You cannot directly add or subtract fractions with unlike denominators without altering them because the denominators represent different parts of a whole. For the fractions to be combined, they must represent the same-sized parts.
### Here's why:
1. **Understanding Denominators**: The denominator of a fraction tells you how many equal parts the whole is divided into. For instance, in the fractions \( \frac{1}{3} \) and \( \frac{1}{4} \), the 3 and 4 represent different sizes of parts. One-third of something is larger than one-fourth of the same thing.
2. **Need for a Common Denominator**: To add or subtract fractions, you need a common denominator so that the fractions represent equivalent values based on the same whole. This is essential because:
- \( \frac{1}{3} \) means you have 1 part out of 3 parts.
- \( \frac{1}{4} \) means you have 1 part out of 4 parts.
These fractions cannot be combined directly since we are adding parts of different-sized wholes.
### Steps to Add/Subtract Fractions:
1. **Find a Common Denominator**: Identify the least common multiple (LCM) of the denominators. For \( \frac{1}{3} \) and \( \frac{1}{4} \), the LCM of 3 and 4 is 12.
2. **Convert Fractions**: Convert each fraction to an equivalent fraction with the common denominator.
- For \( \frac{1}{3} \): \( \frac{1 \times 4}{3 \times 4} = \frac{4}{12} \)
- For \( \frac{1}{4} \): \( \frac{1 \times 3}{4 \times 3} = \frac{3}{12} \)
3. **Add or Subtract**: Now that the fractions have the same denominator, you can add or subtract the numerators.
- To add: \( \frac{4}{12} + \frac{3}{12} = \frac{4 + 3}{12} = \frac{7}{12} \)
- To subtract: \( \frac{4}{12} - \frac{3}{12} = \frac{4 - 3}{12} = \frac{1}{12} \)
### Conclusion:
In summary, while you cannot add or subtract fractions with unlike denominators without changing them directly, you can convert them to a common denominator to perform the operation correctly. This process ensures that you're accurately combining equivalent values, thus maintaining the integrity of the mathematical operation.
### Here's why:
1. **Understanding Denominators**: The denominator of a fraction tells you how many equal parts the whole is divided into. For instance, in the fractions \( \frac{1}{3} \) and \( \frac{1}{4} \), the 3 and 4 represent different sizes of parts. One-third of something is larger than one-fourth of the same thing.
2. **Need for a Common Denominator**: To add or subtract fractions, you need a common denominator so that the fractions represent equivalent values based on the same whole. This is essential because:
- \( \frac{1}{3} \) means you have 1 part out of 3 parts.
- \( \frac{1}{4} \) means you have 1 part out of 4 parts.
These fractions cannot be combined directly since we are adding parts of different-sized wholes.
### Steps to Add/Subtract Fractions:
1. **Find a Common Denominator**: Identify the least common multiple (LCM) of the denominators. For \( \frac{1}{3} \) and \( \frac{1}{4} \), the LCM of 3 and 4 is 12.
2. **Convert Fractions**: Convert each fraction to an equivalent fraction with the common denominator.
- For \( \frac{1}{3} \): \( \frac{1 \times 4}{3 \times 4} = \frac{4}{12} \)
- For \( \frac{1}{4} \): \( \frac{1 \times 3}{4 \times 3} = \frac{3}{12} \)
3. **Add or Subtract**: Now that the fractions have the same denominator, you can add or subtract the numerators.
- To add: \( \frac{4}{12} + \frac{3}{12} = \frac{4 + 3}{12} = \frac{7}{12} \)
- To subtract: \( \frac{4}{12} - \frac{3}{12} = \frac{4 - 3}{12} = \frac{1}{12} \)
### Conclusion:
In summary, while you cannot add or subtract fractions with unlike denominators without changing them directly, you can convert them to a common denominator to perform the operation correctly. This process ensures that you're accurately combining equivalent values, thus maintaining the integrity of the mathematical operation.