Question
Sadi wanted to compare the fractions 2
3
and 5
9
using common denominators. His work is shown below.
The common denominator is 9. Two-thirds times one-third = two-ninths. Two-ninths less-than five-ninths. So, two-thirds is less-than five-ninths.
What errors did Sadi make? Check all that apply.
The common denominator is 6, not 9.
He should have multiplied mr001-1.jpg by mr001-2.jpg, not mr001-3.jpg.
He should have gotten a common numerator, not a common denominator.
The fraction mr001-4.jpg is greater than mr001-5.jpg, not less
than mr001-6.jpg.
Answers
GPT-4o mini
Let's analyze Sadi's work step by step based on the process of comparing the fractions \( \frac{2}{3} \) and \( \frac{5}{9} \) using common denominators:
1. **Correct Common Denominator**: The common denominator Sadi chose is 9, but the correct common denominator for \( \frac{2}{3} \) and \( \frac{5}{9} \) is indeed 9. This part of Sadi's work is correct, so the first statement "The common denominator is 6, not 9." is incorrect.
2. **Multiplication for Equivalent Fractions**: To express \( \frac{2}{3} \) with a denominator of 9, Sadi should have multiplied \( \frac{2}{3} \) by \( \frac{3}{3} \) (which equals 1) to convert it to \( \frac{6}{9} \), not by \( \frac{1}{3} \). So the second statement "He should have multiplied \( \frac{2}{3} \) by \( \frac{1}{3} \), not \( \frac{3}{3} \)" is correct because he did the multiplication wrong.
3. **Common Numerator vs. Common Denominator Concept**: It’s important to find a common denominator when comparing fractions, so the statement "He should have gotten a common numerator, not a common denominator." is not applicable, because using a common denominator is a valid approach.
4. **Inequality Error**: After finding equivalent fractions, Sadi concluded that \( \frac{2}{3} \) is less than \( \frac{5}{9} \) based on his incorrect calculation, stating that "the fraction \( \frac{2}{9} \) is less than \( \frac{5}{9} \)". Since \( \frac{2}{9} \) is not the correct equivalent of \( \frac{2}{3} \), this conclusion is erroneous. The correct fraction \( \frac{2}{3} \) would have been \( \frac{6}{9} \), which is greater than \( \frac{5}{9} \). So, the fourth statement "The fraction \( \frac{2}{9} \) is greater than \( \frac{5}{9} \), not less than \( \frac{5}{9} \)" is incorrect.
In summary, the errors Sadi made are:
- **He should have multiplied \( \frac{2}{3} \) by \( \frac{3}{3} \), not \( \frac{1}{3} \).**
- **The fraction \( \frac{2}{9} \) (wrong calculation) cannot be said to be less than or greater than, since it’s not part of the correct comparison. Ultimately, \( \frac{6}{9} \) is greater than \( \frac{5}{9} \).**
So effectively, the errors in Sadi's reasoning are as follows:
- He should have multiplied by \( \frac{3}{3} \).
- His conclusion about the comparison based on incorrect fractions is flawed.
1. **Correct Common Denominator**: The common denominator Sadi chose is 9, but the correct common denominator for \( \frac{2}{3} \) and \( \frac{5}{9} \) is indeed 9. This part of Sadi's work is correct, so the first statement "The common denominator is 6, not 9." is incorrect.
2. **Multiplication for Equivalent Fractions**: To express \( \frac{2}{3} \) with a denominator of 9, Sadi should have multiplied \( \frac{2}{3} \) by \( \frac{3}{3} \) (which equals 1) to convert it to \( \frac{6}{9} \), not by \( \frac{1}{3} \). So the second statement "He should have multiplied \( \frac{2}{3} \) by \( \frac{1}{3} \), not \( \frac{3}{3} \)" is correct because he did the multiplication wrong.
3. **Common Numerator vs. Common Denominator Concept**: It’s important to find a common denominator when comparing fractions, so the statement "He should have gotten a common numerator, not a common denominator." is not applicable, because using a common denominator is a valid approach.
4. **Inequality Error**: After finding equivalent fractions, Sadi concluded that \( \frac{2}{3} \) is less than \( \frac{5}{9} \) based on his incorrect calculation, stating that "the fraction \( \frac{2}{9} \) is less than \( \frac{5}{9} \)". Since \( \frac{2}{9} \) is not the correct equivalent of \( \frac{2}{3} \), this conclusion is erroneous. The correct fraction \( \frac{2}{3} \) would have been \( \frac{6}{9} \), which is greater than \( \frac{5}{9} \). So, the fourth statement "The fraction \( \frac{2}{9} \) is greater than \( \frac{5}{9} \), not less than \( \frac{5}{9} \)" is incorrect.
In summary, the errors Sadi made are:
- **He should have multiplied \( \frac{2}{3} \) by \( \frac{3}{3} \), not \( \frac{1}{3} \).**
- **The fraction \( \frac{2}{9} \) (wrong calculation) cannot be said to be less than or greater than, since it’s not part of the correct comparison. Ultimately, \( \frac{6}{9} \) is greater than \( \frac{5}{9} \).**
So effectively, the errors in Sadi's reasoning are as follows:
- He should have multiplied by \( \frac{3}{3} \).
- His conclusion about the comparison based on incorrect fractions is flawed.