Question
Manda converted the following repeating decimal a fraction. Her work is shown below.
1. x = 0.ModifyingAbove 27 with bar. 2. 100 x = 27.ModifyingAbove 27 with bar. 3. 100 x = 27. 4. x = StartFraction 27 Over 100 EndFraction
Analyze Manda’s work. Is she correct? If not, what was her mistake?
Yes, she is correct.
In step 2, she needed to multiply both sides by 10.
In step 3, she did not subtract one x from the left side.
In step 4, she did not simplify her fraction.
1. x = 0.ModifyingAbove 27 with bar. 2. 100 x = 27.ModifyingAbove 27 with bar. 3. 100 x = 27. 4. x = StartFraction 27 Over 100 EndFraction
Analyze Manda’s work. Is she correct? If not, what was her mistake?
Yes, she is correct.
In step 2, she needed to multiply both sides by 10.
In step 3, she did not subtract one x from the left side.
In step 4, she did not simplify her fraction.
Answers
Answered by
GPT-4o mini
Manda's work is incorrect, and her mistake can be identified as follows:
1. In step 1, she correctly identifies the repeating decimal as \( x = 0.\overline{27} \).
2. In step 2, she multiplies both sides by 100 correctly, resulting in \( 100x = 27.\overline{27} \). However, because \( 0.\overline{27} \) is equal to \( 27.\overline{27} \), she should actually have \( 100x = 27 + x \) instead of \( 27.\overline{27} \).
3. In step 3, if she correctly set up the equation, she should subtract \( x \) from both sides, leading to \( 100x - x = 27 \) or \( 99x = 27 \).
4. Lastly, in step 4, if she had done the correct operations, she would have arrived at \( x = \frac{27}{99} \), which can be simplified to \( \frac{3}{11} \).
So, the correct answer is \(\frac{3}{11}\), and Manda's mistake was in failing to set up the equation correctly after multiplying by 100. Her mistyped equation led to the incorrect conclusion.
1. In step 1, she correctly identifies the repeating decimal as \( x = 0.\overline{27} \).
2. In step 2, she multiplies both sides by 100 correctly, resulting in \( 100x = 27.\overline{27} \). However, because \( 0.\overline{27} \) is equal to \( 27.\overline{27} \), she should actually have \( 100x = 27 + x \) instead of \( 27.\overline{27} \).
3. In step 3, if she correctly set up the equation, she should subtract \( x \) from both sides, leading to \( 100x - x = 27 \) or \( 99x = 27 \).
4. Lastly, in step 4, if she had done the correct operations, she would have arrived at \( x = \frac{27}{99} \), which can be simplified to \( \frac{3}{11} \).
So, the correct answer is \(\frac{3}{11}\), and Manda's mistake was in failing to set up the equation correctly after multiplying by 100. Her mistyped equation led to the incorrect conclusion.