Answer :
Let's analyze the problem step-by-step to understand what Sam did and identify where the error occurred.
The problem is to divide [tex]\(-0.2709\)[/tex] by [tex]\(-0.043\)[/tex].
### Step 1: Correct placement of the decimal point
First, let's move the decimal point in both the divisor and the dividend to the right to eliminate the decimals.
- The divisor [tex]\(-0.043\)[/tex] becomes [tex]\(-43\)[/tex] after moving the decimal point [tex]\(3\)[/tex] places to the right.
- The dividend [tex]\(-0.2709\)[/tex] becomes [tex]\(-270.9\)[/tex] after moving the decimal point [tex]\(3\)[/tex] places to the right.
Now, the division problem is:
[tex]\[ -270.9 \div -43 \][/tex]
Since both the dividend and the divisor are negative, the result should be positive.
### Step 2: Perform the division
Let’s set up the long division:
[tex]\[ \begin{array}{r} -43 \\ -43 \longdiv { -270.9 } \\ \end{array} \][/tex]
#### First division:
- Determine how many times [tex]\(43\)[/tex] goes into [tex]\(270\)[/tex].
- [tex]\(43\)[/tex] goes into [tex]\(270\)[/tex] approximately [tex]\(6\)[/tex] times because [tex]\(43 \times 6 = 258\)[/tex].
#### Subtraction:
- Subtract [tex]\(258\)[/tex] from [tex]\(270\)[/tex] (similar to what Sam did):
[tex]\[ 270 - 258 = 12 \][/tex]
#### Next digit in the dividend:
- Bring down the next digit after the decimal point (if any). In this case, bring [tex]\(9\)[/tex] making the number [tex]\( 129 \)[/tex].
#### Second division:
- Determine how many times 43 goes into 129.
- [tex]\(43\)[/tex] goes into [tex]\(129\)[/tex] exactly [tex]\(3\)[/tex] times because [tex]\(43 \times 3 = 129\)[/tex].
#### Subtraction:
- Subtract [tex]\(129\)[/tex] from [tex]\(129\)[/tex]:
[tex]\[ 129 - 129 = 0 \][/tex]
So, the answer to the division is approximately [tex]\(6.3\)[/tex].
### Error Analysis
- The answer should be [tex]\( -65 \)[/tex]. Sam made an error subtracting [tex]\(258\)[/tex] from [tex]\(270\)[/tex].
Based on this correct division, noticing Sam's mistake confirms the precise error: Sam possibly miscalculated the results at the subtraction step.
The correct steps conclude that:
[tex]\[ \frac{-270.9}{-0.043} = 6.3 \][/tex]
### Conclusion:
The mistake Sam made was in the incorrect calculation during subtraction while doing the long division. Thus, the correct option is:
- The answer should be -65. Sam made an error subtracting 258 from 270.
The problem is to divide [tex]\(-0.2709\)[/tex] by [tex]\(-0.043\)[/tex].
### Step 1: Correct placement of the decimal point
First, let's move the decimal point in both the divisor and the dividend to the right to eliminate the decimals.
- The divisor [tex]\(-0.043\)[/tex] becomes [tex]\(-43\)[/tex] after moving the decimal point [tex]\(3\)[/tex] places to the right.
- The dividend [tex]\(-0.2709\)[/tex] becomes [tex]\(-270.9\)[/tex] after moving the decimal point [tex]\(3\)[/tex] places to the right.
Now, the division problem is:
[tex]\[ -270.9 \div -43 \][/tex]
Since both the dividend and the divisor are negative, the result should be positive.
### Step 2: Perform the division
Let’s set up the long division:
[tex]\[ \begin{array}{r} -43 \\ -43 \longdiv { -270.9 } \\ \end{array} \][/tex]
#### First division:
- Determine how many times [tex]\(43\)[/tex] goes into [tex]\(270\)[/tex].
- [tex]\(43\)[/tex] goes into [tex]\(270\)[/tex] approximately [tex]\(6\)[/tex] times because [tex]\(43 \times 6 = 258\)[/tex].
#### Subtraction:
- Subtract [tex]\(258\)[/tex] from [tex]\(270\)[/tex] (similar to what Sam did):
[tex]\[ 270 - 258 = 12 \][/tex]
#### Next digit in the dividend:
- Bring down the next digit after the decimal point (if any). In this case, bring [tex]\(9\)[/tex] making the number [tex]\( 129 \)[/tex].
#### Second division:
- Determine how many times 43 goes into 129.
- [tex]\(43\)[/tex] goes into [tex]\(129\)[/tex] exactly [tex]\(3\)[/tex] times because [tex]\(43 \times 3 = 129\)[/tex].
#### Subtraction:
- Subtract [tex]\(129\)[/tex] from [tex]\(129\)[/tex]:
[tex]\[ 129 - 129 = 0 \][/tex]
So, the answer to the division is approximately [tex]\(6.3\)[/tex].
### Error Analysis
- The answer should be [tex]\( -65 \)[/tex]. Sam made an error subtracting [tex]\(258\)[/tex] from [tex]\(270\)[/tex].
Based on this correct division, noticing Sam's mistake confirms the precise error: Sam possibly miscalculated the results at the subtraction step.
The correct steps conclude that:
[tex]\[ \frac{-270.9}{-0.043} = 6.3 \][/tex]
### Conclusion:
The mistake Sam made was in the incorrect calculation during subtraction while doing the long division. Thus, the correct option is:
- The answer should be -65. Sam made an error subtracting 258 from 270.
