Let's analyze the student's work step-by-step and discuss the errors made. The goal is to find point [tex]\( C \)[/tex] on the directed line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] on a number line, such that the segment is partitioned in a ratio of [tex]\( 3 : 4 \)[/tex].
To do this, follow these steps:
1. First, calculate the total number of sections in the ratio.
Since the ratio is [tex]\( 3 : 4 \)[/tex], the total number of sections is [tex]\( 3 + 4 = 7 \)[/tex].
2. Next, we need to find the fraction that corresponds to the partition point [tex]\( C \)[/tex].
Since we want [tex]\( C \)[/tex] to partition the segment into a ratio of [tex]\( 3: 4 \)[/tex], the fraction is given by [tex]\( \frac{3}{7} \)[/tex].
3. Calculate the difference between the points [tex]\( A \)[/tex] and [tex]\( B \)[/tex]:
[tex]\[
B - A = 2 - (-6) = 2 + 6 = 8
\][/tex]
4. Multiply this difference by the fraction [tex]\( \frac{3}{7} \)[/tex]:
[tex]\[
\frac{3}{7} \times 8 = \frac{24}{7}
\][/tex]
5. Now, add this product to the starting point [tex]\( A \)[/tex] to find point [tex]\( C \)[/tex]:
[tex]\[
C = -6 + \frac{24}{7} = -6 + 3.4285714285714284 \approx -2.5714285714285716
\][/tex]
Hence, the correct point [tex]\( C \)[/tex] on the number line is approximately [tex]\( -2.5714285714285716 \)[/tex].
Now, examining the student's work:
1. [tex]\( c = \left(\frac{3}{4}\right)(2-(-6))+(-6) \)[/tex]
- The student used the fraction [tex]\( \frac{3}{4} \)[/tex] instead of [tex]\( \frac{3}{7} \)[/tex], which is incorrect because the total number of sections is [tex]\( 7 \)[/tex], not [tex]\( 4 \)[/tex].
2. [tex]\( c = \left(\frac{3}{4}\right) = 0 - 6 \)[/tex]
- This step seems to misuse the fraction and incorrectly simplifies the expression, indicating a lack of clarity in the subtraction process.
3. [tex]\( C = 6 - 6 \)[/tex]
- This transformation just appears incorrect and doesn't follow logically from previous steps.
4. [tex]\( c = 0 \)[/tex]
- This leads to an incorrect result.
To conclude, the student's solution is incorrect because they did not correctly account for the total number of sections in the ratio and thus used the wrong fraction. The correct fraction to use is [tex]\( \frac{3}{7} \)[/tex], and using this correct fraction and proper calculations leads us to the correct point [tex]\( C \approx -2.5714285714285716 \)[/tex].
