A student wants 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]. Point [tex]\( A \)[/tex] is at [tex]\(-6\)[/tex] and point [tex]\( B \)[/tex] is at [tex]\(2\)[/tex]. The student's work is shown:

1. [tex]\( c = \left( \frac{3}{4} \right)(2 - (-6)) + (-6) \)[/tex]
2. [tex]\( c = \left( \frac{3}{4} \right)(8) - 6 \)[/tex]
3. [tex]\( c = 6 - 6 \)[/tex]
4. [tex]\( c = 0 \)[/tex]

Analyze the student's work. Is the answer correct? Explain.

A. No, the student should have added [tex]\( 3 + 4 \)[/tex] to get the total number of sections and used the fraction [tex]\( \frac{3}{7} \)[/tex] instead of [tex]\( \frac{3}{4} \)[/tex].

B. No, the student should have subtracted [tex]\( 2 \)[/tex] from [tex]\(-6\)[/tex] to find the distance.

C. No, the student should have added [tex]\( 2 \)[/tex] at the end to add to the starting point.

D. Yes, the student's answer is correct.



Answer :

Let’s analyze the student’s work and correct it step by step.

The task is to find point [tex]\( C \)[/tex] on the directed line segment from [tex]\( A \)[/tex] to [tex]\( B \)[/tex] such that the segment is partitioned in the ratio [tex]\( 3:4 \)[/tex]. Given points are [tex]\( A = -6 \)[/tex] and [tex]\( B = 2 \)[/tex].

First, let's determine the correct way to approach this:

1. Determine the Total Number of Sections:
The segment is divided in the ratio [tex]\( 3:4 \)[/tex]:
[tex]\[ \text{Total sections} = 3 + 4 = 7 \][/tex]

2. Calculate the Length of Each Section:
The distance between [tex]\( A \)[/tex] and [tex]\( B \)[/tex] is:
[tex]\[ B - A = 2 - (-6) = 8 \][/tex]

3. Find the Fraction of the Distance for Point [tex]\( C \)[/tex]:
Point [tex]\( C \)[/tex] needs to be located at a position that divides the segment in the ratio [tex]\( 3:4 \)[/tex]. Therefore, we use the fraction [tex]\( \frac{3}{7} \)[/tex]:
[tex]\[ C = A + \left( \frac{3}{7} \right) \times (B - A) \][/tex]

4. Substitute the Values:
Substitute [tex]\( A = -6 \)[/tex], [tex]\( B = 2 \)[/tex], and the fraction [tex]\( \frac{3}{7} \)[/tex]:
[tex]\[ C = -6 + \left( \frac{3}{7} \right) \times (2 - (-6)) \][/tex]
[tex]\[ C = -6 + \left( \frac{3}{7} \right) \times 8 \][/tex]
[tex]\[ C = -6 + \left( \frac{3}{7} \right) \times 8 \][/tex]
[tex]\[ C = -6 + \frac{24}{7} \][/tex]
[tex]\[ C = -6 + 3.4285714285714286 \][/tex]
[tex]\[ C \approx -2.5714285714285716 \][/tex]

According to the correct method, the coordinate of point [tex]\( C \)[/tex] should be approximately [tex]\(-2.5714285714285716\)[/tex].

Now let’s analyze the student’s work:

1. [tex]\[ c = \left( \frac{3}{4} \right)(2 -(-6)) + (-6) \][/tex]
The student incorrectly used the fraction [tex]\( \frac{3}{4} \)[/tex] instead of [tex]\( \frac{3}{7} \)[/tex].

2. [tex]\[ c = \left( \frac{3}{4} \right)(8) - 6 \][/tex]
Correct intermediate step, given their wrong fraction.

3. [tex]\[ c = 6 - 6 \][/tex]
Calculated the term in parenthesis but still based on a wrong fraction.

4. [tex]\[ c = 0 \][/tex]
The final answer is incorrect.

Thus, the student's answer is not correct. They should have added [tex]\(3+4\)[/tex] to get the total number of sections and used the fraction [tex]\(\frac{3}{7}\)[/tex] instead of [tex]\(\frac{3}{4}\)[/tex]. The correct coordinate for point [tex]\( C \)[/tex] is approximately [tex]\(-2.5714285714285716\)[/tex].