Question

a student wants to find point c on the directed line segment from a to b on a number line such that the segment is partitioned in a ratio of 3:4. point a is at -6 and point b is at 22. the students work is shown.

1. ( c=left|\frac{3}{4}

ight|(22-(-6))+(-6) )

1. ( c=left|\frac{3}{4}

ight|(28)-6 )

1. ( c = 21 - 6 )
2. ( c = 15 )

analyze the students work. is the answer correct? explain
no, the student should have added ( 3 + 4 ) to get the total number of sections, and used the fraction ( \frac{3}{7} ) instead of ( \frac{3}{4} )
no, the student should have subtracted 2 from -6 to find the distance
no, the student should have added 2 at the end to add to the starting point
yes, the students answer is correct

Explanation:

Step1: Recall the section formula for a line - segment on a number line

The formula for finding a point \(C\) that divides the line - segment from \(A\) to \(B\) in the ratio \(m:n\) is \(C=\frac{m}{m + n}(B - A)+A\). Here, \(m = 3\) and \(n=4\), so \(m + n=3 + 4=7\).

Step2: Calculate the distance between \(A\) and \(B\)

Given \(A=-6\) and \(B = 2\), then \(B - A=2-(-6)=2 + 6=8\).

Step3: Apply the correct formula

Using the formula \(C=\frac{3}{3 + 4}(2-(-6))+(-6)=\frac{3}{7}\times8-6=\frac{24}{7}-\frac{42}{7}=\frac{24 - 42}{7}=\frac{-18}{7}\approx - 2.57\). The student used \(\frac{3}{4}\) instead of \(\frac{3}{7}\) in the formula \(C=\frac{m}{m + n}(B - A)+A\). The formula for the partition of a line - segment in the ratio \(m:n\) (where \(m:n\) is the ratio of the sub - segments \(AC:CB\)) requires finding the fraction of the total length of the segment \(AB\) that \(AC\) represents. The total number of parts is \(m + n\).

Answer:

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