Question

practice
examples 1 and 3
refer to the number - line.
m
2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19

1. find the coordinate of point b that is 1/4 of the distance from m to j.
2. find the coordinate of point c that is 7/8 of the distance from m to j.
3. find the coordinate of point d that is 7/16 of the distance from m to j.
4. find the coordinate of point x such that the ratio of mx to xj is 3:1.
5. find the coordinate of point x such that the ratio of mx to xj is 2:3.
6. find the coordinate of point x such that the ratio of mx to xj is 1:1.

refer to the number - line.
a b c d e f
-7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7

1. find the coordinate of point g that is 2/3 of the distance from b to d.
2. find the coordinate of point h that is 1/5 of the distance from c to f.
3. find the coordinate of point j that is 1/6 of the distance from a to e.
4. find the coordinate of point k that is 4/5 of the distance from a to f.
5. find the coordinate of point x such that the ratio of ax to xf is 1:3.
6. find the coordinate of point x such that the ratio of bx to xf is 3:2.
7. find the coordinate of point x such that the ratio of cx to xe is 1:1.
8. find the coordinate of point x such that the ratio of fx to xd is 5:3.

Explanation:

1. For the first - type of problem (e.g., find the coordinate of point \(B\) that is \(\frac{1}{4}\) of the distance from \(M\) to \(J\)):

• Let the coordinate of \(M\) be \(x_1\) and the coordinate of \(J\) be \(x_2\). The formula to find the coordinate of a point \(P\) that divides the line - segment from \(x_1\) to \(x_2\) in the ratio \(r\) is \(P=x_1 + r(x_2 - x_1)\).
• Assume \(M = 2\) and \(J = 19\).
• Step1: Calculate the distance between \(M\) and \(J\)
• The distance \(d=x_2 - x_1=19 - 2 = 17\).
• Step2: Find the position of point \(B\)
• Since \(r=\frac{1}{4}\), the coordinate of \(B\) is \(B = 2+\frac{1}{4}(19 - 2)=2+\frac{17}{4}=\frac{8 + 17}{4}=\frac{25}{4}=6.25\).

1. For the problem of ratio \(a:b\) (e.g., find the coordinate of point \(X\) such that the ratio of \(MX\) to \(XJ\) is \(3:1\)):

• The formula to find the coordinate of a point \(X\) that divides the line - segment from \(x_1\) to \(x_2\) in the ratio \(a:b\) is \(X=\frac{bx_1+ax_2}{a + b}\).
• Assume \(x_1 = 2\) (coordinate of \(M\)) and \(x_2 = 19\) (coordinate of \(J\)), \(a = 3\), \(b = 1\).
• Step1: Apply the formula
• \(X=\frac{1\times2+3\times19}{3 + 1}=\frac{2+57}{4}=\frac{59}{4}=14.75\).

We will solve the first problem (finding the coordinate of point \(B\) that is \(\frac{1}{4}\) of the distance from \(M\) to \(J\)) in a more general step - by - step format for all similar problems:

Step1: Identify the endpoints

Let the coordinate of the starting - point \(M\) be \(x_1\) and the coordinate of the ending - point \(J\) be \(x_2\). For the first number - line, if \(M = 2\) and \(J = 19\).

Step2: Calculate the distance between the endpoints

The distance \(d=x_2 - x_1\). Here, \(d=19 - 2=17\).

Step3: Find the position of the dividing point

The coordinate of the point \(B\) that is \(r=\frac{1}{4}\) of the distance from \(M\) to \(J\) is given by \(B=x_1+r(x_2 - x_1)\). Substituting the values, we get \(B = 2+\frac{1}{4}\times17=2 + 4.25=6.25\).

Answer:

6.25

(Note: You can apply the same steps and formulas to solve the other problems in the list. For example, for the problem of finding the coordinate of a point that divides a line - segment in a ratio \(a:b\), use the formula \(X=\frac{bx_1+ax_2}{a + b}\).)