Question

examples 1 and 3
for exercises 1 - 8, refer to the number line.

1. find the coordinate of point n that is $\frac{1}{5}$ of the distance from k to f.
2. find the coordinate of point m such that the ratio of km to mf is 1:2.
3. find the coordinate of point w such that the ratio of kw to wf is 1:1.
4. find the coordinate of point v that is $\frac{2}{3}$ of the distance from k to f.
5. find the coordinate of point g such that the ratio of kg to gf is 4:1.
6. find the coordinate of point a that is $\frac{3}{5}$ of the distance from k to f.
7. find the coordinate of point y that is $\frac{4}{5}$ of the distance from k to f.
8. find the coordinate of point b such that the ratio of kb to bf is 2:3.

locating points on a number line

Explanation:

Step1: Identify the coordinates of K and F

The coordinate of point K is 3 and the coordinate of point F is 18.

Step2: General formula for finding a point on a number - line

If we want to find a point P that is $\frac{m}{n}$ of the distance from point A with coordinate $a$ to point B with coordinate $b$, the coordinate of P is given by $a+\frac{m}{n}(b - a)$. Or, if the ratio of AP to PB is $m:n$, the coordinate of P is $\frac{na+mb}{m + n}$.

1. For point N

$n = 5$, $m = 1$, $a = 3$, $b = 18$.
$3+\frac{1}{5}(18 - 3)=3 + 3=6$

2. For point M

The ratio of KM to MF is $1:2$, so $m = 1$, $n = 2$, $a = 3$, $b = 18$.
$\frac{2\times3+1\times18}{1 + 2}=\frac{6 + 18}{3}=8$

3. For point W

The ratio of KW to WF is $1:1$, so $m = 1$, $n = 1$, $a = 3$, $b = 18$.
$\frac{1\times3+1\times18}{1+1}=\frac{3 + 18}{2}=10.5$

4. For point V

$n = 3$, $m = 2$, $a = 3$, $b = 18$.
$3+\frac{2}{3}(18 - 3)=3+10 = 13$

5. For point G

The ratio of KG to GF is $4:1$, so $m = 4$, $n = 1$, $a = 3$, $b = 18$.
$\frac{1\times3+4\times18}{4 + 1}=\frac{3+72}{5}=15$

6. For point A

$n = 5$, $m = 3$, $a = 3$, $b = 18$.
$3+\frac{3}{5}(18 - 3)=3 + 9=12$

7. For point Y

$n = 5$, $m = 4$, $a = 3$, $b = 18$.
$3+\frac{4}{5}(18 - 3)=3+12 = 15$

8. For point B

The ratio of KB to BF is $2:3$, so $m = 2$, $n = 3$, $a = 3$, $b = 18$.
$\frac{3\times3+2\times18}{2 + 3}=\frac{9+36}{5}=9$

Answer:

1. 6
2. 8
3. 10.5
4. 13
5. 15
6. 12
7. 15
8. 9