Question

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 7. find the coordinate of point g that is 2/3 of the distance from b to d. 8. find the coordinate of point h that is 1/5 of the distance from c to f. 9. find the coordinate of point j that is 1/6 of the distance from a to e. 10. find the coordinate of point k that is 4/5 of the distance from a to f. 11. find the coordinate of point x such that the ratio of ax to xf is 1:3. 12. find the coordinate of point x such that the ratio of bx to xf is 3:2. 13. find the coordinate of point x such that the ratio of cx to xe is 1:1. 14. find the coordinate of point x such that the ratio of fx to xd is 5:3.

Explanation:

Step1: Recall the formula for finding a point on a number - line

If we have two points $a$ and $b$ on a number - line, and we want to find a point $x$ that is $k$ of the distance from $a$ to $b$, the formula is $x=a + k(b - a)$. If the ratio of $AX$ to $XB$ is $m:n$, then the formula for the coordinate of $X$ is $X=\frac{na+mb}{m + n}$.

Step2: Solve problem 7

The coordinate of $B=-5$ and the coordinate of $D = 1$. We want to find the coordinate of point $G$ such that $k=\frac{2}{3}$. Using the formula $G=-5+\frac{2}{3}(1-(-5))$. First, calculate $1-(-5)=6$. Then $\frac{2}{3}\times6 = 4$. So $G=-5 + 4=-1$.

Step3: Solve problem 8

The coordinate of $C=-4$ and the coordinate of $F = 5$. We want to find the coordinate of point $H$ such that $k=\frac{1}{5}$. Using the formula $H=-4+\frac{1}{5}(5-(-4))$. First, calculate $5-(-4)=9$. Then $\frac{1}{5}\times9=\frac{9}{5}=1.8$. So $H=-4 + 1.8=-2.2$.

Step4: Solve problem 9

The coordinate of $A=-7$ and the coordinate of $E = 2$. We want to find the coordinate of point $J$ such that $k=\frac{1}{6}$. Using the formula $J=-7+\frac{1}{6}(2-(-7))$. First, calculate $2-(-7)=9$. Then $\frac{1}{6}\times9=\frac{3}{2}=1.5$. So $J=-7 + 1.5=-5.5$.

Step5: Solve problem 10

The coordinate of $A=-7$ and the coordinate of $F = 5$. We want to find the coordinate of point $K$ such that $k=\frac{4}{5}$. Using the formula $K=-7+\frac{4}{5}(5-(-7))$. First, calculate $5-(-7)=12$. Then $\frac{4}{5}\times12=\frac{48}{5}=9.6$. So $K=-7 + 9.6 = 2.6$.

Step6: Solve problem 11

The coordinate of $A=-7$ and the coordinate of $F = 5$, and the ratio of $AX$ to $XF$ is $1:3$. Using the formula $X=\frac{3\times(-7)+1\times5}{1 + 3}=\frac{-21 + 5}{4}=\frac{-16}{4}=-4$.

Step7: Solve problem 12

The coordinate of $B=-5$ and the coordinate of $F = 5$, and the ratio of $BX$ to $XF$ is $3:2$. Using the formula $X=\frac{2\times(-5)+3\times5}{3 + 2}=\frac{-10 + 15}{5}=1$.

Step8: Solve problem 13

The coordinate of $C=-4$ and the coordinate of $E = 2$, and the ratio of $CX$ to $XE$ is $1:1$. Using the formula $X=\frac{1\times(-4)+1\times2}{1 + 1}=\frac{-4 + 2}{2}=-1$.

Step9: Solve problem 14

The coordinate of $F = 5$ and the coordinate of $D = 1$, and the ratio of $FX$ to $XD$ is $5:3$. Using the formula $X=\frac{3\times5+5\times1}{5 + 3}=\frac{15 + 5}{8}=\frac{20}{8}=2.5$.

Answer:

1. -1
2. -2.2
3. -5.5
4. 2.6
5. -4
6. 1
7. -1
8. 2.5