Question

type the correct answer in the box. use numerals instead of words. if necessary, use / for the fraction bar

on a number line, point d is at -3, and point e is at 6. the point f lies on \\(\overline{de}\\). the ratio of \\(df\\) to \\(fe\\) is 2 : 3. where does point f lie on the number line?

point f is at \\(\boxed{}\\) on the number line

Explanation:

Step1: Find the length of DE

The distance between D (-3) and E (6) is \( 6 - (-3) = 9 \).

Step2: Determine the length of DF

The ratio of DF to FE is 2:3, so the total parts are \( 2 + 3 = 5 \). Each part is \( \frac{9}{5} = 1.8 \). Then DF is \( 2\times1.8 = 3.6 \).

Step3: Find the coordinate of F

Starting from D (-3), moving 3.6 units towards E: \( -3 + 3.6 = 0.6 \) (or \( \frac{3}{5} \) in fraction, but decimal is 0.6). Alternatively, using the section formula: If a point divides the line segment joining \( (x_1,y_1) \) and \( (x_2,y_2) \) in the ratio \( m:n \), the coordinate is \( \frac{mx_2 + nx_1}{m + n} \). Here, \( x_1 = -3 \), \( x_2 = 6 \), \( m = 2 \), \( n = 3 \). So \( F = \frac{2\times6 + 3\times(-3)}{2 + 3} = \frac{12 - 9}{5} = \frac{3}{5} = 0.6 \).

Answer:

0.6