Question

refer to the number line. find the coordinate of point x such that the ratio of fx to xd is 5:3.
(number line with points a at -7, b at -5, c at -4, d at 1, e at 2, f at 5)

Explanation:

Step1: Identify coordinates of F and D

From the number line, \( F \) is at \( 5 \) and \( D \) is at \( 1 \). Let the coordinate of \( X \) be \( x \). The ratio \( FX:XD = 5:3 \). Using the section formula for internal division: if a point \( X \) divides the line segment joining \( F(x_1) \) and \( D(x_2) \) in the ratio \( m:n \), then \( x=\frac{mx_2 + nx_1}{m + n} \). Here, \( m = 5 \), \( n = 3 \), \( x_1 = 5 \), \( x_2 = 1 \).

Step2: Apply section formula

Substitute values into the formula: \( x=\frac{5\times1 + 3\times5}{5 + 3}=\frac{5 + 15}{8}=\frac{20}{8}=2.5 \)? Wait, no, wait. Wait, maybe I mixed up \( F \) and \( D \). Wait, \( F \) is at 5, \( D \) is at 1. Wait, the ratio is \( FX:XD = 5:3 \), so \( F \) to \( X \) to \( D \), so \( X \) is between \( F \) and \( D \)? Wait, no, \( F \) is at 5, \( D \) is at 1, so \( F \) is to the right of \( D \). So the distance from \( F \) to \( X \) over \( X \) to \( D \) is 5:3. So let's let \( FX = 5k \), \( XD = 3k \), so total \( FD = 5k + 3k = 8k \). The coordinate of \( F \) is 5, \( D \) is 1, so \( FD = 5 - 1 = 4 \)? Wait, no, distance is \( |5 - 1| = 4 \). Wait, that can't be. Wait, maybe I misread the number line. Wait, looking at the number line: A is at -7, B at -5, C at -4, D at 1, E at 2, F at 5? Wait, no, the number line has A at -7, then the next mark is -6, then B at -5, then -4, C at -4? Wait, no, the number line: the marks are at -7 (A), then -6, -5 (B), -4 (C), -3, -2, -1, 0, 1 (D), 2 (E), 3, 4, 5 (F), 6, 7. So \( F \) is at 5, \( D \) is at 1. So the distance between \( F \) (5) and \( D \) (1) is \( 5 - 1 = 4 \). But the ratio is 5:3, so total parts 8. But 4 divided by 8 is 0.5, so each part is 0.5? Wait, no, maybe I got the ratio reversed. Wait, maybe \( XD:FX = 5:3 \)? No, the problem says \( FX:XD = 5:3 \). Wait, maybe the points are in the order \( D \), \( X \), \( F \)? Wait, \( D \) is at 1, \( F \) is at 5, so from \( F \) to \( X \) to \( D \), so \( X \) is between \( F \) and \( D \). So the coordinate of \( X \) should be \( x = \frac{3\times5 + 5\times1}{5 + 3} \)? Wait, no, the section formula: if dividing \( F(x_1) \) and \( D(x_2) \) in ratio \( m:n \) ( \( FX:XD = m:n \) ), then \( x = \frac{n x_1 + m x_2}{m + n} \). Wait, let's recall: the section formula is \( x = \frac{m x_2 + n x_1}{m + n} \) where \( m:n \) is the ratio of \( FX:XD \), so \( F \) is \( x_1 \), \( D \) is \( x_2 \), \( m = 5 \), \( n = 3 \). So \( x_1 = 5 \), \( x_2 = 1 \), \( m = 5 \), \( n = 3 \). Then \( x = \frac{5\times1 + 3\times5}{5 + 3} = \frac{5 + 15}{8} = \frac{20}{8} = 2.5 \)? But E is at 2, so 2.5 is between 2 and 3? Wait, but maybe I misread the number line. Wait, D is at 1, E at 2, F at 5? Wait, no, the number line: after 1 (D), next is 2 (E), then 3, 4, 5 (F). So D is 1, E is 2, F is 5. So the distance from D (1) to F (5) is 4. The ratio FX:XD = 5:3, so total parts 8. So each part is 4 / 8 = 0.5. So FX = 5 * 0.5 = 2.5, XD = 3 * 0.5 = 1.5. So starting from F (5), moving left 2.5 units: 5 - 2.5 = 2.5. Wait, but 2.5 is between E (2) and 3. But E is at 2, so 2.5 is correct? Wait, but maybe I made a mistake. Wait, let's check again. Alternatively, use the formula for internal division: if a point divides the segment from \( (x_1) \) to \( (x_2) \) in ratio \( m:n \), then \( x = \frac{m x_2 + n x_1}{m + n} \). Here, \( FX:XD = 5:3 \), so \( m = 5 \), \( n = 3 \), \( x_1 = 5 \) (F), \( x_2 = 1 \) (D). So \( x = \frac{5*1 + 3*5}{5 + 3} = \frac{5 + 15}{8} = \frac{20}{8} = 2.5 \). So the coordinate of X is 2.5, which is 2.5 or \( \frac{5}{2} \). Wait, but let's check the number line again. D is at 1, E at 2, so 2.5 is halfway between 2 and 3? Wait, maybe the number line has D at 1, E at 2, F at 5. So yes, X is at 2.5.

Answer:

\( 2.5 \) (or \( \frac{5}{2} \))