Question

find the point on the number line that is \\(\frac{5}{9}\\) of the distance from b to a. provide the answer as an integer or fraction.

Explanation:

Step1: Find distance from A to B

Point A is at -4, point B is at 6. Distance is \( 6 - (-4) = 10 \).

Step2: Calculate \( \frac{5}{9} \) of the distance

\( \frac{5}{9} \times 10 = \frac{50}{9} \approx 5.555... \)

Step3: Find the point from B

Start at B (6) and move towards A (since it's from B to A? Wait, no: "from B to A"? Wait, the problem says "from B to A"? Wait, the wording: "the point on the number line that is \( \frac{5}{9} \) of the distance from B to A". Wait, A is at -4, B at 6. So direction: from B (6) towards A (-4). The vector from B to A is \( -4 - 6 = -10 \). So \( \frac{5}{9} \) of that vector: \( \frac{5}{9} \times (-10) = -\frac{50}{9} \). Then add to B's coordinate: \( 6 + (-\frac{50}{9}) = \frac{54}{9} - \frac{50}{9} = \frac{4}{9} \)? Wait, no, maybe I messed up direction. Wait, distance from B to A is 10, as calculated. So moving \( \frac{5}{9} \) of the distance from B towards A: the coordinate is \( B + \frac{5}{9} \times (A - B) \). \( A - B = -4 - 6 = -10 \). So \( 6 + \frac{5}{9} \times (-10) = 6 - \frac{50}{9} = \frac{54 - 50}{9} = \frac{4}{9} \)? Wait, that can't be. Wait, maybe I got A and B reversed. Wait, the number line: A is at -4, B at 6. So distance from B to A is \( |-4 - 6| = 10 \). So \( \frac{5}{9} \) of 10 is \( \frac{50}{9} \approx 5.555 \). So starting at B (6) and moving towards A (left) by \( \frac{50}{9} \): \( 6 - \frac{50}{9} = \frac{54 - 50}{9} = \frac{4}{9} \). Wait, but let's check again. Alternatively, maybe the problem is "from B to A" or "from A to B"? Wait, the original problem: "the point on the number line that is \( \frac{5}{9} \) of the distance from B to A". So formula: \( x = B + \frac{5}{9}(A - B) \). \( A = -4 \), \( B = 6 \). So \( x = 6 + \frac{5}{9}(-4 - 6) = 6 + \frac{5}{9}(-10) = 6 - \frac{50}{9} = \frac{54 - 50}{9} = \frac{4}{9} \). Wait, but that seems odd. Wait, maybe I made a mistake in the direction. Let's recast: the distance between A (-4) and B (6) is 10 units. We need a point P such that the distance from B to P is \( \frac{5}{9} \) of the distance from B to A. So \( BP = \frac{5}{9} \times BA \). Since \( BA = 10 \), \( BP = \frac{50}{9} \). Since P is between B and A (assuming), then the coordinate of P is \( B - BP \) (since moving left from B towards A). So \( 6 - \frac{50}{9} = \frac{54 - 50}{9} = \frac{4}{9} \). Wait, but let's check with another approach. The coordinate can also be calculated as \( A + (1 - \frac{5}{9}) \times BA \), but no. Wait, maybe the problem is "from B to A" meaning starting at B and going towards A, so the fraction is \( \frac{5}{9} \) of the total distance. So the position is \( B + \frac{5}{9}(A - B) \). Which is what I did. So \( 6 + \frac{5}{9}(-10) = \frac{4}{9} \approx 0.444 \). But let's verify the distance from B to P: \( |\frac{4}{9} - 6| = |\frac{4 - 54}{9}| = \frac{50}{9} \), and distance from B to A is 10, so \( \frac{50}{9} \) is \( \frac{5}{9} \) of 10 (since \( 10 = \frac{90}{9} \), \( \frac{50}{9} \) is \( \frac{5}{9} \) of that). Yes, that works. So the point is \( \frac{4}{9} \)? Wait, but maybe I misread the problem: is it from B to A or A to B? The problem says "from B to A". So the answer is \( \frac{4}{9} \)? Wait, no, wait the number line: A is at -4, B at 6. Let's plot: B is at 6, A at -4. The distance between them is 10. So \( \frac{5}{9} \) of 10 is about 5.555. So moving 5.555 units from B towards A (left) from 6: 6 - 5.555 ≈ 0.444, which is \( \frac{4}{9} \approx 0.444 \). So that's correct.

Answer:

\(\frac{4}{9}\)