Question

find the coordinate of p that represents the weighted average of points f and h when point f has a weight of \\(\frac{3}{4}\\) and point h has a weight of \\(\frac{1}{4}\\).
number line with f at -2 and h at 5
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Explanation:

Step1: Identify coordinates of F and H

From the number line, point \( F \) is at \( -2 \) and point \( H \) is at \( 5 \). The weights are \( w_F = \frac{3}{4} \) and \( w_H = \frac{1}{4} \).

Step2: Apply weighted average formula

The formula for the weighted average (coordinate of \( P \)) is \( P = \frac{w_F \cdot F + w_H \cdot H}{w_F + w_H} \). Since \( w_F + w_H = \frac{3}{4} + \frac{1}{4} = 1 \), it simplifies to \( P = w_F \cdot F + w_H \cdot H \).

Substitute the values: \( P = \frac{3}{4} \cdot (-2) + \frac{1}{4} \cdot 5 \).

Step3: Calculate each term

First term: \( \frac{3}{4} \cdot (-2) = \frac{-6}{4} = -\frac{3}{2} \).

Second term: \( \frac{1}{4} \cdot 5 = \frac{5}{4} \).

Step4: Add the terms

\( P = -\frac{3}{2} + \frac{5}{4} \). Convert \( -\frac{3}{2} \) to fourths: \( -\frac{6}{4} + \frac{5}{4} = \frac{-6 + 5}{4} = \frac{-1}{4} \)? Wait, no, wait—wait, did I mix up? Wait, no, let's recheck. Wait, \( F \) is at -2, \( H \) at 5. Wait, maybe I made a mistake. Wait, weighted average formula is \( \frac{w_1x_1 + w_2x_2}{w_1 + w_2} \). Since \( w_1 + w_2 = 1 \), it's \( w_1x_1 + w_2x_2 \). So \( \frac{3}{4}(-2) + \frac{1}{4}(5) = \frac{-6}{4} + \frac{5}{4} = \frac{-1}{4} \)? No, that can't be. Wait, no, wait the number line: F is at -2, H is at 5. Wait, maybe I misread the weights? Wait, the problem says F has weight 3/4, H has 1/4. Wait, but let's recalculate:

\( \frac{3}{4} \times (-2) = \frac{-6}{4} = -1.5 \)

\( \frac{1}{4} \times 5 = 1.25 \)

Adding them: \( -1.5 + 1.25 = -0.25 \)? Wait, that seems off. Wait, maybe I misread the coordinates. Wait, looking at the number line: F is at -2 (between -3 and -1, the first tick after -3 is -2, so F is at -2). H is at 5 (the tick at 5). Wait, maybe the formula is correct, but let's check again. Wait, weighted average is (w1x1 + w2x2)/(w1 + w2). Since w1 + w2 = 3/4 + 1/4 = 1, so it's (3/4(-2) + 1/45)/1 = 3/4(-2) + 1/45 = (-6/4) + (5/4) = (-1)/4 = -0.25? But that seems between F and H? Wait, F is at -2, H at 5. The weighted average with more weight on F should be closer to F. -0.25 is between -2 and 5, closer to F? Wait, -2 to 5 is 7 units. 3/4 of the way from F to H? Wait, no, weighted average: if weight on F is 3/4, then it's 3/4 from F towards H? Wait, no, weighted average formula: the coordinate is (w_F x_F + w_H x_H)/(w_F + w_H). So with w_F = 3/4, w_H = 1/4, x_F = -2, x_H = 5. So:

\( P = \frac{\frac{3}{4} \times (-2) + \frac{1}{4} \times 5}{\frac{3}{4} + \frac{1}{4}} = \frac{\frac{-6 + 5}{4}}{1} = \frac{-1}{4} = -0.25 \). Wait, but that's -1/4. But let's check again. Wait, maybe I misread the coordinates. Wait, the number line: F is at -2, H is at 5. Let's confirm the positions. The leftmost tick is -3, then -2 (F), -1, 0, 1, 2, 3, 4, 5 (H), 6, etc. So F: -2, H:5. Then weighted average: (3/4)(-2) + (1/4)(5) = (-6/4) + (5/4) = (-1)/4 = -0.25. So the coordinate of P is -0.25, which is -1/4. Wait, but let me check again. Maybe I made a mistake in the formula. The weighted average formula is correct: (w1x1 + w2x2)/(w1 + w2). Since the sum of weights is 1, it's just w1x1 + w2x2. So yes, that's correct.

Answer:

\( -\frac{1}{4} \) (or -0.25)