Question

gennita is creating a histogram based on the table below.

salary range	number of people
$20,000 - $39,999	30
$40,000 - $59,999	25

which scale can she use for the vertical axis such that the difference in the heights of the bars is maximized?

• 0–50
• 0–40
• 10–50
• 25–40

Explanation:

Step1: Analyze the data

The number of people for each salary range is 40, 30, and 25. We need to find a vertical axis scale that maximizes the difference in bar heights.

Step2: Evaluate each option

- Option 0 - 50: The range is 50, and the values are within this range. The differences between 40, 30, 25 will be shown with a relatively large vertical space, but let's check others.
- Option 0 - 40: The maximum value (40) is at the top, so 30 and 25 will be lower, but the top is at 40, so the height difference might be less than if the scale starts lower? Wait, no. Wait, if the scale is 25 - 40, then the minimum is 25, so 40 - 25 = 15, 30 - 25 = 5, 25 - 25 = 0. Wait, no, let's think again.

Wait, the vertical axis represents the number of people. The values are 25, 30, 40. To maximize the difference in bar heights, we want the scale to start as low as possible (but above or at the minimum) and end at the maximum, or a scale that compresses the lower end? Wait, no. Let's list the options:

1. 0 - 50: The bars will be at 25, 30, 40. The heights are 25, 30, 40 units (if each unit is 1). The differences are 5 (30 - 25), 10 (40 - 30).

2. 0 - 40: The maximum value is 40, so the bar for 40 is at the top. The bar for 30 is 30/40 of the height, and 25 is 25/40. The differences in terms of height (proportional) would be less than if the scale starts higher? Wait, no, maybe I got it wrong. Wait, the vertical axis is a linear scale. So if the scale is from 25 to 40, then the minimum value (25) is at the bottom, so the bar for 25 is at the bottom (height 0), 30 is 5 units up, 40 is 15 units up. So the differences between 30 and 25 is 5, between 40 and 30 is 10. In the scale 0 - 50, the differences are 5 and 10 as well, but the actual height (in pixels or length) would be larger in the 25 - 40 scale because the total range is 15 (40 - 25) vs 50 (50 - 0) in 0 - 50. Wait, no, the scale's range affects the height per unit. For example, if the vertical axis length is L, then in 0 - 50, each unit is L/50. In 25 - 40, each unit is L/15. So the difference between 40 and 30 is 10 units in 0 - 50: 10*(L/50) = L/5. In 25 - 40, the difference between 40 and 30 is 10 units: 10*(L/15) = 2L/3. Wait, that can't be. Wait, no, the number of people is 25, 30, 40. So the values are 25, 30, 40. Let's compute the height for each scale:

- Scale 0 - 50: The height of each bar is (value / 50) * total_height. So 25: 0.5*H, 30: 0.6*H, 40: 0.8*H. Differences: 0.1H (30-25), 0.2H (40-30).

- Scale 0 - 40: 25: (25/40)*H = 0.625H, 30: 0.75H, 40: H. Differences: 0.125H, 0.25H.

- Scale 10 - 50: 25: (15/40)*H = 0.375H, 30: 0.5H, 40: 0.75H. Differences: 0.125H, 0.25H.

- Scale 25 - 40: 25: 0*H, 30: (5/15)*H = (1/3)H, 40: (15/15)*H = H. Differences: (1/3)H, (2/3)H. Wait, no, 40 - 25 = 15, so the range is 15. So 30 - 25 = 5, 40 - 30 = 10. So the height for 30 is 5/15 of the total height (H), and 40 is 15/15 of H. So the differences between 30 and 25 is 5/15 H = H/3, between 40 and 30 is 10/15 H = 2H/3. In the 0 - 50 scale, the differences were 0.1H and 0.2H (since 5/50=0.1, 10/50=0.2). In 25 - 40, the differences are larger (H/3 ≈ 0.333H, 2H/3 ≈ 0.666H) compared to 0 - 50 (0.1H and 0.2H). Wait, but the option is 25 - 40? Wait, the options are 0-50, 0-40, 10-50, 25-40.

Wait, let's check the values: the number of people is 25, 30, 40. So the minimum is 25, maximum is 40. So if we use a scale from 25 to 40, then the bar for 25 is at the bottom (height 0), 30 is 5 units up, 40 is 15 units up. So the differences between 30 and 25 is 5, between 40 and 30 is 10. In the scale 0 - 50, the bar for 25 is 25 units up, 30 is 30, 40 is 40. So the differences are 5 and 10, but the actual length (if the axis is, say, 100 pixels) in 25 - 40: 100 pixels for 15 units (40-25), so each unit is 100/15 ≈ 6.666 pixels. So 5 units is 33.33 pixels, 10 units is 66.66 pixels. In 0 - 50: 100 pixels for 50 units, each unit is 2 pixels. So 5 units is 10 pixels, 10 units is 20 pixels. So the differences are larger in the 25 - 40 scale. Therefore, the scale 25 - 40 will maximize the height differences because it starts at the minimum value (25) and ends at the maximum (40), so the bars are spread out over the entire axis, making the differences more pronounced.

Wait, but let's check the options. The options are:

A. 0 - 50

B. 0 - 40

C. 10 - 50

D. 25 - 40

So the correct option is D. 25 - 40.

Answer:

D. 25 - 40