Question

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1 multiple choice 1 point
suppose that it takes 2 people 8 hours to mow a 15 - acre plot of grass. which of the following is not a proportional relationship?
the relationship between the number of people and the number of acres they mow in 8 hours.
the relationship between the number of people and the number of hours it takes to mow a 15 - acre plot.
the relationship between number of hours and the number of acres mown when 2 people work together.

Explanation:

Step1: Analyze proportional relationships

A proportional relationship has a constant ratio (or unit rate) and can be expressed as \( y = kx \) (direct proportion) or \( xy = k \) (inverse proportion).

Step2: Evaluate each option

- **Option 1 (people vs. acres in 8h):** If more people work, more acres are mowed in 8h (direct proportion, \( \text{acres} = (\frac{15}{2}) \times \text{people} \) in 8h).
- **Option 2 (people vs. hours for 15 acres):** More people mean fewer hours (inverse proportion, \( \text{people} \times \text{hours} = 2 \times 8 = 16 \) for 15 acres).
- **Option 3 (hours vs. acres with 2 people):** 2 people mow \( \frac{15}{8} \) acres per hour. But "number of hours" and "acres mown" here—wait, no: with 2 people, acres mown in \( t \) hours is \( \frac{15}{8}t \) (direct proportion). Wait, no—wait, the question is which is **NOT** proportional. Wait, re-examine:

Wait, the third option: "number of hours and the number of acres mown when 2 people work together." Wait, no—wait, the first option: people and acres in 8h (direct, as acres per person is constant). Second: people and hours (inverse, product constant). Third: hours and acres with 2 people—acres = (15/8) * hours (direct, constant rate). Wait, no—wait, maybe I misread. Wait, the problem: 2 people take 8h to mow 15 acres.

Wait, let's re-express:

- Let \( P \) = people, \( A \) = acres, \( H \) = hours.

1. \( A \) vs. \( P \) in 8h: In 8h, 2 people mow 15 acres, so rate per person per 8h is \( \frac{15}{2} \) acres. So \( A = \frac{15}{2}P \) (proportional, direct).

2. \( H \) vs. \( P \) for 15 acres: Total work = \( P \times H = 2 \times 8 = 16 \) (constant), so \( H = \frac{16}{P} \) (proportional, inverse).

3. \( A \) vs. \( H \) with \( P = 2 \): Rate is \( \frac{15}{8} \) acres per hour, so \( A = \frac{15}{8}H \) (proportional, direct). Wait, but the options—wait, maybe the third option is misphrased? Wait, no—wait, the question is which is **NOT** proportional. Wait, maybe I made a mistake. Wait, no—wait, the second option: "number of people and the number of hours it takes to mow a 15 - acre plot"—is this proportional? Let's check: If \( P_1 = 2 \), \( H_1 = 8 \); if \( P_2 = 4 \), \( H_2 = 4 \) (since 4*4=16=2*8). So \( P \times H = 16 \), which is inverse proportion (proportional in the sense of inverse variation).

Wait, maybe the third option is not proportional? No, \( A = \frac{15}{8}H \) is direct proportion. Wait, maybe the first option: "number of people and the number of acres they mow in 8 hours"—if you have 1 person, in 8h, they mow 7.5 acres; 2 people, 15; 3 people, 22.5. So \( A = 7.5P \), which is proportional.

Wait, maybe the second option is not proportional? No, inverse proportion is still a type of proportional relationship (inverse variation). Wait, the question is which is **NOT** proportional. Wait, maybe I misread the options. Let me check again:

Options:

1. Relationship between number of people and acres mowed in 8h. (Direct proportion: more people → more acres in fixed time.)

2. Relationship between number of people and hours to mow 15 acres. (Inverse proportion: more people → fewer hours.)

3. Relationship between number of hours and acres mown when 2 people work together. (Direct proportion: more hours → more acres, since 2 people have a constant rate.)

Wait, but the problem says "which of the following is NOT a proportional relationship". Wait, maybe the second option is not proportional? No, inverse proportion is proportional. Wait, maybe the third option? No, \( A = rt \) (rate * time) is proportional. Wait, maybe the first option? No. Wait, perhaps the error is in my analysis. Let's think differently: Proportional relationships have a constant ratio (for direct) or constant product (for inverse).

Wait, the second option: people (P) and hours (H) for 15 acres. The product \( P \times H = 2 \times 8 = 16 \) (constant), so it's an inverse proportion (proportional).

The third option: hours (H) and acres (A) with 2 people. The ratio \( \frac{A}{H} = \frac{15}{8} \) (constant), so direct proportion (proportional).

The first option: people (P) and acres (A) in 8h. The ratio \( \frac{A}{P} = \frac{15}{2} \) (constant), so direct proportion (proportional).

Wait, this can't be. There must be a mistake. Wait, maybe the third option is "number of hours and the number of acres mown"—but when 2 people work, the rate is 15 acres / 8 hours = 1.875 acres per hour. So acres = 1.875 * hours, which is proportional.

Wait, maybe the second option is not proportional? No, inverse proportion is a type of proportional relationship. Wait, the question must have the second option as NOT proportional? No, that doesn't make sense. Wait, maybe I misread the options. Let me check again:

Wait, the second option: "the relationship between the number of people and the number of hours it takes to mow a 15 - acre plot". Let's take P=2, H=8; P=1, H=16; P=4, H=4. So H = 16 / P. This is an inverse variation, which is a proportional relationship (since it can be written as H = k / P, where k=16).

The third option: "the relationship between number of hours and the number of acres mown when 2 people work together". Acres = (15/8) * hours. So A = k * H, k=15/8, direct proportion.

The first option: "the relationship between the number of people and the number of acres they mow in 8 hours". Acres = (15/2) * people. So A = k * P, k=15/2, direct proportion.

Wait, this suggests all are proportional, which can't be. There must be a mistake in my analysis. Wait, maybe the third option is "number of hours and the number of acres mown"—but maybe the question is about "number of hours" and "acres mown" but with 2 people, but maybe the "number of hours" and "acres mown"—no, that's direct. Wait, maybe the second option is not proportional? No, inverse is proportional. Wait, perhaps the answer is the second option? No, that's inverse proportion. Wait, maybe the question is misworded. Alternatively, maybe the third option is not proportional. Wait, no.

Wait, let's recall: A proportional relationship is one where \( y = kx \) (direct) or \( xy = k \) (inverse). So both direct and inverse are proportional. So which of the three is not of this form?

Wait, maybe the first option: "number of people and acres mown in 8h"—if you have 0 people, 0 acres, which fits \( A = kP \).

Second option: "people and hours for 15 acres"—0 people would take infinite hours, which doesn't fit \( H = k/P \) (since division by zero is undefined), but in real - world terms, we consider positive values. Mathematically, inverse variation is still a proportional relationship (just not direct).

Third option: "hours and acres with 2 people"—0 hours, 0 acres, fits \( A = kH \).

Wait, maybe the answer is the second option. Because when we talk about proportional relationships in the context of direct proportion (which is often the default in basic problems), inverse proportion might be considered non - proportional. But strictly speaking, inverse proportion is a type of proportional relationship. However, in many basic math contexts, proportional relationship is taken to mean direct proportion.

If we consider direct proportion ( \( y = kx \), constant ratio), then:

- First option: direct proportion ( \( A = \frac{15}{2}P \) ).
- Second option: inverse proportion ( \( H=\frac{16}{P} \) ), not direct.
- Third option: direct proportion ( \( A = \frac{15}{8}H \) ).

Answer:

The relationship between the number of people and the number of hours it takes to mow a 15 - acre plot.