When you see that one quantity is "proportional" to another, this means they have a direct relationship where one equals a constant times the other. Here, profit equals some constant times the number of lawns: $$P = k \cdot n$$, where $$k$$ is the constant rate.

To find this constant rate, use the given information: when $$n = 24$$ lawns, $$P = \960$$. Substituting: $$960 = k \cdot 24$$, so $$k = 960 ÷ 24 = 40$$ dollars per lawn. This means the company earns $$\40$$ profit for each lawn they service.

Now you can set up the equation for their goal: if they want $$\1400$$ profit, then $$1400 = 40n$$, where $$n$$ is the unknown number of lawns needed.

Choice A is wrong because proportional relationships are multiplicative ($$P = k \cdot n$$), not additive ($$P = n + \text{constant}$$). The "+40" suggests adding a fixed amount rather than multiplying by a rate.

Choice B incorrectly uses 24 as the multiplier, but 24 was the number of lawns in the given example, not the profit rate. This confuses the input with the constant.

Choice C uses 960 as the multiplier, but 960 was the profit amount from the example, not the rate per lawn. This treats the output as the constant rate.

Choice D correctly identifies 40 as the dollars earned per lawn serviced, making $$1400 = 40n$$ the right equation.

Study tip: In proportional relationships, always find the constant rate by dividing the given output by the given input, then use that rate in your equation.

When you encounter proportional relationships, remember that the equation $$C = kn$$ means the constant $$k$$ represents the unit rate - in this case, the cost per pencil. To find the correct equation, you need to determine this unit rate from the given data.

The student's original equation $$C = 0.75n$$ suggests each pencil costs $0.75. However, the new information tells us that 30 pencils cost $18 total. To find the actual cost per pencil, divide the total cost by the number of pencils: $$\frac{18}{30} = 0.60$$ dollars per pencil. Therefore, the correct equation is $$C = 0.60n$$.

Let's examine why the other answers are incorrect. Answer A ($$C = 30n$$) incorrectly uses the number of pencils as the rate, which would mean each pencil costs $30 - clearly unreasonable. Answer B claims the original equation was correct, but if you substitute: $$C = 0.75(30) = 22.50$$, not $18 as given. Answer C ($$C = 18n$$) mistakenly uses the total cost as the rate, meaning each pencil would cost $18.

Answer D correctly calculates $$\frac{18}{30} = 0.60$$ and recognizes this as the cost per individual pencil.

Study tip: In proportional relationships, always identify what the constant represents. When given a specific data point, use it to calculate the actual unit rate by dividing the total amount by the number of units. This will help you catch errors in proposed equations and write correct ones.

The correct answer is A. Since the cost is proportional to miles driven at $$\0.25$$ per mile, the equation is $$T = 0.25m$$. We can verify: $$T = 0.25(150) = 37.50$$, which matches the given information. Choice B uses the total payment as the rate. Choice C uses the miles driven as the rate. Choice D adds a constant term, making it non-proportional.

The correct answer is B. Using $$p = 18t$$ with $$t = 2.5$$: $$p = 18(2.5) = 45$$ pages. This demonstrates that the constant rate of $$18$$ pages per minute applies to any time interval, including fractional times. Choice A gives the wrong calculation ($$18 × 2 = 36$$). Choice C gives an incorrect sum ($$18 + 2.5$$). Choice D uses incorrect multiplication ($$18 × 4$$) and makes a false claim about whole numbers.

The correct answer is A. Since flour is $$1.5$$ times the sugar, $$f = 1.5s$$. With $$s = \frac{2}{3}$$: $$f = 1.5 × \frac{2}{3} = \frac{3}{2} × \frac{2}{3} = 1$$ cup. Choice B reverses the relationship. Choice C has the right equation but wrong calculation ($$\frac{4}{9}$$ instead of $$1$$). Choice D uses addition instead of multiplication and gets $$\frac{2}{3} + 1.5 = \frac{13}{6}$$.

The correct answer is C. Both equations represent the same proportional relationship. Student A's equation $$d = 28g$$ shows distance as a function of gallons ($$28$$ miles per gallon). Student B's equation $$g = \frac{d}{28}$$ is the inverse, showing gallons as a function of distance. These are equivalent: solving $$d = 28g$$ for $$g$$ gives $$g = \frac{d}{28}$$. Choice A and B incorrectly claim only one is right. Choice D makes a false statement about proportional relationships.

This question tests writing equations y=kx for proportional relationships from tables, graphs, contexts, or verbal descriptions, identifying k and defining variables contextually. Proportional equation y=kx: k is constant of proportionality (unit rate, ratio y/x). From table: calculate k from any pair (14/2=7, k=7 gives y=7x), from graph: k=slope (or read y when x=1: if graph through (1,7), k=7), from context: stated rate is k ("$3 per pound" → k=3, equation c=3p where c=cost, p=pounds). Variables: choose meaningful (c for cost, n for number, d for distance) and define in context. For example, context "apples $3/lb" write c=3p (c=cost dollars, p=pounds), k=3 from $/lb rate; or table x:2,4,6 y:10,20,30 find k=10/2=5, write y=5x; or graph through origin with slope 8 write y=8x. The correct equation is f=2.5b with proper k=2.5 and variables f for flour and b for batches. A common error is reversing variables like b=2.5f instead of f=2.5b, wrong form like f=b+2.5 not proportional, or including intercept like f=2.5b+1. To write the equation: (1) identify proportional relationship (context says "2.5 cups per batch"), (2) find k (stated rate of 2.5), (3) choose variables (f for flour, b for batches), (4) write f=2.5b, (5) define variables (f=cups of flour, b=number of batches), (6) verify (b=1, f=2.5×1=2.5, yes✓). Multiple representations: equation f=2.5b matches table of multiples of 2.5, graph with slope 2.5, verbal "2.5 per batch"—all show k=2.5. Mistakes: wrong form (additive), variables reversed, k wrong, undefined variables.

This question tests writing equations y=kx for proportional relationships from tables, graphs, contexts, or verbal descriptions, identifying k and defining variables contextually. Proportional equation y=kx: k is constant of proportionality (unit rate, ratio y/x). From table: calculate k from any pair (14/2=7, k=7 gives y=7x), from graph: k=slope (or read y when x=1: if graph through (1,7), k=7), from context: stated rate is k ("$3 per pound" → k=3, equation c=3p where c=cost, p=pounds). Variables: choose meaningful (c for cost, n for number, d for distance) and define in context. For example, recipe 2 cups flour per batch, write f=2b (f=cups of flour, b=batches), k=2 from cups per batch; or table x:2,4,6 y:10,20,30 find k=10/2=5, write y=5x; or graph through origin with slope 8 write y=8x. The correct equation is f=2b, with k=2 and variables f for flour and b for batches. A common error is reversing like b=2f, using additive f=b+2, or including intercept f=2b+2. To write: (1) identify proportional from "2 cups for each batch," (2) find k=2 as rate, (3) choose f and b, (4) write f=2b, (5) define f as cups and b as batches, (6) verify b=1, f=2. Multiple representations: f=2b matches table multiples of 2, graph slope 2, verbal "2 per batch"—all k=2. Mistakes: reversed variables, wrong form, added constants.

This question tests writing equations y=kx for proportional relationships from tables, graphs, contexts, or verbal descriptions, identifying k and defining variables contextually. Proportional equation y=kx: k is constant of proportionality (unit rate, ratio y/x). From table: calculate k from any pair (14/2=7, k=7 gives y=7x), from graph: k=slope (or read y when x=1: if graph through (1,7), k=7), from context: stated rate is k ("$3 per pound" → k=3, equation c=3p where c=cost, p=pounds). Variables: choose meaningful (c for cost, n for number, d for distance) and define in context. For example: context "apples $3/lb" write c=3p (c=cost dollars, p=pounds), k=3 from $/lb rate; or table x:2,4,6 y:10,20,30 find k=10/2=5, write y=5x; or graph through origin with slope 8 write y=8x. The correct equation is f=(3/2)b, where f is the cups of flour and b is the number of batches, with k=3/2 from 3 cups per 2 batches. A common error is reversing the ratio like f=(2/3)b, reversing variables like b=(3/2)f, or using additive form like f=b+(3/2) instead of multiplicative. To write the equation: (1) identify proportional relationship (context says "3 cups for every 2 batches"), (2) find k (ratio 3/2), (3) choose variables (f for flour, b for batches), (4) write f=(3/2)b, (5) define variables (f=cups of flour, b=number of batches), (6) verify (for b=2, f=(3/2)×2=3, matches✓). Multiple representations: equation f=(3/2)b matches a table with ratios of 3/2, a graph through origin with slope 3/2, and verbal "3 cups per 2 batches"—all show same k=3/2.

This question tests writing equations y=kx for proportional relationships from tables, graphs, contexts, or verbal descriptions, identifying k and defining variables contextually. Proportional equation y=kx: k is constant of proportionality (unit rate, ratio y/x). From table: calculate k from any pair (14/2=7, k=7 gives y=7x), from graph: k=slope (or read y when x=1: if graph through (1,7), k=7), from context: stated rate is k ("$3 per pound" → k=3, equation c=3p where c=cost, p=pounds). Variables: choose meaningful (c for cost, n for number, d for distance) and define in context. For example, in the context of movie tickets at $9 each, write c=9t (c=total cost in dollars, t=number of tickets), where k=9 from the dollars per ticket rate; or table x:2,4,6 y:10,20,30 find k=10/2=5, write y=5x; or graph through origin with slope 8 write y=8x. The correct equation is c=9t, with k=9 and variables c for total cost and t for tickets. A common error is using the wrong form like c=t+9 which is not proportional, or reversing variables like t=9c, or including an intercept like c=9t+9 when it should pass through the origin. To write the equation: (1) identify the proportional relationship from the context "charges $9 per ticket," (2) find k=9 as the stated rate, (3) choose variables c for cost and t for tickets, (4) write c=9t, (5) define c as total cost in dollars and t as number of tickets, (6) verify by substituting t=1, c=9×1=9, which is reasonable. Multiple representations: equation c=9t matches a table where costs are multiples of 9, a graph through origin with slope 9, and the verbal "$9 per ticket"—all show k=9. Mistakes include using additive forms like c=t+9 instead of multiplicative, reversing variables, or adding unnecessary constants.