Question

d. suppose you know how many cats are at the shelter in any week. how can you find the number of cans of cat food x cats will eat? e. at the shelter, there is a proportional relationship between the quantities. when a group of ratios are equivalent, it means there is a proportional relationship between the number of cats and the number of cans of cat food. ask: every time the number of cats increases by 1, what does the number of cans of cat food increase by? share: i notice that when you multiply the number of cats by the unit rate... discuss it lesson 3 understand proportional relationships learning targets: • identify proportional relationships. • find the constant of proportionality of a proportional relationship. • write an equation to represent a proportional relationship. copying is not permitted. ©curriculum associates, llc

Explanation:

Step1: Identify the proportional - relationship

The number of cats and the number of cans of cat food are in a proportional relationship. If we assume the number of cats is $x$ and the number of cans of cat food is $y$, and the constant of proportionality is $k$. The general form of a proportional relationship is $y = kx$.

Step2: Find the constant of proportionality

From the given information, when $x = 7$ cats, $y = 56$ cans of cat food. We can find $k$ by substituting these values into the equation $y=kx$. So, $k=\frac{y}{x}=\frac{56}{7} = 8$.

Step3: Answer the question in part d

If there are $x$ cats, and the constant of proportionality $k = 8$, then the number of cans of cat food $y$ that the cats will eat in a week is given by the formula $y = 8x$.

Step4: Analyze the proportional - relationship in part e

In a proportional relationship between two quantities (the number of cats and the number of cans of cat food), the ratios of the two quantities in any group are equivalent. That is, if we have two pairs $(x_1,y_1)$ and $(x_2,y_2)$ that follow the proportional relationship $y = kx$, then $\frac{y_1}{x_1}=\frac{y_2}{x_2}=k$. When the number of cats increases by 1, the number of cans of cat food increases by the value of the constant of proportionality. Since $k = 8$, when the number of cats increases by 1, the number of cans of cat food increases by 8.

Answer:

d. The number of cans of cat food $y$ that $x$ cats will eat in a week is $y = 8x$.
e. The number of cans of cat food increases by 8 when the number of cats increases by 1 because the constant of proportionality between the number of cats and the number of cans of cat food is 8.