Question

Marie is cleaning her clothes out of the closet in her spare bedroom. She started with 650 of her items in the closet. She has been able to continually decrease the number of her items in the closet at a rate of one-half per month. Her brother, Dustin, has started putting 5 items of his clothing in the closet each month as Marie cleans it out. Marie wants to know how many months it will take before the number of items Dustin adds will be equal to the number of her items in the closet.

Create a system of equations to model the situation above, and use it to determine if there are any solutions. If there are any solutions, determine if they are viable or not.

Answer 1

D. there is only one solution, and it is viable

Explanation:

Answer 2

Wording is everything. Here, there are some issues. "... at the rate of 1/2 per month" can be interpreted to mean that at the end of the first month, there are 649 1/2 items in Marie's closet (decreased by 1/2 from 650).

"The number of items Dustin adds" could mean 5 items, the number he adds each month. The wording should specify the time period or whether we're talking about the total number Dustin has added.

We assume your description means that the number of items in Marie's closet at the end of each month is 1/2 what it was at the beginning. (As opposed to decreasing by 1/2 item each month.) We assume we're interested in the total number of items of Dustin's that are in the closet.

Marie's quantity can be modeled by ...

... m = 650·(1/2)^t . . . . . t = time in months

Dustin's quantity can be modeled by ...

... d = 5t

There will be one solution for d=m, at about t = 4.8. At that point, Dustin will have added about 24 items, which will be the number Marie is down to.

There is a viable solution for d=m at about t = 4.8.