Calculating break-even number of days or months

My daughter has two problems that ask her to find the number of days or months for two different series to equal. I don't remember how to figure it out algebraically - only manually. In one case we're told that Verizon gives 4 free months and then $25 per month thereafter while Sprint gives 3 free months and charges $20 thereafter. It wants to know how many months it will take before two subscribers - one on Verizon and one on Sprint would have paid exactly the same total. The answer is that they would both pay $100 at the end of the 8th month.

In the second problem we're told that Blockbuster charges $2.25 for the first day and $.15 per day thereafter. Hollywood charges $5 for the first two days and then $ .05 per day thereafter. It wants to know the same thing - after how many days would a renter under each system spend the same amount.

As I said, I used a spreadsheet to set up a table and calculate a running balance under each scenario. But since my daughter is in pre-algebra I assume there must be some algebraic expression that would calculate this easier than doing the brute force method.

Any ideas are appreciated!

Thank you
rfbrenner

Hello, rfbrenner!

I hope your daughter has had some algebra . . .

Verizon gives 4 free months and then charges $25 per month thereafter
while Sprint gives 3 free months and charges $20 per month thereafter.
How many months will it take for the two services to have the same total cost?

Click to expand...

Suppose you use Verizon for \(\displaystyle x\) months.
The first 4 months are free, so you are charged $25 per month for the other \(\displaystyle x-4\) months.
. . \(\displaystyle \text{Your total Verizon bill will be: }\:25(x-4)\text{ dollars.}\)

Suppose you use Sprint for \(\displaystyle x\) months.
The first 3 months are free, so you are charged $20 per month for the other \(\displaystyle x-3\) months.
. . \(\displaystyle \text{Your total Sprint bill will be: }\:20(x-3)\text{ dollars.}\)

The two bills are equal: .\(\displaystyle 25(x-4) \:=\:20(x-3)\)

Solve for \(\displaystyle x\) and get: .\(\displaystyle x \,=\,8\)

Blockbuster charges $2.25 for the first day and $0.15 per day thereafter.
Hollywood charges $5 for the first two days and then $ 0.05 per day thereafter.
After how many days would a renter under each system spend the same amount?

Click to expand...

Suppose you rent a movie for \(\displaystyle x\) days.

Blockbuster charges $2.25 for the first day and $0.15 per day for the other \(\displaystyle x-1\) days.
. . \(\displaystyle \text{Your total Blockbuster bill will be: }\:2.25 + 0.15(x-1)\text{ dollars.}\)

Hollywood charges $5 for the first two days and $0.05 per day for the other \(\displaystyle x-2\) days.
. . \(\displaystyle \text{Your total Hollywood bill will be: }\:5.00 + 0.05(x-2)\text{ dollars.}\)

Got the idea?

She's in 7th grade - not quite sure how much Algebra she has had. This makes perfect sense to me. Now I just have to be able to explain it to her!

Thank you soooo much. Terrific!

Rick