Finding 2 missing numbers (Spilled Ink Story Problem)

[geekily]

Hi there,
I have the homework problem "Someone spilled ink on a bill for 36 sweatshirts. If only the first and last digits were covered and the other three digits were, in order, 8, 3, 9 as in ?83.9?, how much did each cost? Other than arbitrarily plugging in numbers, I have no idea how to figure this out. I guessed 183.96 because divided by 36 is 5.11, which is a nicer decimal that most of the ones I'd been getting. However, I have no way of knowing if that's right. Is there a formula behind this? Thanks for your help!

 

[skeeter]

since the total price is divisible by 36, it is also divisible by 4 and 9.

a number divisible by 4 must have its last two digits divisible by 4

the next to last digit is 9, there are two choices for the last digit ... 2 and 6.

since the number is also divisible by 9, the sum of all the digits must be divisible by 9.

if the last four digits are ?83.96 whose sum is 26 ... the only digit which will yield a sum divisible by 9 is 1 ... 183.96, making a digit sum of 27.

if the last four digits are ?83.92 whose sum is 22 ... the only digit which will yield a sum divisible by 9 is 5 ... 583.92, also making a digit sum of 27.

so ... 183.96 works and so does 583.92

the question is ... do the sweats cost 5.11 each or 16.22 each?

... either a Walmart price or a Macy's price.

 

[geekily]

Thanks so much for explaining that! I worked the problem out both ways so my teacher can see my work, but I ended up going with $16.22, since that sounds a little more sweatshirt-y of a price. Thanks again! This forum is a lifesaver!