Question

Owen made 100 sandwiches which
she sold for exactly $100. She sold
caviar sandwiches for $5.00 each, the
bologna sandwiches for $2.00, and
the liverwurst sandwiches for 10 cents.
How many of each type of sandwich
did she make?

Answer 1

Tricky.
Set up your equations:
C = # of caviar
B = # bologna
L = # of liverwurst

C + B + L = 100
5C + 2B + L/10 = 100

multiply the 2nd equation by 10 and subtract the two
49C + 19B = 900

Now, there are lots of ways to do that, but you need both C and B to be integers. Notice that 900 is divisible by 10. Divide the whole equation by 10
$4.9 C + $1.9 B = $90

Now, we know she need to make exactly $90 selling just caviar and bologna. Each time she sells a sandwich, she'll have to give a dime in change. To get back to an exact dollar figure, she'll need to sell some multiple of 10 sandwiches. So, make the substitution

C + B = 10 n

Now, solve for C and substitute
49 n - 3 B = 90

Remembering that n and B are both integers, n must be a multiple of 3 (divide that equation by 3 and see). So try, n=3, 6, 9, etc.
For n=3, B = 19
For n=6, B = 68
For n=9, B = 117 (which is impossible.)

If B = 19, then C = 11 and L = 70.
If B = 68, then C = -8 which is impossible.

So, the only possible answer is 11 caviar sandwiches, 19 bologna sandwiches and 70 Liverwurst sandwiches.

Answer 2

Why post a question if you know the answer?

Answer 3

Win.

Answer 4

@hba Sometimes people want an answer checked.

Answer 5

@e.mccormick Well that's something which is to be mentioned in the question.

Answer 6

Yup

Answer 7

True.