SOLUTION: A school orders $99$ textbooks, all for the same price. When the bill for the total order comes, the first and last digits are obscured. What are the missing digits? $_38,254.

Algebra ->  Decimal-numbers -> SOLUTION: A school orders $99$ textbooks, all for the same price. When the bill for the total order comes, the first and last digits are obscured. What are the missing digits? $_38,254.      Log On


   

Question 1207681: A school orders $99$ textbooks, all for the same price. When the bill for the total order comes, the first and last digits are obscured. What are the missing digits?
$_38,254.1_
Found 2 solutions by Edwin McCravy, greenestamps:
Answer by Edwin McCravy(20054) About Me  (Show Source):
You can put this solution on YOUR website!

Let the price of a book be ABCD.EF.

Then 100 books would cost ABCDEF.00

So we subtract the price of 1 book from ABCDEF.00 and get G38254.1H,
the price of 99 books.

So we have this subtraction

     A B C D E F.0 0
       - A B C D.E F 
     ---------------
     G 3 8 2 5 4.1 H

So G=A, 10-F=H and 9-E=1, so E=8

     A B C D 8 F.0 0
       - A B C D.8 F 
     ---------------
     A 3 8 2 5 4.1 H

F cannot be 0, since that would make H be 10.

Depending on whether we will have to borrow from the 8 above the C,
C must be either 2 or 3, so those two cases are:

    A B 2 D 8 F.0 0             A B 3 D 8 F.0 0
      - A B 2 D.8 F               - A B 3 D.8 F 
    ---------------             ---------------
    A 3 8 2 5 4.1 H             A 3 8 2 5 4.1 H

Either way, we know we'll have to borrow from the B to make the 2 a 12.
So 12-A=8 so A is 4.  And since we borrow from B, B=4

So these two cases become

    4 4 2 D 8 F.0 0             4 4 3 D 8 F.0 0
      - 4 4 2 D.8 F               - 4 4 3 D.8 F 
    ---------------             ---------------
    4 3 8 2 5 4.1 H             4 3 8 2 5 4.1 H

Now we see that D can only be 6:

    4 4 2 6 8 F.0 0             4 4 3 6 8 F.0 0
      - 4 4 2 6.8 F               - 4 4 3 6.8 F 
    ---------------             ---------------
    4 3 8 2 5 4.1 H             4 3 8 2 5 4.1 H

That tells us that we did have to borrow from the 8,
so that eliminates the second case and makes F having to
have been scratched with a zero over it, which makes F=1

    4 4 2 6 8 1.0 0            
      - 4 4 2 6.8 1                
    ---------------             
    4 3 8 2 5 4.1 H

So we finally see that H=9.

    4 4 2 6 8 1.0 0            
      - 4 4 2 6.8 1                
    ---------------             
    4 3 8 2 5 4.1 9

So the missing digits are 4 and 9.

[Can you believe that a textbook would cost $4426.81?]  

Edwin

Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


The amount on the bill is "A38,254.1B", where A and B are digits that are not known.

The bill is for 99 textbooks that all have the same price, so A382541B is divisible by 99.

The number is divisible by 99 if and only if it is divisible by both 9 and 11.

To be divisible by 9, the sum of the digits must be divisible by 9:

A+3+8+2+5+4+1+B = 23+(A+B)

For that sum to be divisible by 9, A+B can only be either 4 or 13.

(1) A+B = 4 or A+B = 13

The number is divisible by 11 if and only if the two sums of alternating digits are either equal or differ by a multiple of 11.

A+8+5+1 = A+14; 3+2+4+B = B+9

If A and B are positive single digit numbers with A+B=4, those two sums can't be equal. So A+B is 13.

We need to have A+14 = B+9, and A+B=13:

A+B=13 --> B=13-A
A+14 = B+9
A+14 = (13-A)+9
A+14 = 22-A
2A = 8
A = 4
B = 13-A = 9

ANSWER: The obscured digits are A=4 and B=9

CHECK: $438,254.19/99 = $4426.81