Hi Grace.
You have three different denominations and you know how many bills there are in total (500) and how much
they add up to ($50,000).
Let's use some logic now.
First, the number of ones must be divisible by 100, because the total value ends in "00". That means there are only six choices for the number of ones: zero, 100, 200, 300, 400 or 500, but if he has 500 ones, then he doesn't have enough total money, so we can ignore that.
Now there are five possibilities for the values and the number of bills (a is the number of $100 bills and b is the number of $500 bills):

values equation 
number of bills equation 
i) 
$400 + $100a + $500b = $50,000 
400 + a + b = 500 
ii) 
$300 + $100a + $500b = $50,000 
300 + a + b = 500 
iii) 
$200 + $100a + $500b = $50,000 
200 + a + b = 500 
iv) 
$100 + $100a + $500b = $50,000 
100 + a + b = 500 
v) 
$0 + $100a + $500b = $50,000 
0 + a + b = 500 
(Remember that a and b are whole numbers!)
Let's look at them in turn:
i) Divide the values equation by $100 and you get:
4 + a + 5b = 500
a + 5b = 496
and the number of bills equation becomes:
a + b = 100
a = 100  b
We can substitute the bottom value for a into the equation above. Let's see:
a + 5b = 496
(100  b) + 5b = 496
4b + 100 = 496
4b = 396
b = 99
(that's a whole number)
So if a = 100  b, then a = 100  99 = 1. That means in option (i) there are 400 $1 bills, 1 $100 bill and 99 $500 bills.
Let's check: 400 + 1 + 99 = 500. Ok. 400($1) + 1($100) + 99($500) = $50,000. Yes this works.
(ii) Since we got a right answer we could stop, but let's follow the same procedure for the second option since there may be more than one solution:
$300 + $100a + $500b = $50,000
3 + a + 5b = 500
a + 5b = 497
and
300 + a + b = 500
a + b = 200
a = 200  b
Substituting:
a + 5b = 497
(200  b) + 5b = 497
200 + 4b = 497
4b = 297
But 4 doesn't divide evenly into 297, so we can't get a whole number for b. This means option (ii) doesn't work.
Now you try possibilities iii), iv) and v).
In Canada, we have no $1 bills, no $500 bills and $100 bills are hard to cash, so hopefully Daddy Warbucks can buy some travellers' cheques before he visits!
Hope this helps,
Stephen La Rocque>
