Expert: Scott A Wilson - 3/27/2011

Question
Abe has a bag containing 340 coins and 160 bills totaling $1809.  There are pennies and nickels for coins. There are fives, tens, and twenties for bills. The total value of the tens is equal to the total value of the twenties. There are the same number of fives and twenties.  What is the total value of the coins and bills bearing Lincoln's portrait?

Answer
Contents
340 (pennies/nickels), 160 ($5/$10/$20)bills, $1809 total.

Let x be the number of $20's.
Total value of tens is the same as the total value of the twenties mean that there are half as many twenties, for every twenty is worth two tens.  From this, the number of $10's is 2x.

There are the same number of $5's and $20's, so the number of $5's is also x.

What can be said from this is that whatever x is, there are x $20's, 2x $10's, and x $5's,
This means the value of x is 20 + 2*10 + 5 = $45.  It also means the number of bills is 4x.
Since there are 160 bills, 4x = 160, so x=40.

Since we know the value is 45x and x=40, the value is 45*40 = $1800.
Looking at the value of the bag, this only leaves $9 in change.

If there are y pennies, there are 340-y nickels.
The value of this would be 0.01y + (340-y)0.05 = $9.
Multiplying out the () gives 0.01y + $17 - 0.05y = $9.
The y's combine to $17 - 0.04y = $9.
This means that 0.04y = $8.
That says that y = 200, and that is how many pennies are involved.
That means that x = 340 - 200 = 140, and that is how many nickels are involved.
The value of 140 nickels is $7 and the values of 200 pennies is $2, for a total of $9.
That was the amount we were short in the bills, so that's the answer.

That is, 40 #20's, 80 $10's, $40 $5's, 140 nickels, and 200 pennis.
The value is $800 in twenties, $800 in tens, $200 in fives, $7 in nickels, and $2 in pennies.
That total is $800 + $800 + $200 + $7 + $2 = $1809, so that is the right answer.