Money denomination problem

Question

Otis Taylor has a box of coins that he uses to play poker with. The box contains 44 coins, consisting if pennies, dimes, and quarters. The number of pennies is equal to the number of dimes, and the total value is $4.37. How many of each demonination are there?

Answer 1

Hi Steve,

 

Let's use P to be the number of pennies, D the number of dimes, and Q the number of quarters. With the information you've provided, we can write two equations. First, the total number of coins is 44, so:

 

   P + D + Q = 44

 

Second, the total value is $4.37. So:  0.01 * P + 0.10 * D + 0.25 * Q = 4.37

 

Let's multiple the second equation by 100 on each side to get:  P + 10 * D + 25 * Q = 437

 

Let's solve the first equation for Q. We subtract P and D from each side of the equation, and get:

 

  Q = 44 - P - D

 

We can now substitute this into our second equation, to eliminate the Q variable:

 

  P + 10 * D + 25 * Q becomes:  P + 10 * D + 25 * (44 - P - D)

 

so we get:  P + 10 * D + 25 * (44 - P - D) = 437

or:  P + 10 * D + 25 * 44 - 25 * P - 25 * D = 437

or:  P + 10 * D + 1100 - 25 * P - 25 * D = 437

or:  P * (1 - 25) + D * (10 - 25) + 1100 - 437 = 0

or:  -24 * P - 15 * D + 663 = 0

 

To make it a little clearer, I'll multiply each side by -1, and move the 663 to the right to get:

 

  24 * P + 15 * D = 663

 

And we HAVE A BIG PROBLEM !! One equation, but two unknowns. This tells us that there probably is not a single (unique) answer. We also know that P, D, and Q must be whole numbers (can't have half a penny, for example). But still, we don't have enough information. But let's go a bit further. Let's take our equation with two unknowns, and split it so we have one unknown on each side. For example, let's move P over to the right side:

 

  15 * D = 663 - 24 * P

or:  D = (663 - 24 * P) / 15

 

Now we see something interesting -- the division by 15 -- because for a number to be divisible by 15, and the result be a whole number, then the numerator (that is, 663 - 24 * P) must end with "0" or "5". If not you can't divide by 15! How do we use this information ... well, we know that P cannot be 1 (a single penny), because (663 - 24 * P) = (663 - 24) = 639, and that does not end with 0 or 5. Could P be 2 ? We test (663 - 24 * 2) = (663 - 48) = 615, and sure enough, it ends in 5. Checking if 615 is divisible by 15, we get exactly 41 (that is D=41). Therefore, we have a solution as

 

  P = 2 and D = 41

 

and remember from way at the top, Q = 44 - P - D. So, Q = 44 - 2 - 41 = 1 quarter. If we check the total value of P=2, D = 41, and Q = 1, sure enough, it's $4.37

 

How about P = 17 ? Checking (663 - 25 * P) = (663 - 24 * 17) = 663 - 408 = 255. 255 end with "5", so it's a possibility. For P = 17, we have D = (663 - 24 * 17) / 15 = 255 / 15 = 17. And Q = 44 - P - D. So, Q = 44 - 17 - 17 = 44 - 34 = 10 quarters. If we check the total value of P=17, D = 17, and Q = 10, sure enough, it's $4.37. (Always check your work...always). This is the solution Desiree gave us.

 

So far, we have two combinations that work, P=2, D = 41, and Q = 1; and P=17, D = 17, and Q = 10. As I wrote earlier, your problem does not have a unique answer.

 

So far, we've listed two combinations ... there are MORE. Your job is to follow the pattern and get the others!

 

I hope this helps you.

Answer 2

Using the trial and error method I started with 7 pennies and 7 dimes.

$ .07 + $ 0.70 = % 0.77

 

However there is a remaining 30 coins that are quarters.

$ 0.25 x 30 = $7.50

 

Together that totals:

$ 0.77 + $ 7.50 = $ 8.27

 

That is over the suggested value of $ 4.37

 

However, a second attempt in which I tried:

17 pennies= $ 0.17

17 dimes= $1.70

34 coins= $ 2.37

 

That leave the final ten quarters which equals $2.50

 

Together they total up to:

$2.37 + $2.50= $4.37

 

So the answer is 17 pennies, 17 dimes and 10 quarters.