Part 1.

A piggybank contains only coins. All of the coins are half-dollars except two; all the coins are quarters except two; and all of the coins are dimes except two.  What is the total amount of money in the piggy bank?

Try to figure this out on your own before looking at the solution below!

85 cents is in the piggy bank:

one half dollar =  50 cents

one quarter      =  25 cents

one dime         =  10 cents

total                 =  85 cents

An algebraic solution:

Let h = the number of half dollars.

Let q = the number of quarters.

Let d = the number of dimes.

Let t = the total number of coins = h + q + d.

All but two of the coins are half dollars.  This means

t - 2 = h

All but two of the coins are quarters. This means

t - 2 = q

All but two of the coins are dimes.  This means

t - 2 = d.

We add these three equations, and we get

3*t - 6 = h + q + d = t

So

3*t - 6 = t

2*t      = 6

    t      = 3

Therefore h = q = d = 1.

Part  2.

I have some pennies, nickels, dimes, and quarters in my piggybank.

All but 32 are pennies.

All but 55 are nickels.

All but 42 are dimes.

All but 57 are quarters.

How many of each coin are in the piggybank?

Let p = number of pennies, n = number of nickels, d = number of dimes, and q = number of quarters.

Let t = total number of coins = p + n + d + q.

Then

t - 32 = p

t - 55 = n

t - 42 = d

t - 57 = q

Then

4*t - 186 = p + n + d + q = t

3*t = 186

t = 62

p = 62 - 32 = 30 pennies

n = 62 - 55 = 7 nickels

d = 62 - 42 = 20 dimes

q = 62 - 57 = 5 quarters

So I have .30 + .35 + 2.00 + 1.25 = $3.90 in my piggybank.

Part 3.

Now make up your own problem with this pattern.

Hint:  Work backwards.  Decide how many of each coin you have first.

More hints:

Suppose you have n different categories of coins.  For example, if you have pennies, nickels, dimes, and quarters, n = 4.

Then let t be the total number of coins.  We see that

n*t  - (sum of numbers specified) = t

(In the above example, the sum of the numbers specified = 32 + 55 + 42 + 57 = 186.)

So

t*(n - 1) = (sum of specified numbers)

So the sum of the specified numbers must be divisible by n-1. 

Above, we saw that the sum of the specified numbers is 186, and 186 is divisible by 3.