A wallet contains $1 and $10 coins amounting to $45.00. How much money of each kind will produce the given amount?

Answer 1

Let x denote the number of $1 coins, and y denoted the number of $10 coins in the wallet.

Since the total amount is $45, we get the linear equation

1x+10y = 45.

Solving this equation for x, we see that

x = 45-10y.

Since only non-negative integer solutions make sense, solving this can quickly be done by choosing a value for y, and finding the corresponding value for x:

If y=0, then x=45.

If y=1, then x=35.

Etc.

Based on the tags you put on the question, you may want a more "Diophantine" sort of answer form:

The gcd of 1 and 10 is 1. Since 45 is clearly a multiple of 1, there is a solution.

Other solutions must have the form (x+kv,y-ku), where (x,y) is one solution of the equation, and u,v are the quotients of 1 and 10, respectively, when divided by 1 (the gcd of 1 and 45), and k is an integer (not arbitrary, since we must have a non-negative number of coins).

If we take (45,0) as the reference solution, and note that u=1 and v=10, we get that a general solution must take the form

(45+10k,-k), for k=0,-1, -2,-3, and -4, which also gives the solutions found above.