Question

I cant seem to solve this problem someone help:
I have a total of $300 in coins of domination $1, $2, $5 coins . the total number of coins is 160. How many coins of each denomination are with me ?

Answer 1

Answer:

There is not a unique solution to this problem as stated.

For instance, one solution is 20 $1 coins, 140 $2 coins, 0 $5 coins.

(Check: total number of coins is then 20 + 140 + 0 = 160, and total value is $20 + $280 = $300.)

Another solution is 110 $1 coins, 20 $2 coins and 30 $5 coins.

(Check: total number of coins is then 110 + 20 + 30 = 160, and total value is $110 + $40 + $150 = $300.)

Actually, there are 36 different solutions, corresponding to choosing 0 to 35 $5 coins to start with.

Step-by-step explanation:

Let the number of $1, $2, $5 coins be a, b, c, respectively.

Then we need a + b + c = 160 and a + 2b + 5c = 300.

Subtracting the first from the second to eliminate a, we have

b + 4c = 140.

As long as we choose c to be one of the 36 values 0, 1, 2,..., 35, this equation determines a non-negative value for b.  Then as b + c ≤ b + 4c = 140 < 160, the relation a + b + c = 160 determines a positive value for a.