7 coins in bag which add up to $1.25

Question

So I came across this question, there are 7 coins in bag of only nickels, dimes and quarters and we know it totals to $1.25. So, how many of each coin are there?

Just by looking, one solution is 4 quarters, 2 dimes and a nickel, but are there other ways? I tried writing some equations and using a matrix but I don't think it worked out properly (I most likely made a mistake)

Answer 1

Hint:

• If you have exactly four quarters, how many other coins might you have? How many in total?

• If you have exactly three quarters, how many other coins might you have? How many in total?

• If you have fewer than three quarters, how many other coins might you have? How many in total?

• If you have exactly five quarters, how many other coins might you have? How many in total?

• If you have more than five quarters, how many other coins might you have? How many in total?

Answer 2

There must be exactly four quarters, since if there are three the most you can have is $3\times 0.25+4\times 0.10=1.15$, whereas if there are five quarters you have at least $1.35$. Now the three other coins are dimes or nickels, so if there are $n$ nickels the three coins total $(3-n)\times 0.10+n\times 0.05=0.30-0.05n$, so $n=1$.

Answer 3

If there are at least 5 quarters, total sum exceeds $5\times .25=1.25$. If there are at most 3 quarters, total sum cannot exceed $3\times .25+4\times .1=1.15$. So there must be 4 quarters. And there should be 2 dimes and 1 nickel to make up $.25$.