How exactly does the response "infintely many" answer the question of "how many"? I admit that the level of this question is roughly about middle school, but this is what the question asks:

The ratio of nickels to dimes to quarters is 3:8:1. If all the coins were dimes, the amount of money would be the same. Show that there are infinitely many solutions to this problem.


let $n,d,q$ be the quantity of nickels, dimes and quarters respectively.
The fact they are in ratio 3:8:1 means for every 12 coins:  3 are nickels , 8 are dimes and 1 is a quarter.
So if we have 12 coins exactly there are 3 nickels, 8 dimes and 1 quarter (1.20 dollars total). On the other hand if we had 12 dimes it would be the same.
So the case with 12 works. What about the case with 24? Well we would just end up having twice of each coin and the cash would add up to 2.40 dollars. And 24 dimes also give 2.40 dollars.
We can use this method for any multiple of 12. And there are infinite of those.
Hope this helps, Regards.
