Step-by-step explanation:
x = number of quarters.
y = number of dimes.
z = number of pennies.
0.25x + 0.1y + 0.01× = 1.05
but it is not possible to find a combination of a subset of the coins to create exactly 1 (= the change for $1).
like to have only 3 quarters and 3 dimes (and no pennies). that makes $1.05, and no combination can create exactly $1.
as a consequence
x < 4 (because 4 quarters is $1, and more than 4 quarters is already more than $1.05).
z < 5 (because with 5 or more pennies, since the total sum is $1.05, if I simply remove 5 pennies, I get $1 with the other coins, no matter what combination they are).
with 1 or 2 or 3 or 4 pennies I cannot build a total sum of $1.05, as the other coins will can only build a sum that ends in 0 or in 5.
so, the sum of these other coins and 1, 2, 3, 4 pennies can never end in 5.
that means, we have 0 pennies.
that rules out to have 0 or 2 quarters, because then together with the dimes the sum can only end in 0 (and actually also give change for $1).
so, we can have only 1 or 3 quarters.
that means the solutions are
1 quarters and 8 dimes
or
3 quarters and 3 dimes.
1 quarter and 8 dimes (and 0 pennies) is the desired solution, because it contains more coins.
for the maximum amount of money built by coins not providing change for $1 we need to remember :
4 quarters are $1 (and would provide change for $1). so, we always need less than 4 quarters. but also at least 1 quarter to keep the sum from ending with a 0 (which would allow again to build $1).
10 dimes are $1 (and provide change for $1). we always need less than 10 dimes.
5 or more pennies would always allow to take away or fill up to create a sum that ends in 0 and therefore build $1. so, we always need less than 5 pennies.
that means again
1 or 3 quarters
max. 9 dimes
4 pennies.
if the the maximum possible sum would be higher than 1.19, then we could only increase in steps of 0.10 (dimes) : 1.29, 1.39, ...
if the sum is 1.29, I could remove 1 quarter and 4 pennies to make $1.
if it is 1.39, I could remove 1 quarter, 1 dime and 4 pennies to make $1.
and so on.
any sum >= 1 created incl. 4 or more quarters or 10 or more dimes or 5 and more pennies can always be reduced to a full $1.
$1.19 can only be built by 1 quarter and 9 dimes and 4 pennies, so that we cannot build $1 out of a subset, or by 3 quarters, 4 dimes and 4 pennies.
and 1 quarter, 9 dimes and 4 pennies has more coins.
