20 coins, all nickels, is a dollar.
Since there is exactly 1 nickel, she has to have
the remaining 95 cents with 19 coins.
Obviously the number of pennies has to be 0, 5, 10, or 15
She can't have 0 pennies because even 19 dimes
is way more than 95 cents, let alone having any
higher coins.
If she had only 5 pennies, then those and the nickel
make 10 cents with 6 coins, and the remaining 90 cents
must be made with 14 coins. But even 14 dimes is way
more than 90 cents, let alone having any higher
coins.
If she had only 10 pennies, then those and the nickel
make 15 cents with 11 coins, and the remaining 85 cents
must be made with 9 coins. But even 9 dimes is more
than 85 cents, let alone any higher coins.
So she must have exactly 15 pennies, then those and the
nickel make 20 cents with 16 coins, and the remaining 80
cents must be made with 4 coins.
If she had exactly 1 quarter, she'd have to have 55
cents with 3 coins. There is no way she could have that
without having another nickel.
If she had exactly 2 quarters, she'd have to make 30
cents with 2 coins. There is no way she could have that
without having another nickel.
If she had exactly 3 quarters, she'd have to have 5
cents with 1 coin. But she can't have another nickel.
So she can't have any quarters.
So the only way she can have that is with 1 half dollar
and 3 dimes. So that's the answer:
She has 1 half dollar, 3 dimes, 1 nickel and 15 pennies.
[Half dollars aren't in circulation anymore, but Mary
in the problem has one! I have a few myself. I'll
bet you've got one stashed in a drawer someplace in
your house! :) ]
Edwin
