Coin word problem help please
There is $68.50. It is made up of 440 US coins.
There are three types of coins. What are the coins?
I will accept the premise that whoever created this problem believed
that there is only one set of coins, and the combinations of those coins,
that satisfy the stated requirements.
It falls to me to prove that assumption false. There are five U.S. coins
in circulation; the half-dollar (H), the quarter (Q), the dime (D),
the nickel (N) and the penny (P). Taking any three of them, in certain
combinations, we can have 440 coins that add up to $68.50 as I will show.
I will begin each list with the three letters repesenting the coins, as
inidcated above. The largest coin is listed first, the mid-value coin second,
and the smallest coin last. For each group, I will give the first combination
and the last combination. Then, I will give the increments/decrements for each
of the coins in order to progress to the next combination that satisfies the
problem requirements.
HQD
There are 20 combinations.
1) 2/158/280
20) 59/6/375
To move to the next combination, H increases by 3, Q decreases by 8, D increases by 5.
HQN
There are 25 combinations.
1) 6/219/215
25) 102/3/335
For each step, H increases by 4, Q decreases by 9, N increases by 5.
HQP
There are 5 combinations.
1) 26/214/200
5) 122/18/300
For each step, H increases by 24, Q decreases by 49, P increases by 25.
HDN
There are 42 combinations.
1) 62/372/6
42) 103/3/334
For each step, H increases by 1, D decreases by 9, N increases by 8.
HDP
There are 7 combinations.
1) 68/342/30
7) 122/48/270
For each step, H increases by 9, D decreases by 49, P increases by 40.
HNP
There are 7 combinations.
1) 106/304/30
7) 130/10/300
For each step, H increases by 4, N decreases by 49, P increases by 45.
QDN
There are 69 combinations.
1) 164/274/2
69) 232/2/206
For each step, Q increases by 1, D decreases by 4, N increases by 3.
QDP and QNP have no combinations that equal $68.50, and for DNP there
are no combinations that even reach $44.00.
As you can see, all five types of coins can be used, in the appropriate
combinations, to have 440 coins add up to $68.50, making the question,
"What are the coins?" unanswerable, or at least, not unique.