Luanne bought a sandwich from the vending machine for $1.65. The vending machine takes nickels, dimes, and quarters. She dropped 19 coins into the vending machine to pay for her sandwich.
1. How many of each kind of coin did Luanne use to pay for the sandwich?
2. What strategy did you use to investigate the problem?
3. Are you sure you are correct? How do you know?
4. Is there only one answer to the problem? How do you know?

Question

mertsj · Answer

There is a piece of information missing.

anonymous · Answer

this was the only information given...i was able to come up with 2 equations but could get no farther
$$n+d+q=19$$ and $$.05n+.10d+.25q=1.65$$

anonymous · Answer

that isn't 165 though

anonymous · Answer

and thats only 12 coins and i have to have 19 coins total...

anonymous · Answer

ok i think i have it
2 quarters, 6 dimes, 11 nickels

anonymous · Answer

thats one solution, i also come up with 14 nickels, 2 dimes, and 3 quarters. so now i know there are multiple solutions but how do i explain the strategy used to solve this problem?

anonymous · Answer

still not 19 coins though

anonymous · Answer

do u guys have every thing under controle ?

anonymous · Answer

ok now i am on a real computer.  lets try this 
$$Q+D+N=19$$
$$25Q+10D+5N=165$$ now lets get rid of the nickels by multiplying the first equation by -5 and adding to second one

anonymous · Answer

get
$$20Q+5D=70$$ or 
$$D=14-4Q$$

anonymous · Answer

now we are looking for integer solutions, so you know Q = 1, 2, 3, ... etc

anonymous · Answer

now get rid of the dimes in a similar fashion

anonymous · Answer

im with you so far

anonymous · Answer

btw we really can stop here, because we already know that we have only 3 choices for quarters, 
1, 2 and 3 and that determines the number of dimes, so we really don't have to repeat and solve for nickels in terms of quarters

anonymous · Answer

since we know that 
$$D=14-4Q$$ if there is one quarter, there are ten dimes, and the rest nickels right?

anonymous · Answer

similarly if there are 2 quarters there are 6 dimes and 11 nickels

anonymous · Answer

the last one doesnt have 19 coins

anonymous · Answer

that was wrong. 3 quarters, 2 dimes 14 nickels

anonymous · Answer

yeah i added wrong sorry.

anonymous · Answer

ok that sounds better

anonymous · Answer

so there are three solutions, and they all work.   
$$\checkmark$$

anonymous · Answer

thanks so much

anonymous · Answer

yw