Need to understand how to solve the following question:

You have 50 coins that add up to 1 dollar. What are the possible combinations of pennys, nickels, dimes, and quarters that will satisfy these limits?

P + N + D + Q = 50

.01P + .05N + .10D + .25Q = $1

I know the answers are available online, but I want to know how I would go about solving it. Any suggestions?


multiply the second equation by $100$, and substract first from second. you get $$4N+9D+24Q=50$$ So, for $Q$ , you have three choises. $0$,$1$, or $2$.

if $Q=2$ then $4N+9D=2$ this is impossible.

if $Q=1$ then $4N+9D=26$, $D$ must be $2$mod $4$. trying $D=2$ we get $N=2$ and $P=45$

if $Q=0$ then $4N+9D=50$. again $D$ must be $2$mod $4$. trying $D=2$ we get $N=8$ so $P=40$. we cant choose $D=6$ because then $N$ is negative. so solution is $$\begin{matrix} P & N & D & Q \\ 45 & 2 & 2 & 1 \\ 40 & 8 & 2 & 0 \\ \end{matrix}$$

  • $\begingroup$ Thanks for your response, getting 4N + 9D + 24Q = 50 makes sense to me, but I'm not really understanding how Q has 3 choices only. Did you just decide to use Q = 0,1,2 because its the largest value (.25)? Also why must D be 2mod4? $\endgroup$ – thedeepfield Oct 24 '13 at 7:26
  • $\begingroup$ (1) if $Q>2$ was correct then $4N+9D$ would be negative. for example if $Q=3$ then $4N+9D=-22$ this is impossible................ (2) $4N+9D=26$. writing as mod$4$, $0N+1D=2$mod $4$ so $D$ is $2$ mod$4$. $\endgroup$ – mert Oct 24 '13 at 8:01

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