systems of equation and matrice

Question

A piggy bank contains only nickels, dimes, and quarters. There is a total of 105 coins and the total value of those coins is $10.75.

1. Find a general solution of the number of each coin.
2. List three possible specific solutions.

Show your work for full credit.

Answer 1

5N+10D+25Q=1075

N+D+Q=105

Let Q = a maximum 1075/25=43. One solution is Q=43, N=D=0 But 43 does not equal 105

Let Q=40 That equals 25x40=1000 15 nickels add 75 but 40+15 is less than 105

Q has to be less than 40

Let Q=30 30x25=750 1075-750=325 65 nickels would add 5x65=325

Q=30 D=0 and N=65 is one solution

Other solutions would have have Q<30 Q=29 means one quarter has to be replaced with one coin either N or D, and switch some Ns for D. 29x25=725 1075-725=350. 63=N 5xx63=315 add 3 dimes

3x10+315+725 is 5 cents short of 1075

Q=27 27x25=675 Need N&D worth 400 where N+D=78 That's two equations 5N+10D=78

also 5N=5D=390 subtract to get 5D=390-78=312 D= a fraction, it won't work

(Q,D,N)=(30,0,65) is one solution

In general, N+D+Q=105, 5N+10D+25Q=1075 5N+5D+5Q=525 D+4Q=110

Pick a Q, it has to fit in that equation and get an integer result for D.

D=110-4Q=2(55-2Q) Q has to be <27

Try Q=26, 425 left for N&D. N+D=79 and 5N+10D=425 5N+5D=5(=7079)=395 425-395=30 5D=30, D=6

Another solution is (Q,D,N)=(26,6,7)

(Q,D,N) = (30,0,65), (26,6,7)

A third solution would necessarily involve a Q < 26

Try Q=25, then D+4Q=110 means D=10 N=70

25x25 + 10x10 + 5x70 = 625+ 100 + 350 = 1075

another solution is (Q,D, N) = (25, 10, 70)