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Beatrice has at least one quarter, one dime, one nickel, and one penny. She has exactly 13 coins total. Beatrice also has a total value of 92 cents. How many different sets of coins can she have?​

Respuesta :

The number of different sets of coins that she can have is; Two sets which are;  (0, 7, 4, 2) or (1, 3, 7, 2)

What is the combination of coins?

Let the numbers of coins be;

  • Quarter; Q
  • Dimes: D
  • Nickel; N
  • Penny; P

We have:

Q + D + N + P = 13

We know that;

25 quarters make 1 cent

10 dimes makes one cent

5 nickels make one cent

1 penny makes one cent

Thus;

25Q + 10D + 5N + P = 92

P will have to be 2 because the others will give a value with the unit digit as 0. Thus;

Q + D + N = 11  ----(1)

25Q + 10D + 5N + 2 = 92

25Q + 10D + 5N = 90

Divide through by 5 to get;

5Q + 2D + N = 18      (2)

Subtract eq (1) from eq (2) to get;

4Q + D = 7

Let us put D = 7 and Q = 0, and so from eq (1), N = 4

The set here is now; (Q, D, N, P) = (0, 7, 4, 2)

Let us put D = 3, Q = 1, and from eq (1), N = 7

The set here is now; (Q, D, N, P) = (1, 3, 7, 2)

If D = 6, 5, 4, 2, 1, 0, then Q is not an integer, these can be ignored.

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