Can someone help me with this word problem!?

Question

mark gets $1.85 a week for an allowance. he always gets 16 coins. the coins are always nickels dimes and quarters. how many of each type of coin does he receive?

--i have to have 2 ways to solve this problem, the first way i used was guess and check..but i have no clue how to solve it any other way!! any ideas?

from guessing and checking i came up with: 9 dimes, 3 quarters, and 4 nickels by the way

thank you so much :)

Answer 1

N + D + Q = 16   (number of coins)

5 N + 10 D + 25 Q = 185  (total value of coins)

We can eliminate one of them:

5 N + 5 D + 5 Q = 80  (5 sets of coins)

5D + 20 Q =105  (subtract the last two)

Divide by 5:

D + 4 Q = 21

Now you could graph that with, for example, D on the y-axis

and Q on the x-axis and then from points where both are

integers, try values of N.

That would be equivalent to the following:

Note that we have  (4 of something) + (some more) = multiple of 4 + 1

Since D and Q must be integers, then

D must be a "multiple of 4 + 1".  <-- important to know why this is true

So D can be 1, 5, 9, 13, or 17  (if D = 21, we go over $1.85).

From those you can calculate what Q could be:

(21-1)/4, (21-5)/4, (21-9)/4, (21-13)/4, or (21-17)/4 or

5, 4, 3, 2, or 1

So we have for D and Q:

1 5

5 4

9 3

13 2

17 1

Total coins is 16, so we would have:

D Q N

1 5 10

5 4 7

9 3 4

13 2 1

17 1 -1 ... strike this one

Then figure the total value for each of those:

10 + 125 + 50 = 185

50 + 100 + 35 = 185

90 + 75 + 20 = 185

130 + 50 + 10 = 185

So any of those are possible.

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