jamie has a collection of 78 nickles,dimes,and quarters worth $12.40. If the number of quarters is doubled, the value becomes $22.15. How many of each type of coin does JAmie have?

Answer 1

x nickels, y dimes and 78 - x - y quoters;

1x + 10y + 25(78 - x - y) = 1240;

1x + 10y + 50(78 - x - y) = 2215.

If we subtract we get: 25(78 - x - y) = 975; 78 - x - y = 39; x + y = 39; x = 39 - y;

39 - y + 10y + 975 = 1240; 9y = 225; y = 25 dimes; x = 39 - 25 = 14 nickels; 78 - 14 - 25 = 39 quoters

Answer 2

Let "n" be the number of nickels she has, let "d" be the number of dimes she has, and let "q" be the number of quarters she has.

Since each nickel is worth $0.05 then the value of her "n" nickels is 0.05n (for instance, if she has 10 nickels, it will be worth 0.05•10 = $0.50 or 50 cents). In the same way, the value of her dimes is 0.10d and the value of her quarters is 0.25q. Adding these together, we get:

0.05n + 0.10d + 0.25q = 12.40

You can make this equation a little easier to work with if you multiply both sides of the equation by 20:

20(0.05n + 0.10d + 0.25q) = 12.40(20) or:

n + 2d + 5q = 248

We are also told that she has a total of 78 coins so:

n + d + q = 78

Finally we are told that if we double the number of quarters (so "2q") the value is $22.15 so:

0.05n + 0.10d + 0.25(2q) = 22.15

0.05n + 0.10d + 0.50q = 22.15 and, again, multiplying by 20 gives:

n + 2d + 10q = 443

Using these 3 equations:

(1) n + 2d + 5q = 248

(2) n + d + q = 78

(3) n + 2d + 10q = 443

you can use the process of elimination to solve. First, modify equation (2) by multiplying both sides by "-1" to get:

(2A) -n - d - q = -78

Then add equations (1) and (2A):

(1) n + 2d + 5q = 248

(2A) -n - d - q = -78

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(4) d + 4q = 170

Then add (2A) and (3) to get:

(2A) -n - d - q = -78

(3) n + 2d + 10q = 443

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(5) d + 9q = 365

Now, multiply equation 4 by "-1" to get:

(4A) -d - 4q = -170

and add equations (4A) and (5) to get:

(4A) -d - 4q = -170

(5) d + 9q = 365

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5q = 195

or q = 39

Plug that value of "q" back into equation (5) to get:

d + 9(39) = 365

d = 14

Then plug in d = 14 and q = 39 into equation (2) to get:

n + 14 + 39 = 78

n = 25