Find the number of each coin

Question

Owne has a collection of nickels and quaters worth $8.10.If the nickels were quaters and the quaters were nickels,the value would be $17.70.

Answer 1

Value of each nickel = 5 cents = 0.05

Value of each quarter = 25 cents = 0.25

nickels = n

quarters = q

total $ from nickels + total $ from quarters = total $

First equation:

0.05n + 0.25q = 8.10

Second equation: now we want to pretend the nickels that we have are quarters and the quarters that we have are nickels —> we are swapping x and y

0.05q + 0.25n = 17.70

I am going to rewrite the equations on top of each other and line up the n’s and q’s

0.05n + 0.25q = 8.10

0.25n + 0.05q = 17.70

Now we can solve using elimination. We want one pair of matching terms, either the n’s or the q’s, to be equal and opposite so we can eliminate them. Let’s get the n’s to match. We need the least common multiple of 0.05 and 0.25.

Count by 0.05’s

0.05, 0.10, 0.15, 0.20, 0.25, 0.30

Count by 0.25’s

0.25, 0.50, 0.75, 1.00

Both lists meet at 0.25 so this is the least common multiple

0.05 * 5 = 0.25 —> multiply the whole top equation by 5 so the 0.05n turns into a 0.25n. The bottom equation already has 0.25n so nothing has to happen to it.

5*(0.05n + 0.25q = 8.10)

0.25n + 1.25q = 40.50

0.25n + 0.05q = 17.70

Now our n terms are equal but they are not opposite. We will make them opposite by taking one equation and multiplying everything by -1 —> flipping the signs of each term. I am going to pick the second equation but it doesn’t matter which one you choose.

0.25n + 0.05q = 17.70 becomes

-0.25n - 0.05q = -17.70

Now we can set up elimination

0.25n + 1.25q = 40.50

-0.25n - 0.05q = -17.70

Add downward as if they are in columns

0.25n - 0.25n cancels out

1.25q - 0.05q = 1.2q

=

40.50 - 17.70 = 22.80

We are left with

1.2q = 22.80 divide by 1.2 to get q

q = 19 quarters! Plug back in to any equation solve for n

0.25n + 1.25q = 40.50

0.25n + 1.25(19) = 40.50

0.25n + 23.75 = 40.50 subtract 23.75

0.25n = 16.75 divide by 0.25

n = 67 nickels

19 quarters and 67 nickels

Answer 2

We can set N as the number of nickels and Q as the number of quarters.

Each Nickel is worth .05 dollars and each quarter is worth 0.25 dollars. As such, adding up the worth of Owne's collection would be

.05N + .25Q = 8.10

If we switched the number of nickels and quarters, we would get

.25N + .05Q = 17.70

We now have two equations to work with. We can use them together in a system of equations to find the answer.

Equation 1: .05N + .25Q = 8.10

Equation 2: .25N + .05Q = 17.70

One thing we could do is multiply the top equation by 5 so that the coefficients of N match up.

(.05N + .25Q = 8.10) * 5, becoming

.25N + 1.25Q = 40.50

We can then use this new equation to subtract equation 2.

.25N + 1.25Q = 40.50

- .25N + 0.05Q = 17.70

0N + 1.20Q = 22.80

1.20Q = 22.80

Divide both sides by 1.20

Q = 19 <---19 Quarters

We can then use this information by putting it back into any of our previous equations to solve for N, the number of Nickels.

For example, our original equation is

.05N + .25Q = 8.10

We can replace Q with 19 now

.05N + .25(19) = 8.10

.05N + 4.75 = 8.10

Subtract 4.75 from both sides

.05N = 3.35

Divide both sides by .05

N = 67, or 67 Nickels

19 Quarters and 67 Nickels.