Verified answer

This is just a system of equations.

ok so you know there is some number of Quarters. lets call this number Q. Also there are Dimes. Lets call this number D.

you have a total of 19.75 in some amount of Q and D. Thus you want to set the equation to equal 19.75.

19.75=...

because quarters are worth .25 and dimes are worth .10 you set it up like this:

19.75=.25Q+.1D

for the other equation we have:

100=Q+D

this is because there are a total of 100 coins.

using the second equation we can subtract D from both sides to get:

100-D=Q

Now we can substitute (100-D) for Q in the other equation:

19.75=.25(100-D)+.1D

Distribute the .25:

19.75=.25x100-.25D+.1D

Combine like terms and multiply:

19.75=25-.15D

subtract 25 from both sides:

-5.25=-.15D

divide both sides by (-.15):

35=D

Which means, Q=65 because we know 100=Q+D

Say D is a dime, and Q is a quarter. You know there are 100 total coins, and there are only dimes and quarters, so:

Q+D=100

Now you have to use the fact that they add up to $19.75. You know dimes are $0.10 and quarters are $0.25. Use this to make the equation:

.25*Q+.10*D=19.75

Now you just9.759. have 2 equations and 2 unknowns that you have to solve for. It turns out that you will have 65 quarters and 35 dimes.

The proper way to solve this is to set it up as a system of two equations. Let D and Q equal the number of dimes and quarters, respectively. You know that there are 100 coins in all, so let D + Q = 100. You also know that the total value of the coins is $19.75. Dimes are worth 0.10 each, and quarters are worth 0.25 each, so you would set the second equation up as 0.10D + 0.25Q = 19.75. And simply substitute and solve.

D + Q = 100

D = 100 - Q

0.10D + 0.25Q = 19.75

0.10(100-Q)+0.25Q=19.75

10-0.10Q+0.25Q = 19.75

0.15Q = 9.75

Q = 65

D + 65 = 100

D = 35

Check this

0.10(35) + 0.25(65) = 19.75

3.50 + 16.25 = 19.75

19.75 = 19.75

So there are 35 dimes and 65 quarters.

Well

a * 0.10 + b * 0.25 = 19.75

Where a is the amount of dimes, and b the amount of quarters.

In surplus, you know that

b = 100 - a

Thus the formula is now :

a * 0.10 + (100 - a) * 0.25 = 19.75

0.10 * a - 0.25 * a + 25 = 19.75

- 0.15 a = - 5.25

a = 35 (dimes)

And thus b = 100 - a = 100 - 35 = 65 (quarters)

There is 65 quarter coins and 35 Dime coins because 65 x $0.25 = $16.25 + 35 x $0.10 = $3.50 so $3.50 + $16.25 = $19.75 so there is 35 dimes in total

0.10x + 0.25y = 19.75

x + y = 100

from x + y = 100, we see that y = 100 - x.

therefore,

0.10x + 0.25(100 - x) = 19.75

0.10x + 25.00 - 0.25x = 19.75

-0.15x = -5.25

x = 5.25/0.15 = 35 dimes

y = 100 - x = 65 quarters.

35(0.10) + 65(0.25) = 3.50 + 16.25 = 19.75