SOLUTION: The value of the dimes is exactly half the value of the quarters. The value of the nickels is 50 cents less than the value of the dimes. The value of the pennies is exactly half th

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Question 1053338: The value of the dimes is exactly half the value of the quarters. The value of the nickels is 50 cents less than the value of the dimes. The value of the pennies is exactly half the value of the nickels. There are 97 coins in all with a combined value of $6. How much in quarters?
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
pennies,nickels, dimes, quarters;
p, n, d, q;



Notice the description indicates five equations but only needs four variables to be solved.

First goal is to simplify the system.
This is one possible system found along the way:
system%284d=5q%2Cn=2d-10%2C2p=5n%2Cp%2Bd%2Bn%2Bq=97%2Cp%2B5n%2B10d%2B25q=600%29

A possible next step could be substitute for n in four of the equations and obtain this possible system:
system%284d=5q%2Cp=5d-25%2Cp%2B11d%2Bq=147%2Cp%2B60d%2B25q=850%29

...exactly which direction of steps you take anywhere in the process is up to you. Continued substitution? Elimination? Matrix row operations?

Continuing the solution process here, substituting for p,
system%284d=5q%2C16d%2Bq=172%2C13d%2B5q=35%29

Answer by ikleyn(52781) About Me  (Show Source):
You can put this solution on YOUR website!
.
The value of the dimes is exactly half the value of the quarters. The value of the nickels is 50 cents less than the value of the dimes.
The value of the pennies is exactly half the value of the nickels. There are 97 coins in all with a combined value of $6.
How much in quarters?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

Let D be the value of dimes, in cents.  (Notice: D is the VALUE, not the number of dimes !)

Then the value of quarters is 2D, according to the condition.

Then the value of nickels is (D-50), according to the condition.
Then the value of nickels is %28D-50%29%2F2.

Then the combined value of all coins is 


   D   +   2D     +   (D-50)  +   D-50%29%2F2 = 600.
dimes    quarters     nickels     cents


You have ONE equation for only ONE single unknown D.

To solve it, multiply both sides by 2. You will get

2D + 4D + 2(D-50) + (D-50) = 1200.

Simplify it:

9D - 150 = 1200  --->  9D = 1200 + 150  --->  9D = 1350  --->  D = 1350%2F9 = 150.

Thus we found that the value of dimes is 150 cents. Hence, there are 150%2F10 = 15 dimes in the collection.

Then the value of quarters is twice of it, i.e. 300 cents. Hence, the number of quarters is 300%2F25 = 12.

So, we are ready to formulate the answer.

Answer.  The number of quarters is 12.

Check.  The value of nickels is D-50 = 150-50 = 100. Hence, the number of nickels is 100%2F5 = 20.

        The value of cents is %28D-50%29%2F2 = %28150-50%29%2F2%29 = 50. Hence, there are 50 1-cent coins.

        In total, there are 15 + 12 + 20 + 50 = 97 coins.   Correct!

The problem is solved.

The lessons to learn from the solution are:

1.  This problem is for ONE unknown, not for 4 or 5.

2.  The condition regarding the total number of coins is, actually, excessive. We can solve the problem without this condition.

3.  The way proposed by the person "josgarithmetic" is the way to nowhere.
     It is not the way to solve the problem.
     Simply ignore his post.