SOLUTION: Moolah has 27 coins that are all dimes and quarters. The value of the coins is $4.35. How many dimes and how many quarters does Moolah have?

Question 1065268: Moolah has 27 coins that are all dimes and quarters. The value of the coins is $4.35. How many dimes and how many quarters does Moolah have?
Answer by KMST(5328) (Show Source): You can put this solution on YOUR website!
THE FIFTH GRADER APPROACH:
We know that 17 quarters make $4.25,
so 17 quarters + 1 dime = 18 coins makes $4.35.
However, with that combination we are
27-18=9 coins short of the required number of coins.
We also know that $0.50 can be made with 5 dimes or with 2 quarters.
Replacing 2 quarters with 5 dimes increases the number of coins by
5-2=3 , but does not alter the value of the set of coins.
Since the combination of 17 quarters and 1 dime was 9 coins short,
we can make 9�3=3 swaps of 5 dimes for 2 quarters and end up with
2�3=6 less quarters, and
5�3=15 more dimes.
Then we would have
17-6= quarters and
1+15= dimes.
The total amount would still be $4.35,
but the number of coins would be the required
11+16=27 .

AS AN ALGEBRA EXERCISE:
You start by defining one variable (or two):
= number of quarters
If you would like to use two variables you would also define
= number of dimes,
and then you would write
as one of your equations.
From there, solving for you would have
as the number of dimes.
If you rather use just one variable,
you would just say directly that
= number of dimes.
Your other equation would be
the one relating the numbers of coins
to the value in $ :
if you use two variables,
or .
The equation above is what you get substituting
for in the equation with two variables.
It is also what you would write directly
if you are using just one variable.
Solving:

Multiplying both sides of the equal sign times 20,
we get ,and we do not have to deal with decimals.
If we had multiplied times 10p, we would have
, with is equivalent.
From either one, we get
for the number of quarters.
After that, we calculate the number of dimes as
,
or as .
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