SOLUTION: In her purse Anita has 16 coins (nickels,dimes,and quarters), worth $2.05. She has three times as many dimes as nickels. How many of each coin does she have?

Question 437114: In her purse Anita has 16 coins (nickels,dimes,and quarters), worth $2.05. She has three times as many dimes as nickels. How many of each coin does she have?
Found 2 solutions by mananth, oberobic:
Answer by mananth(16946) (Show Source): You can put this solution on YOUR website!
--------------value-----------numbers
nickels 5 x
dimes 10 3x
quarters 25 16-4x

Value= 205

5x+30x-100x+400 =205
-65x=-195
-65x=-195
/-65
x=3nickels, 9 dimes,4 quarters
(15+90+100)= 205 15

Answer by oberobic(2304) (Show Source): You can put this solution on YOUR website!
With coin problems, you have to keep track of the counts of each coin denomination as well as the value. For example, let d=number of dimes. Then the value of the number of dimes = 10d. Let n=number of nickels, 5n = value of nickels. Similarly, q=number of quarters and 25q=value of the quarters.
.
We're told she has nickels, dimes, and quarters worth $2.05 or 205 cents.
5n + 10d + 25q = 205
.
We're also told she has 16 coins, so
n + d + q = 16
.
And we're told she has 3 times as many dimes as nickels:
3n = d
or
d = 3n
.
We can use simultaneous equations.
.
5n + 10d + 25q = 205
Divide by 5
n + 2d + 5q = 41
.
Substitute d = 3n in the equations
.
n + 2(3n) + 5q = 41
n + 6n + 5q = 41
7n + 5q = 41
.
n + 3n + q = 16
4n + q = 16
.
So we have two equations with two unknowns.
7n + 5q = 41
4n + q = 16
.
Multiply the second equation by 5
20n + 5q = 80
.
7n + 5q = 41
20n + 5q = 80
.
Subtract the second equation from the first.
-13n = -39
n = 3
.
Therefore d = 3n = 9
.
Therefore q = 16 - 3 -9 = 4
.
Our tentative answer is:
n = 3
d = 9
q = 4
.
Always check your answer. In this case, we have to use the "value" equation since we used the "count" equation to figure out how many quarters.
What is the total value?
3*5 = 15 cents
9*10 = 90 cents
4*25 = 100 cents
45 + 90 + 100 = 205 cents
Correct.
.
Answer: She has 3 nickels, 9 dimes, and 4 quarters.
.
Done.
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