Sarah has a collection of nickels, dimes, and quarters worth $9.25. She has 10 more dimes than nickels and twice as many quarters as dimes. How many coins of each kind does she have?
Well, let's play with some numbers and solve this puzzle!
Let's start by assigning variables to the number of nickels, dimes, and quarters Sarah has.
Let N be the number of nickels,
Let D be the number of dimes,
And let Q be the number of quarters.
According to the information given, we know three things:
1. The total value of the coins is $9.25.
2. Sarah has 10 more dimes than nickels.
3. Sarah has twice as many quarters as dimes.
First, let's convert everything to cents to make the math a bit easier.
The value of N nickels is 5N cents.
The value of D dimes is 10D cents.
The value of Q quarters is 25Q cents.
So from the first piece of information, we know that 5N + 10D + 25Q = 925 cents.
Now, let's work on the second and third pieces of information.
Sarah has 10 more dimes than nickels, so D = N + 10.
Sarah has twice as many quarters as dimes, so Q = 2D.
Now we can substitute these expressions into the first equation to solve for N, D, and Q.
5N + 10D + 25Q = 925.
Substituting D = N + 10 and Q = 2D:
5N + 10(N + 10) + 25(2(N + 10)) = 925.
Simplifying:
5N + 10N + 100 + 50N + 500 = 925.
Combining like terms:
65N + 600 = 925.
Subtracting 600 from both sides:
65N = 325.
Dividing both sides by 65:
N = 5.
Now we know there are 5 nickels.
Using D = N + 10, we find D = 5 + 10 = 15.
Using Q = 2D, we find Q = 2 * 15 = 30.
So Sarah has 5 nickels, 15 dimes, and 30 quarters.
Now, it's time to help Sarah become a successful coin collector!
Let's solve this problem step by step.
Step 1: Assign variables.
Let's assume the number of nickels as "n", the number of dimes as "d", and the number of quarters as "q".
Step 2: Translate the given information into equations.
According to the problem, the value of all the coins is $9.25.
The value of a nickel is $0.05, the value of a dime is $0.10, and the value of a quarter is $0.25.
The equation for the total value of all the coins can be written as:
0.05n + 0.10d + 0.25q = 9.25 -- Equation 1
According to the problem, Sarah has 10 more dimes than nickels.
So, the equation for the number of dimes can be written as:
d = n + 10 -- Equation 2
According to the problem, Sarah has twice as many quarters as dimes.
So, the equation for the number of quarters can be written as:
q = 2d -- Equation 3
Step 3: Solve the system of equations.
To solve this system of equations, we will use the substitution method.
Substitute Equation 2 and Equation 3 into Equation 1:
0.05n + 0.10(n+10) + 0.25(2d) = 9.25
0.05n + 0.10n + 1.00 + 0.50d = 9.25
0.15n + 0.50d = 8.25 -- Equation 4
Now, substitute Equation 2 into Equation 4:
0.15n + 0.50(n+10) = 8.25
0.15n + 0.50n + 5.00 = 8.25
0.65n = 3.25
n = 5
Substitute the value of n into Equation 2:
d = n + 10
d = 5 + 10
d = 15
Substitute the value of d into Equation 3:
q = 2d
q = 2 * 15
q = 30
Step 4: Check the solution.
Let's check if our solution is correct by using the values of n = 5, d = 15, and q = 30 in Equation 1:
0.05n + 0.10d + 0.25q = 9.25
0.05(5) + 0.10(15) + 0.25(30) = 9.25
0.25 + 1.50 + 7.50 = 9.25
9.25 = 9.25
Since both sides of the equation are equal, our solution is correct.
Step 5: Answer the question.
Sarah has 5 nickels, 15 dimes, and 30 quarters.
To solve this problem, we can use a system of equations.
Let's start by assigning variables:
Let the number of nickels be N
Let the number of dimes be D
Let the number of quarters be Q
We can now translate the given information into equations.
1) The total value of the coins is $9.25:
The value of N nickels is 5N cents.
The value of D dimes is 10D cents.
The value of Q quarters is 25Q cents.
The sum of the values is 5N + 10D + 25Q.
Therefore, the equation is:
5N + 10D + 25Q = 925 (because $9.25 is equal to 925 cents)
2) Sarah has 10 more dimes than nickels:
The number of dimes is 10 more than the number of nickels.
Therefore, the equation is:
D = N + 10
3) Sarah has twice as many quarters as dimes:
The number of quarters is twice the number of dimes.
Therefore, the equation is:
Q = 2D
Now we have a system of three equations:
1) 5N + 10D + 25Q = 925
2) D = N + 10
3) Q = 2D
We can solve this system of equations using substitution or elimination.
Let's use the substitution method to solve this system:
Substitute D in terms of N from equation (2) into equations (1) and (3):
1) 5N + 10(N + 10) + 25Q = 925
2) Q = 2(N + 10)
Simplify equation (1):
5N + 10N + 100 + 25Q = 925
15N + 25Q = 825
3N + 5Q = 165
Multiply equation (2) by 5:
5Q = 10N + 50
Substitute the value of Q from equation (2) into equation (1):
3N + 5(10N + 50) = 165
3N + 50N + 250 = 165
53N = 165 - 250
53N = -85
N = -85/53 (rounded to the nearest whole number)
Since the number of coins cannot be negative, we know that N must be equal to 0.
Therefore, the number of nickels (N) is 0.
Substitute the value of N back into equation (2):
D = N + 10
D = 0 + 10
D = 10
Substitute the value of D back into equation (3):
Q = 2D
Q = 2(10)
Q = 20
So, Sarah has 0 nickels, 10 dimes, and 20 quarters.
nickels --- x
dimes ---- x+10
quarters --- 2(x+10) = 2x + 20
now for the value of your coints
5x + 10(x+10) + 25(2x + 20) = 925
solve for x , easy to solve