rose counted her money and found that her 25 coins which were nickels, dimes, and quarters were worth $3.20. The number of dimes exceeded the number of nickels by 4. How many coins of each kind did she have?
I know that the expression for nickles would be 5x, for dimes it would be
10x + 40, but i don't understand how to do the expression for quarters.....can someone help me with the answer
It is not clear what your x means, and where you came up with 10x + 40 for the value of the dimes. Let the numbers of nickels, dimes and quarters be N, D and Q.
5 N + 10 D + 25 Q = 320
D = N + 4
N + D + Q = 25
Now solve those three equations in three unknowns.
5 N + 10(N + 4) + 25 Q = 320
15 N + 25 Q = 280
2 N + 4 + Q = 25
Multiply that last equation by 25
50 N + 25 Q = 525
Now eliminate the Q
35 N = 245
N = 7
You do the rest
To find the number of quarters, let's assume that Rose has x quarters.
Now, let's use the given information to form two equations.
First, we know that the total value of the coins is $3.20. So, we can write the equation:
0.05n + 0.1d + 0.25q = 3.20
Second, we know that the number of dimes exceeds the number of nickels by 4. So, we can write the equation:
d = n + 4
We can substitute the value of d from the second equation into the first equation. Therefore, the equation becomes:
0.05n + 0.1(n + 4) + 0.25q = 3.20
Now, we can simplify and solve for n.
0.05n + 0.1n + 0.4 + 0.25q = 3.20
0.15n + 0.4 + 0.25q = 3.20
Rearranging the terms:
0.15n + 0.25q = 3.20 - 0.4
0.15n + 0.25q = 2.80
We can divide the whole equation by 0.05 to simplify it further:
3n + 5q = 56 (equation 1)
Now, let's go back to the second equation:
d = n + 4
Since we know that d is the number of dimes, we can substitute the value of d with 10x + 40 (from before):
10x + 40 = n + 4
10x - n = -36 (equation 2)
Now, we can solve these two equations simultaneously to find the values for n and q.
First, let's multiply equation 1 by -10:
-30n - 50q = -560
Now, let's add equation 2 to this equation:
-30n - 50q + 10x - n = -560 - 36
Simplifying,
-31n - 50q + 10x = -596 (equation 3)
Now, let's multiply equation 2 by 3:
30x - 3n = -108
Now, let's add equation 3 to this equation:
30x - 3n - 31n - 50q = -108 - 596
Simplifying,
30x - 34n - 50q = -704 (equation 4)
Now, we have a system of equations:
-31n - 50q + 10x = -596 (equation 3)
30x - 34n - 50q = -704 (equation 4)
Using any method of solving systems of equations (substitution, elimination, or matrices), you can find the values for n and q.
To solve this problem, let's break it down step by step:
1. Let's assume that the number of nickels Rose has is "x."
2. Since the number of dimes exceeds the number of nickels by 4, we can express the number of dimes as "x + 4."
3. Now, let's calculate the value of these nickels and dimes. The value of x nickels would be 5x cents, and the value of (x + 4) dimes would be 10(x + 4) cents.
4. Finally, Rose also has some quarters, but we don't know the exact number. Let's represent the number of quarters as "y."
5. The value of y quarters would be 25y cents.
6. According to the problem, the total value of all these coins (nickels, dimes, and quarters) is $3.20 or 320 cents.
7. Now, we can create an equation using the values we found: 5x + 10(x + 4) + 25y = 320.
8. Simplify the equation: 5x + 10x + 40 + 25y = 320. Combining like terms, we get 15x + 40 + 25y = 320.
9. Rearrange the equation: 15x + 25y = 280.
10. We need more information or another equation to solve for x and y simultaneously. However, we can still solve for the possible values of x and y using trial and error.
11. Start by assuming x = 0. Then, substitute it into the equation: 15(0) + 25y = 280, which simplifies to 25y = 280.
12. Solve for y: y = 280 / 25, which gives y = 11.2. Since the number of coins must be a whole number, y cannot be 11.2, so we discard this possibility.
13. Now, let's assume x = 1. Substituting it into the equation, we get 15(1) + 25y = 280, which simplifies to 15 + 25y = 280.
14. Solve for y: 25y = 280 - 15, which gives 25y = 265. Dividing both sides by 25, we find y = 10.6. Again, since y must be a whole number, we discard this possibility as well.
15. Let's try x = 2. Substituting into the equation, we get 15(2) + 25y = 280, which simplifies to 30 + 25y = 280.
16. Solve for y: 25y = 280 - 30, which gives 25y = 250. Dividing both sides by 25, we find y = 10.
17. Now that we have found a whole number solution for y, we can substitute it back into one of the original equations to find x: x + 4 = 2 + 4, which gives x = 2.
18. Therefore, when Rose counted her money, she had 2 nickels, 6 dimes, and 10 quarters.
Hope this helps!
N = number of nickles
D = number of dimes = N+4
Q = number of quarters
= 25-N-D
= 25-N-(N+4)
= 21-(2N)
Add the coins to get $3.20
(N*$0.05) + (D*$0.10) + (Q*$0.25) = $3.20
Expressing D and Q in terms of N:
(N*$0.05) + ((N+4)*$0.10) + (21-2N)*$0.25=$3.20
Solve for N. Then plug N into the equations for D and Q