In a collection of nickels, dimes, and quarters worth &6.90, the ratio of the number of nickels to dimes is 3:8. The ratio of the number of dimes to quarters is 4:5. Find the number of each type of coin. Use a coin value chart to help.
well, 4:5 is the same as 8:10
That makes the ratio of N, D, Q 3:8:10
Value = N * .05 + D*.10 + Q*.25
let 3x be the number of nickels, then 8x is the number of dimes, and 10 x is the number of q
value= 3x * .05 + 8x*.10 + 10x*.25
you know value, solve for x, then solve for the number of nickels (3x), dimes, and quarters. Nice problem.
Since the value is $6.90, we have the equation:
6.90 = 3x * 0.05 + 8x * 0.10 + 10x * 0.25
Now, we simplify the equation:
6.90 = 0.15x + 0.80x + 2.50x
6.90 = 3.45x
Now, we solve for x:
x = 6.90 / 3.45
x = 2
Now, we find the number of nickels, dimes, and quarters:
Nickels (3x) = 3 * 2 = 6
Dimes (8x) = 8 * 2 = 16
Quarters (10x) = 10 * 2 = 20
So, there are 6 nickels, 16 dimes, and 20 quarters.
Let's solve for x first:
Value = 3x * 0.05 + 8x * 0.10 + 10x * 0.25
6.90 = 0.15x + 0.80x + 2.50x
6.90 = 3.45x
x = 2
Now substitute x back into the equation to find the number of each type of coin:
Number of nickels = 3x = 3 * 2 = 6
Number of dimes = 8x = 8 * 2 = 16
Number of quarters = 10x = 10 * 2 = 20
Therefore, there are 6 nickels, 16 dimes, and 20 quarters.
To solve this problem, let's use the information given and set up equations to find the number of each type of coin.
Let's assign variables to represent the number of nickels, dimes, and quarters.
Let N represent the number of nickels.
Let D represent the number of dimes.
Let Q represent the number of quarters.
From the given information, we have the following ratios:
The ratio of N to D is 3:8.
The ratio of D to Q is 4:5.
Now, let's use the ratios to set up equations.
1) The ratio of N to D is 3:8, which means that N/D = 3/8.
Cross-multiplying, we get 8N = 3D.
2) The ratio of D to Q is 4:5, which means that D/Q = 4/5.
Cross-multiplying, we get 5D = 4Q.
Now, we also know the value of the coins. The total value is $6.90.
The value of N nickels is 0.05N.
The value of D dimes is 0.10D.
The value of Q quarters is 0.25Q.
Therefore, the equation for the total value is:
0.05N + 0.10D + 0.25Q = 6.90.
Now we have a system of equations:
8N = 3D (Equation 1)
5D = 4Q (Equation 2)
0.05N + 0.10D + 0.25Q = 6.90 (Equation 3)
To solve this system of equations, we will use substitution or elimination.
First, let's solve Equation 1 (8N = 3D) for N:
N = (3/8)D
Now, substitute this value for N in Equation 3 (0.05N + 0.10D + 0.25Q = 6.90):
0.05(3/8)D + 0.10D + 0.25Q = 6.90
Next, simplify and convert the decimals to fractions:
(3/160)D + 0.10D + 0.25Q = 6.90
To eliminate the decimals, let's multiply the entire equation by 160:
3D + 16D + 40Q = 1104
Combine like terms:
19D + 40Q = 1104 (Equation 4)
Now, we have two equations to work with:
5D = 4Q (Equation 2)
19D + 40Q = 1104 (Equation 4)
We can now solve this system of equations to find the values of D and Q.
Using Equation 2, solve for D in terms of Q:
D = (4/5)Q
Now, substitute this value for D in Equation 4:
19((4/5)Q) + 40Q = 1104
Simplify and combine like terms:
(76/5)Q + 40Q = 1104
To eliminate the fractions, multiply the entire equation by 5:
76Q + 200Q = 5520
Combine like terms:
276Q = 5520
Divide both sides by 276:
Q = 5520/276
Q = 20
Now that we know Q = 20, we can substitute this value back into Equation 2 to solve for D:
5D = 4(20)
5D = 80
D = 80/5
D = 16
Finally, substitute the values of D = 16 and Q = 20 back into Equation 1 to solve for N:
8N = 3(16)
8N = 48
N = 48/8
N = 6
Therefore, the number of nickels is 6, the number of dimes is 16, and the number of quarters is 20.