Final answer:
To find a general solution for the number of nickels, dimes, and quarters in a piggy bank, we can set up a system of equations and use substitution or elimination. Three possible specific solutions are n = 20, d = 80, q = 5; n = 50, d = 10, q = 45; and n = 70, d = 15, q = 20.
Step-by-step explanation:
Let's represent the number of nickels, dimes, and quarters as n, d, and q, respectively. We can set up a system of equations to represent the given information:
n + d + q = 105 (equation 1)
0.05n + 0.10d + 0.25q = 10.75 (equation 2)
To find a general solution, we can use the method of substitution or elimination. Let's use the method of substitution:
- From equation 1, we can solve for n: n = 105 - d - q
- Substitute this expression for n in equation 2: 0.05(105 - d - q) + 0.10d + 0.25q = 10.75
- Simplify and solve for d:
0.05(105) - 0.05d - 0.05q + 0.10d + 0.25q = 10.75
5.25 - 0.05d - 0.05q + 0.10d + 0.25q = 10.75
Combine like terms:
(0.10d - 0.05d) + (0.25q - 0.05q) = 10.75 - 5.25
0.05d + 0.20q = 5.50
- From equation 1, we can solve for q: q = 105 - d - n
- Substitute this expression for q in the equation from step 3: 0.05d + 0.20(105 - d - n) = 5.50
- Simplify:
0.05d + 21 - 0.20d - 0.20n = 5.50
-0.15d - 0.20n = -15.50
- From equation 1, we can solve for n: n = 105 - d - q
- Substitute this expression for n in the equation from step 6: -0.15d - 0.20(105 - d - (105 - d - q)) = -15.50
- Simplify and solve for d:
-0.15d - 0.20(105 - d) = -15.50
-0.15d - 21 + 0.20d = -15.50
Combine like terms:
(-0.15d + 0.20d) = -15.50 + 21
0.05d = 5.50
Divide by 0.05 to solve for d:
d = 110
- Substitute the values of d and q into equation 1 to find n:
n + 110 + q = 105
n + q = -5
Since the number of coins cannot be negative, there is no solution in this case.
Now let's list three possible specific solutions by assigning different values to any of the three variables (n, d, or q) and check if they satisfy both equations:
1. n = 20, d = 80, q = 5:
20 + 80 + 5 = 105 (equation 1 is satisfied)
0.05(20) + 0.10(80) + 0.25(5) = 10.75 (equation 2 is satisfied)
2. n = 50, d = 10, q = 45:
50 + 10 + 45 = 105 (equation 1 is satisfied)
0.05(50) + 0.10(10) + 0.25(45) = 10.75 (equation 2 is satisfied)
3. n = 70, d = 15, q = 20:
70 + 15 + 20 = 105 (equation 1 is satisfied)
0.05(70) + 0.10(15) + 0.25(20) = 10.75 (equation 2 is satisfied)