Solve the problem. Joe has a collection of nickels and dimes that is worth $6.00. If the number of dimes were doubled and the number of nickels were increased by 6, the value of the coins would be $9.90. How many dimes does he have?
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To solve this problem, let's assign variables to the unknowns. Let's say Joe has x number of nickels and y number of dimes.
From the given information, we can create two equations to represent the total values of the coins:
Equation 1: 0.05x + 0.10y = 6.00
This equation represents the original value of the collection.
Equation 2: 0.05(x + 6) + 0.10(2y) = 9.90
This equation represents the value of the collection when the number of nickels is increased by 6 and the number of dimes is doubled.
Now let's solve the system of equations to find the values of x and y.
Step 1: Rearrange Equation 2 to isolate x:
0.05x + 0.30 + 0.20y = 9.90
0.05x + 0.20y = 9.60
Step 2: Multiply Equation 1 by 0.05 to get rid of the decimal:
0.05(0.05x + 0.10y) = 0.05(6.00)
0.0025x + 0.005y = 0.30
Step 3: Multiply Equation 2 by 20 to get rid of the decimals:
20(0.05x + 0.20y) = 20(9.60)
x + 4y = 192
Step 4: Multiply Equation 1 by 4 to eliminate the x term:
4(0.0025x + 0.005y) = 4(0.30)
0.01x + 0.02y = 1.20
Step 5: Subtract Equation 4 from Equation 3 to eliminate the x term:
x + 4y - 0.01x - 0.02y = 192 - 1.20
0.99x + 3.98y = 190.80
Step 6: Solve for y by multiplying Equation 5 by 99 and subtracting it from Equation 6:
(0.99x + 3.98y) - (0.99x + 1.98y) = 190.80 - 118.80
2y = 72
y = 36
Joe has 36 dimes.
Therefore, the answer to the question is 36 dimes.