Jeff has 12 coins that have a total value of $2.25.If there are only quarters and dimes, how many of each kinds are there?
To solve this problem, let's define variables for the number of quarters and dimes.
Let q represent the number of quarters.
Let d represent the number of dimes.
Based on the given information, we can create the following equations:
1. The number of coins: q + d = 12
2. The total value of all coins: 0.25q + 0.10d = 2.25
To solve these equations, we can use the method of substitution.
From equation 1, we can derive q = 12 - d. We can substitute this value of q into equation 2:
0.25(12 - d) + 0.10d = 2.25
Expanding and simplifying:
3 - 0.25d + 0.10d = 2.25
-0.15d = -0.75
d = -0.75 / (-0.15)
d = 5
Now, substitute the value of d back into equation 1 to find q:
q + 5 = 12
q = 12 - 5
q = 7
Therefore, there are 7 quarters and 5 dimes in Jeff's collection of 12 coins.
To find out how many quarters and dimes there are, you can set up a system of equations based on the given information.
Let's assume that Jeff has q quarters and d dimes.
The number of quarters multiplied by their value (25 cents) plus the number of dimes multiplied by their value (10 cents) should equal the total value of the coins, which is $2.25, or 225 cents.
So, the equation becomes:
25q + 10d = 225
We also know that the total number of coins Jeff has is 12. Therefore, the equation for the total number of coins is:
q + d = 12
Now, we have a system of two equations that can be solved simultaneously. Let's solve it using substitution:
From the second equation, we can rewrite it as q = 12 - d.
Substituting this value of q into the first equation:
25(12 - d) + 10d = 225
300 - 25d + 10d = 225
-15d = -75
d = 5
Now, we can substitute the value of d back into the equation q + d = 12:
q + 5 = 12
q = 12 - 5
q = 7
Therefore, Jeff has 7 quarters and 5 dimes.
Q = 2.5D
10D + 25Q = 225
Substitute 2.5D for Q in second equation and solve for D. Insert that value into the first equation and solve for Q. Check by inserting both values into the second equation.