Each traveler gives the cab driver a tip using the same nine coins. The total tip is $8.41. How many dimes did the driver get?
To find out how many dimes the driver got, we need to solve a system of equations.
Let's use some variables to represent the number of each type of coin:
Let's say the number of dimes is 'd', and we know that the total tip is $8.41. We can set up an equation using this information:
0.10d + 0.25q + 0.01p = 8.41
We also know that there are nine coins in total, so we can set up another equation:
d + q + p = 9
Now we have a system of equations to solve:
0.10d + 0.25q + 0.01p = 8.41
d + q + p = 9
There are various methods to solve a system of equations, such as substitution or elimination. In this case, let's use the elimination method:
To eliminate 'p' from the system, we can multiply the second equation by 0.01 and subtract it from the first equation:
(0.10d + 0.25q + 0.01p) - (0.01d + 0.01q + 0.01p) = 8.41 - 0.09
0.09d + 0.24q = 8.32
Now we have a new equation:
0.09d + 0.24q = 8.32
d + q + p = 9
Next, let's multiply the first equation by 100 to make the coefficients whole numbers:
9d + 24q = 832
d + q + p = 9
Now we can solve the system of equations using the elimination or substitution method:
Let's multiply the second equation by 9 to make the coefficients of 'd' the same in both equations:
9d + 9q + 9p = 81
d + q + p = 9
Now we can subtract the second equation from the first equation:
(9d + 24q) - (d + q + p) = 832 - 81
8d + 23q - p = 751
Next, we can substitute 'p' using the second equation:
8d + 23q - (9 - d - q) = 751
8d + 23q - 9 + d + q = 751
9d + 24q = 760
Now we have a new equation:
9d + 24q = 760
From here, we can solve this equation to find the values of 'd' and 'q'.