Please help....struggling with word problems and please explain how to get the answer....
Max has a collection of 210 coins consisting of nickels, dimes, and quarters. He has twice as many dimes as nickels, and 10 more quarters than dimes. How many coins of each kind does he have?
D = 2N, Q = D+10
N + D + Q = 210
N + 2N + (2N + 10) = 210
Solve for N, then D and Q.
To solve this word problem, we can break it down into smaller steps. Let's go through each step and explain how to get the answer.
Step 1: Assign variables
Start by assigning variables to the unknown values in the problem. In this case, let's call the number of nickels "n", the number of dimes "d", and the number of quarters "q".
Step 2: Write equations
Now we can write equations based on the information given in the problem. We are told that Max has a total of 210 coins, so we can write our first equation as:
n + d + q = 210
We are also told that Max has twice as many dimes as nickels. Mathematically, this can be written as:
d = 2n
Finally, we are told that Max has 10 more quarters than dimes. This can be expressed as:
q = d + 10
Step 3: Solve the system of equations
We now have a system of three equations with three variables. We can solve this system by substitution or elimination. Let's use substitution.
Substitute the second equation (d = 2n) into the first equation (n + d + q = 210):
n + 2n + q = 210
3n + q = 210
Substitute the second equation (d = 2n) into the third equation (q = d + 10):
q = 2n + 10
We now have a system of two equations with two variables:
3n + q = 210
q = 2n + 10
Step 4: Solve for the variables
To solve this system, we can substitute the value of q from the second equation into the first equation:
3n + (2n + 10) = 210
Simplify the equation:
5n + 10 = 210
Subtract 10 from both sides:
5n = 200
Divide both sides by 5:
n = 40
Now we can substitute the value of n back into one of the equations to find d:
d = 2n = 2(40) = 80
Substitute the values of n and d back into the third equation to find q:
q = d + 10 = 80 + 10 = 90
Step 5: Check the solution
To check if the solution is correct, substitute the values of n, d, and q back into the original equations:
n + d + q = 210
40 + 80 + 90 = 210
The equation is true, which means our solution is correct.
Therefore, Max has 40 nickels, 80 dimes, and 90 quarters.