Joe and Alan had a total of 33 marbles at first. After Joe lost 3 of his marbles and Alan lost 1/2 of his marbles, they had an equal number of marbles left. How many marbles did Alan have in the end?
J-3 = (33-J)/2
J=13
Alan had 20
Let's solve this problem step-by-step:
Step 1: Start with the given information.
Joe and Alan had a total of 33 marbles at first.
Step 2: Set up equations for the problem.
Let's assume Joe had x marbles initially, and Alan had y marbles initially. We know that:
x + y = 33 (equation 1)
Step 3: Determine what happened next.
After Joe lost 3 marbles and Alan lost 1/2 of his marbles, they had an equal number of marbles left.
Joe had x - 3 marbles left.
Alan had y - (1/2)y = (1/2)y marbles left.
Step 4: Write another equation based on the given information.
Since they had an equal number of marbles left, we can write the equation:
x - 3 = (1/2)y (equation 2)
Step 5: Solve the system of equations.
To solve this system of equations, we can use substitution or elimination method. Let's use the substitution method to solve for y.
From equation 1, we have:
x = 33 - y
Substitute this value of x into equation 2:
33 - y - 3 = (1/2)y
Simplify the equation:
30 = (3/2)y
Multiply both sides by 2/3 to isolate y:
y = 20
Step 6: Determine the number of marbles Alan had in the end.
Therefore, Alan had 20 marbles in the end.
Answer: Alan had 20 marbles in the end.
To solve this problem, let's break it down step by step.
Step 1: Set up equations
Let's assume Joe had x marbles initially, and Alan had y marbles initially. We are told that their total number of marbles is 33, so we can set up the equation: x + y = 33.
Step 2: Determine the remaining marbles
After Joe lost 3 marbles, he now has x - 3 marbles left. After Alan lost 1/2 of his marbles, he now has y - (1/2)y marbles left. We are given that Joe and Alan have an equal number of marbles left, so we can set up the equation: x - 3 = y - (1/2)y.
Step 3: Simplify the equation
To solve for y, we need to eliminate the terms with x. Start by multiplying the second equation by 2 to get rid of the fraction: 2(x - 3) = 2(y - (1/2)y). Simplifying this equation, we get: 2x - 6 = 2y - y.
Step 4: Solve the equation
Now, we can combine like terms: 2x - 6 = y. Rearranging the equation, we get: y = 2x - 6.
Step 5: Substitute back into the first equation
Substitute the expression for y from step 4 into the first equation: x + (2x - 6) = 33. Simplifying this equation, we get: 3x - 6 = 33.
Step 6: Solve for x
To solve for x, add 6 to both sides of the equation: 3x - 6 + 6 = 33 + 6, which simplifies to 3x = 39. Dividing both sides by 3, we get: x = 13.
Step 7: Find y
Substituting the value of x into the expression for y from step 4, we get: y = 2(13) - 6 = 20.
So, Alan had 20 marbles in the end.