Tim and Joe had 354 marbles altogether. Tim lost 84 marbles to Joe in the first game. Joe lost 52 marbles to Tim in the second game. Joe then had twice as many marbles as Tim. How many marbles did Joe have at first?
t+j = 354
2(t-84+52) = j+84-52
solve for j
t = Tim's marbles at the beginning
j = Joe 's marbles at the beginning
Tim and Joe had 354 marbles altogether, means:
t + j = 354
Subtract j to both sides
t = 354 - j
When Tim lost 84 marbles, Tim has:
t - 84 marbles
Replace t by 354 - j
354 - j - 84 = 270 - j
Tim has 270 - j marbles
Joe has:
j + 84 marbles
When Joe lost 52 marbles Joe has:
j + 84 - 52 = j + 32 marbles
Tim has:
270 - j + 52 = 322 - j marbles
Joe then had twice as many marbles as Tim, means:
( j + 32 ) / ( 322 - j ) = 2
Multiply both sides by 322 - j
j + 32 = 2 ( 322 - j )
j + 32 = 644 - 2 j
Add 2 j to both sides.
3 j + 32 = 644
Subtract 32 to both sides.
3 j = 612
j = 612 / 3
j = 204
Joe have 204 marbles at first.
Check result.
t = 354 - j = 354 - 204 = 150
Tim have 150 marbles at first.
When Tim lost 84 marbles to Joe, Tim has:
150 - 84 = 66 marbles
Joe has:
204 + 84 = 288 marbles
When Joe lost 52 marbles Joe has:
288 - 52 = 236 marbles
Tim has:
66 + 52 = 118 marbles
236 / 118 = 2
Let's break down the given information step by step:
1. Tim and Joe had 354 marbles altogether.
- Let's say Tim had x marbles and Joe had y marbles.
- So, we have the equation x + y = 354.
2. Tim lost 84 marbles to Joe in the first game.
- After the first game, Tim's marbles will be reduced by 84: x - 84.
- Joe's marbles will be increased by 84: y + 84.
3. Joe lost 52 marbles to Tim in the second game.
- After the second game, Tim's marbles will be increased by 52: x - 84 + 52 = x - 32.
- Joe's marbles will be reduced by 52: y + 84 - 52 = y + 32.
4. Joe then had twice as many marbles as Tim.
- According to the given information, we have the equation y + 32 = 2(x - 32).
Now, we can solve these two equations simultaneously to find the values of x and y, which represent the initial number of marbles Tim and Joe had:
Equation 1: x + y = 354
Equation 2: y + 32 = 2(x - 32)
Simplifying Equation 2: y + 32 = 2x - 64
Rearranging Equation 1: x = 354 - y
Substituting Equation 1 into Equation 2: y + 32 = 2(354 - y) - 64
Simplifying Equation 2 further: y + 32 = 708 - 2y - 64
Combining like terms: 3y = 612
Solving for y: y = 612 / 3 = 204
Substituting y = 204 into Equation 1: x + 204 = 354
Solving for x: x = 354 - 204 = 150
Therefore, Joe had 204 marbles at first.
To find out how many marbles Joe had at first, we'll need to follow the given information step by step.
Let's assume that Tim had x marbles at first.
So, Joe had (354 - x) marbles at first (since together they had 354 marbles altogether).
In the first game, Tim lost 84 marbles to Joe.
So, Tim had x - 84 marbles after the first game.
And Joe had (354 - x) + 84 marbles after the first game.
In the second game, Joe lost 52 marbles to Tim.
So, Tim had x - 84 + 52 marbles after the second game.
And Joe had (354 - x) + 84 - 52 marbles after the second game.
According to the given information, Joe then had twice as many marbles as Tim.
So, we can write the equation: (354 - x) + 84 - 52 = 2 * (x - 84 + 52).
Let's simplify the equation to solve for x:
(354 - x) + 84 - 52 = 2 * (x - 84 + 52)
(354 - x) + 32 = 2 * (x - 32)
(354 - x) + 32 = 2x - 64
386 - x = 2x - 64
Now, let's solve for x:
386 + 64 = 2x + x
450 = 3x
x = 150
Therefore, Tim had 150 marbles at first.
To find out how many marbles Joe had at first, we can substitute the value of x into the equation:
Joe's initial marbles = 354 - x
= 354 - 150
= 204
Therefore, Joe had 204 marbles at first.