There were 90 marbles in 3 bags, P, Q, and R. Jane moved 6 marbles from Bag P to Bag Q and 5 marbles from Bag Q to Bag R. He then moved 8 marbles from Bag R to Bag P. There were an equal number of marbles in each bag in the end. How many marbles were there in Bag Q at first?

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Equation Setup: Let's denote the initial number of marbles in Bags P, Q, and R as P, Q, and R respectively. We know that the total number of marbles is 90, so we can write the equation: P + Q + R = 90Move Marbles: P to Q: Jane moved 6 marbles from Bag P to Bag Q, so now Bag P has P - 6 marbles, and Bag Q has Q + 6 marbles.Move Marbles: Q to R: Then, Jane moved 5 marbles from Bag Q to Bag R, so Bag Q now has Q + 6 - 5 = Q + 1 marbles, and Bag R has R + 5 marbles.Move Marbles: R to P: Finally, Jane moved 8 marbles from Bag R to Bag P, so Bag R now has R + 5 - 8 = R - 3 marbles, and Bag P has P - 6 + 8 = P + 2 marbles.Equal Marbles in Bags: After all the movements, each bag has an equal number of marbles. Therefore, we can set the expressions for the number of marbles in each bag equal to each other: P + 2 = Q + 1 = R - 3Total Marbles Calculation: Since the total number of marbles is still 90, we can use the equation from the first step and substitute the expressions for P, Q, and R: (P + 2) + (Q + 1) + (R - 3) = 90Marbles Distribution: Simplifying the equation, we get: P + Q + R = 90 P + 2 + Q + 1 + R - 3 = 90 P + Q + R = 90 - 2 - 1 + 3 P + Q + R = 90 This confirms that the total number of marbles remains unchanged.Find Initial Marbles in Q: Now, we know that each bag has an equal number of marbles, so we can divide the total number of marbles by 3 to find out how many marbles each bag has after the movements: 90 marbles / 3 bags = 30 marbles per bagWe can now express the initial number of marbles in Bag Q (which we are trying to find) in terms of the final number of marbles in each bag: Q + 1 = 30Solving for Q, we subtract 1 from both sides of the equation: Q = 30 - 1 Q = 29

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