Andy was on his way to a marble tournament when he met Bob. Andy's bag of marbles was heavy, so he gave Bob 1/2 of his marbles plus 2. Then he met Alex and gave him 1/4 of his remaining marbles plus 5. Just before arriving at the tournament, he met Emily and gave her 1/2 of the marbles he had left. When he got to the tournament, he had 2 marbles left. HOW MANY MARBLES DID HE START WITH?
Let's use the same approach of backward reasoning to arrive at the answer:
When Andy arrived at the tournament, he had 2 marbles left. Just before that, he gave 1/2 of his remaining marbles to Emily. So, if he had 2 marbles left after giving half to Emily, he must have had 4 marbles before giving half to Emily.
Before meeting Emily, Andy gave 1/4 of his remaining marbles plus 5 to Alex. So, if he had 4 marbles left after giving 1/4 to Alex, he must have had (4 - 5) / (1/4) = 16 marbles before giving 1/4 to Alex.
Before meeting Alex, Andy gave Bob 1/2 of his marbles plus 2. So, if he had 16 marbles left after giving 1/2 plus 2 to Bob, he must have had (16 - 2) x 2 = 28 marbles before giving 1/2 plus 2 to Bob.
Therefore, Andy started with 28 marbles.
Good job! That's the correct answer. Using backward reasoning can be a helpful strategy to solve problems like this one.
To find the number of marbles Andy started with, we need to work backwards from the number of marbles he had in the end.
Let's break down the problem step by step:
1. Andy gave Bob 1/2 of his marbles plus 2. This means Andy had (1/2)*x + 2 marbles left after giving some to Bob, where x represents the initial number of marbles Andy had.
2. Andy then gave Alex 1/4 of his remaining marbles plus 5. So, after giving marbles to Alex, he had (3/4)*((1/2)*x + 2) + 5 marbles remaining.
3. Finally, Andy gave Emily 1/2 of the marbles he had left. This means he had 1/2*[(3/4)*((1/2)*x + 2) + 5] marbles left before arriving at the tournament.
4. We are given that Andy had 2 marbles left at the tournament, so we can set up the equation: 1/2*[(3/4)*((1/2)*x + 2) + 5] = 2.
Now we can solve the equation to find the initial number of marbles Andy had:
1/2*[(3/4)*((1/2)*x + 2) + 5] = 2.
We can start by simplifying the equation step by step:
1/2 * ((3/4)*((1/2)*x + 2) + 5) = 2
1/2 * (3/4 * (1/2 * x + 2) + 5) = 2
(3/8 * (1/2 * x + 2) + 5/2) = 4 [multiply both sides by 2 to get rid of 1/2]
3/8 * (1/2 * x + 2) + 5/2 = 4 [distribute 3/8 into the brackets]
Now, let's simplify the equation further:
(3/8) * (1/2 * x + 2) + 5/2 = 4
(3/16 * x + 3/4) + 5/2 = 4 [multiply 3/8 and 1/2]
(3/16 * x) + (6/16) + (40/16) = 4 [add 3/4 to 5/2 and find common denominator]
(3/16 * x) + (46/16) = 4 [add fractions and simplify]
Next, we need to isolate the variable x:
(3/16 * x) + (46/16) = 4
(3/16 * x) = 4 - (46/16)
(3/16 * x) = (64/16) - (46/16)
(3/16 * x) = 18/16
To solve for x, we need to cancel out the (3/16) by multiplying both sides by the reciprocal of (3/16), which is (16/3):
(16/3) * (3/16 * x) = (16/3) * (18/16)
(1/1 * x) = (288/48) [multiply both sides by (16/3)]
x = 288/48
Finally, let's simplify the fraction:
x = 6
Therefore, Andy started with 6 marbles.