One night, during a sleep over, kolyn couldnt sleep so he went down into his kitchen where he found a bowl full of mangoes. being hungry he took 1/6 of the mangoes. later the same night his parent was hungry and couldnt sleep. they too found the mangoes and took 1/5 of what kolyn had left. still later, his friend ger woke up went to the kitchen and ate 1/4 of the remaining mangoes. even later, kolyns other friend, adriana ate 1/3 of what was then left. finally his last friend bhavana ate 1/2 of what was left leaving only three mangoes for katrina. how many mangoes were originally in the bowl?

18

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Explanation:

There are many ways to do this. One is through trial and error (basically guess a fairly large value and work your way down) but you might be working a while. Another way is to work backwards starting with the info about Katrina and move to Bhavana, etc etc; however, that might be confusing to work backwards. 

The best way I think is to set up a chart like you see in figure 1 (attached as an image). What I've done is set up the 6 rounds in which the fruit is eaten. Round 1 corresponds to Kolyn, round 2 for Kolyn's parent, round 3 for Ger, etc etc. In each round (row), I have the starting amount, the amount eaten, and the ending amount. 

Let's go through the rounds one by one

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Round 1: Kolyn

We start with x as the number of mangoes originally. We won't know what x is until round 6. Kolyn eats 1/6 of this amount so he eats (1/6)*x = x/6 mangoes. After eating x/6 mangoes, there are x-x/6 = 6x/6-x/6 = 5x/6 mangoes left over

The expression 5x/6 will be the starting amount carried to the next round

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Round 2: Kolyn's parent

At the start of this round, we have 5x/6 mangoes. This is after Kolyn eats his share and before the parent gets their share. The parent eats 1/5 of the expression 5x/6. So, (1/5)*(5x/6) = x/6 is the amount the parent eats. How much is left over? Subtract the amount eaten by the starting amount for the current round: 5x/6 - x/6 = 4x/6 = 2x/3

The expression 2x/3 will carry over to the next round

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Round 3: Ger

We start with 2x/3 and Ger eats 1/4 of this amount, so,
(1/4)*(2x/3) = x/6

Subtract this from the starting amount for the current round
2x/3 - x/6 = 4x/6 - x/6 = 3x/6 = x/2

The expression x/2 carries over to the next round

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Round 4: Adriana

Starting amount for the round = x/2

Take 1/3 of this
(1/3)*(x/2) = x/6

Subtract that from the starting amount for the round
x/2 - x/6 = 3x/6 - x/6 = 2x/6 = x/3

The expression x/3 carries over to the next round

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Round 5: Bhavana

Starting amount for the round = x/3

Take 1/2 of this
(1/2)*(x/3) = x/6

Subtract that from the starting amount for the round
x/3 - x/6 = 2x/6 - x/6 = x/6

The expression x/6 carries over to the next round

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Round 6: Katrina

Starting amount for this round = x/6

We know that there are 3 left over by the time Katrina gets to the bowl of mangoes, so x/6 must be equal to 3. 

Solve for x
x/6 = 3
6*x/6 = 6*3
x = 18

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So there are 18 mangoes originally. To confirm this, check out figure 2 (also attached as an image). In this table, I break down the starting amount, the amount eaten and the ending amount. The amounts are done on a per round basis. 

Round 1: Start with 18 mangoes. Kolyn eats 3 mangoes which is 1/6 of 18. There are 18-3 = 15 left over. The 15 carries over to the next round.

Round 2: Start with 15 mangoes. Parent eats 3 mangoes which is 1/5 of 15. There are 15-3 = 12 left over. The 12 carries over to the next round.

Round 3: Start with 12 mangoes. Ger eats 3 mangoes which is 1/4 of 12. There are 12-3 = 9 left over. The 9 carries over to the next round.

Round 4: Start with 9 mangoes. Adriana eats 3 mangoes which is 1/3 of 9. There are 9-3 = 6 left over. The 6 carries over to the next round.

Round 5: Start with 6 mangoes. Bhavana eats 3 mangoes which is 1/2 of 6. There are 6-3 = 3 left over. The 3 carries over to the next round.

Round 6: Start with 3 mangoes. Katrina eats 3 mangoes. There are 3-3 = 0 left over.

The fact that we end up with 3 mangoes by round 6 verifies the answer. 

Side Note: Every person mentioned eats 3 mangoes

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