Please help with workings. Thanks

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# Question

# Answer

Let u be the number of cards Ben has at first.

So John has (240 – u) cards at first.

After the first round, John has 2/3(240 – u) = 160 – (2/3)u cards left.

Ben has u + 1/3(240 – u) = (2/3)u + 80 cards.

After the 2nd round, John has 160 – (2/3)u + 1/4((2/3)u + 80)

= 160 + 20 – (2/3)u +(2/12)u

= 180 – (2/3)u + (1/6)u

= 180 – (4/6)u + (1/6)u

= 180 – (3/6)u

= 180 – (1/2)u

Ben has 3/4((2/3)u + 80)

= (1/2)u + 60

After the 3rd round, John has 5/6(180 – (1/2)u)

= 150 – (5/12)u

Ben has (1/2)u + 60 + 1/6(180 – (1/2)u)

= 60 + 30 + (1/2)u – (1/12)u

= 90 + (6/12)u – (1/12)u

= 90 + (5/12)u

Since they both have the same number of cards after 3 rounds,

150 – (5/12)u = 90 + (5/12)u

150 – 90 = (5/12)u + (5/12)u

60 = (10/12)u

u = (12/10)60 = 72

Ben has 72 cards at the beginning.

So John must have 240-72 = 168 cards at the beginning.

The title of the worksheet says “WORK BACKWARDS” so this is what we are going to do.

First make a table to record for each round what happened and the number of card. Everything is happening from John’s perspective (he wins or loses cards) so it is easier.

B | J | Reasoning | |

Start | |||

-1/3 J | |||

Round 1 | |||

+1/4 B | |||

Round 2 | |||

– 1/6 J | |||

Round 3 | 120 | 120 | Equal number so 240/2 = 120 |

Since the cards are just being passed around, the total is still 240 at the end. Therefore each player has 120 each at Round 3.

Next, it is just a matter of filling up the table from the bottom row up.

**WORKING**

B | J | Reasoning | |

Start | 72 | 168 | |

-1/3 J | +56 | -56 | 2/3 J = 112 so 1/3 J = 112/2 = 56 |

Round 1 | 128 | 112 | |

+1/4 B | -32 | +32 | 3/4 B = 96 so 1/4 B = 96/3 = 32 |

Round 2 | 96 | 144 | |

– 1/6 J | +24 | -24 | 5/6 J = 120 so 1/6 J = 120/5 = 24 |

Round 3 | 120 | 120 | Equal number so 240/2 = 120 |

**So Ben had 72 at first and John had 168 at first.**

Things to do when double checking:

- Make sure B + J = 240 at each round.
- Make sure that the signs of the numbers in J column match what happened. E.g. if he -1/3 J, then that row must be negative because he lost the cards.
- The numbers adds up correctly. 72 + 56 = 128, 128 – 32 – 96, etc.