# Question

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.

0 Replies 1 Like ✔Accepted Answer

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.

0 Replies 2 Likes