explain step by step to middle schoolers:

Question

Three girls play three rounds of a game. In each round, there are two winners and one loser. The girl who loses the round has to double the number of chips that each of the other girls has by giving up some of her own chips. Each girl loses one round. At the end of three rounds, each girl has 40 chips. How many chips did each girl have at the beginning of the game?

Answer 1

This does not seem like a problem for a middle schooler, but let's give it a shot.

Let x = #starting chips for girl who loses the 1st round (1st girl)

Let y = #starting chips for girl who loses the 2nd round (2nd girl)

Let z = #starting chips for girl who loses the 3rd round (3rd girl)

At the end of round 1: y and z will double their chips so they will have respectively 2y and 2z.

x will lose y and z chips (to help her opps double)

At the end of round 1 the chips look like this:

x - y - z , 2y, 2z

For round 2:

1st girl doubles: 2 (x - y - z) ; the expression in parentheses is part of what 2nd girl loses

3rd girl doubles: 2 (2z)

2nd girl takes away the amounts that double the other girls: 2y - (x - y - z) - 2z = -x +3y - z

At the end of round 2:

2x -2y - 2z, -x +3y - z, 4z

Ready for round 3:

1st girl doubles current chips: 2 ( 2x - 2y -2z)

2nd girl doubles: 2 (-x + 3y -z)

3rd girl loses the chips that double the others: 4z - (2x - 2y - 2z) - (-x +3y - z)

At the end of the final round:

1st girl: 4x - 4y - 4z = 40

or, x - y - z = 10

2nd girl: -2x +6y - 2z = 40

or, -x +3y -z = 20

3rd girl (let's simplify: 4z - 2x + 2y +2z + x - 3y + z = -x - y +7z

so:

-x - y +7z = 40

Now we have three equations and three unknowns:

1) x - y - z = 10

2) -x +3y -z =20

3) -x -y +7z =40

Eliminate x from 1 and 2 by adding those two equations:

2y - 2z = 30, y - z = 15

Eliminate x from 1 and 3 by adding those two equations:

-2y +6z = 50; -y +3z = 25

Down to two equations and two unknowns:

y - z = 15

-y + 3z = 25

Add to eliminate y:

2z = 40

z = 20

So, y - 20 = 15; y = 35

And:

x - 35 - 20 = 10

x = 65

Check:

Start: 65, 35, 20

Round 1 (65 loses 35 + 20)

10, 70 40

Round 2 (70 loses 10 + 40)

20, 20, 80

Round 3 (80 loses 20 + 20)

40, 40, 40

That does NOT seem to be a middle school problem!