I’m addicted to Free Cell, a game on our computer.

At the moment I’ve won 3300 games, which is 60% of the games I've played.

I’d like to be able to say that I had won two-thirds of the games that I had played.

How many games would I have to win in a row to get to the two-thirds winning mark?

To solve this problem the students must read carefully the information they are given, and decide how to use this to find an unknown amount.

An algebraic approach is desirable, but it may not be intuitively used by students.

The algebraic solution given will provide a foundation for a valuable discussion with students.

### The Problem

I’m addicted to Free Cell, a game on our computer. At the moment I’ve won 3300 games, which is 60% of the games I've played. I’d like to be able to say that I had won two-thirds of the games that I had played. How many games would I have to win in a row to get to the two-thirds winning mark?

#### Teaching sequence

- Write 2/3 on the board and ask the students to tell you all they know about it. (For example: it’s a fraction; it means 2 out of 3 parts; it’s the same as 4/6; …)
- Pose the problem to the class. Ask them to retell the problem using their own words to ensure that they all understand what is required.
- As the students solve the problem ask questions that focus on their understanding of percentages and fractions:
*What is a percentage?*

Would you rather have 2/3 or 60% of a chocolate bar? Why?

Show me how you calculate 2/3 of 120 using the calculator.

Tell me how you started the problem. Why did you start in that way?

Are you convinced that you have the correct answer? Why? - Tell the students that they need to record their solutions for display.
- Share solutions.

#### Other Contexts

other electronic games

### Solution

The information given is:

I have won 60% of my games. This number is 3300 so I can find the number of games that I have played. This is because 3300/games played = 60/100. So *games played* = (3300 x 100)/60 = 5500.

I’m going to play some more games and I’m going to win them all. So 2/3 = (3300 + more games) / (5500 + more games). In the table below, I’ll let the fraction on the right be F. This is now set up for a ‘guess and improve’ strategy and the use of a table.

More games played | F | Compared to 2/3 |

100 | 0.6071 | too small – increase games played |

1000 | 0.6615 | too small – increase games played |

2000 | 0.7067 | too large – decrease games played |

1500 | 0.6857 | too large – decrease games played |

1300 | 0.6765 | too large – decrease games played |

1200 | 0.6716 | too large – decrease games played |

1100 | 0.6667 | Bang on! |

So if I play another 1100 games and win them all, then I shall have won two-thirds of all the games I have played.

**Algebraic approach**: We already know that 2/3 = (3300 + more games) / (5500 + more games). So let *m* = *more games* and we then have 2/3 = (330 + *m*)/ (5500 + *m*). Multiplying both sides by 3 and 5500 + *m* gives 2(5500 + *m*) = 3(3300 + *m*),

so 11000 + 2*m* = 9900 + 3*m*,

or *m* = 1100.