Question

Yesterday, teddy played a game repeatedly on his computer and won exactly 85% of the games he played. today, teddy played the same game repeatedly, winning every game he played, until his overall winning percentage for yesterday and today was exactly 94%. what is the minimum number of games that he played today?

Answer 1

The equation we use here is:

`(0.85)x +(1.00)y = (0.94)(x + y)`
x is for the number of games he played on first day, y is the games on second day. The coefficients are the decimal form of percentages.

Simplify:

`0.85x + y = 0.94x + 0.94y \\ \\ 0.06y = 0.09x \\ \\ (6y)/(100) = (9x)/(100) \\ \\ 6y = 9x \\ \\ 2y = 3x`

To find x we need to use the ratio. We have 85/100 as the first day's ratio and because the games needs to be whole, we can simplify this as 17/20. So in first day, he played at least 20 games and won 17 of them.

When we plug the 20 into the equation as x:

`2y = 3 * 20 \\ \\ 2y = 60 \\ \\ y = 30`
So he needs to play at least 30 games in order to get his score to 94%.

Answer 2

30 games

Explanation:

As a reduced fraction 85% is equal to 85/100 = 17/20. Therefore, the number of games Teddy played yesterday must have been a multiple of 20. Suppose Teddy played 20m games, so he won 17m games.

Also, suppose Teddy played n games today. As a reduced fraction, 94% is equal to 94/100 = 47/50. Since Teddy won every game he played today, we have

(17m + n)/(20m + n) = 47/50

Cross-multiplying, we get 850m + 50n = 940m + 47n. Then 3n = 90m, so n = 30m. Therefore, the minimum number of games that Teddy played today is 30.