Four students at a computer fair competed against each other in a space-travel game. Chris scored 2/3 as many points as the winner Pat. Rich scored 1/2 as many points as Chris. Tome scored 100 points, which was 90 points less than Rich's score. How many points did Pat score?

what do we know?

c = 2/3 p
r = 1/2 c
t = 100 = r-90

So, what is p?

p = 3/2 c
= 3/2 (2r) = 3r
= 3(t+90)
= 3(190)
= 570

Pat - x

Chris - 2/3x
Rich - 1/2*2/3x = 1/3x
Tom - 1/3x - 90 = 100

1/3x = 190
x = 570

To find the number of points Pat scored, let's go step by step.

First, we need to determine Rich's score. We are told that Tome scored 100 points, which was 90 points less than Rich's score. So, Rich's score would be 100 + 90 = 190 points.

Next, we are told that Rich scored 1/2 as many points as Chris. We can calculate Chris's score by multiplying Rich's score by 2. Therefore, Chris's score is 190 * 2 = 380 points.

Now, we know that Chris scored 2/3 as many points as the winner Pat. Let's represent Pat's score as P. According to the given information, Chris's score is 2/3 of Pat's score. So, 2/3 * P = 380.

To solve for P, we can multiply both sides of the equation by 3/2 to isolate P:

(P * 2/3) * (3/2) = 380 * (3/2)
P = (380 * 3) / 2
P = 1140 / 2
P = 570

Therefore, Pat scored 570 points in the space-travel game.