In tennis, a player must win at least 6 games to win a set. If g is the number of games the player won and s is the number of sets the player won, which of the following inequalities must be true?

A. s ≤ 6g
B. s ≥ 6g
C. 6s ≤ g
D. 6s ≥ g

I thought the answer C and D would cancel out since g is games and number of games is 6. So then I chose A but that answer was wrong. Could you explain the reason for that?

This is kinda a bad question because A isn’t actually wrong, it just misses the point they’re going for in the question. You will always win more games than sets, so s<g will always be true, which means s\leq 6g will also be true. The problem isn’t that it’s not true, it’s that it doesn’t really model what the question wants you to model. And this question would have been fine if it just asked which of inequality is the best model instead of which must be true.

To see what the question is really getting at, it helps to plug in. Let’s say a player won 2 sets, which means they must have won at least 12 games; let’s use 13. So s=2 and g=13.

Pop those into the answer choices:

A. 2\leq 6(13)

B. 2\geq 6(13)

C. 6(2)\leq 13

D. 6(2)\geq 13

As expected, A is true but for the reasons above ignore it. C is also true, and that’s what we’re looking for. Because you need to win at least 6 games to win a set, you can multiply the number of sets you win by 6 and you’ll always be less than or equal to the number of games you won.

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