CAT Question: Numberball

[CoralT842]

In the game of Numberball, teams can score in increments of 3, 4, or 7 points at a time. If the final score of a game was 39 – 34, and the majority of the total points were scored by way of seven-point baskets, then what is the range of the possible number of four-point baskets made by the winning team?

What is the algebraic way of doing this problem?

Thanks!

[Sage Pearce-Higgins]

There is no straightforward algebraic way to solve this problem. You could translate the facts into algebra, i.e. defining:
number of 3-point baskets scored by winning team: a
number of 4-point baskets scored by winning team: b
number of 7-point baskets scored by winning team: c
number of 3-point baskets scored by losing team: d
number of 4-point baskets scored by losing team: e
number of 7-point baskets scored by losing team: f
So, 3a + 4b + 7c = 39, and 3d + 4e + 7f = 34. Also, 7c + 7f > 36. We're asked what ? < b < ?

You can already see that it's become pretty complicated and we're likely to get mixed up trying to substitute these various equations. Importantly, we'd have to go through a lot of the logic that we'd do more easily by keeping the information in words. E.g. if 7c + 7f > 36, then c + f > 5.14, so there must have been at least 6 of the 7-point baskets scored in total. Well, I didn't need the algebra to work that out.

Actually, this question is really there to trap people who can only use algebra. Getting used to working with the answer choices is one of the keys to a good score on quant. Finally, when a problem has integer constraints (i.e. all the variables above are integers), it's likely that algebra isn't going to be productive.