Newman2019 wrote:

In a 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 7 point baskets, then what is the range of the possible number of 4 point baskets made by the winning team?"

A) 0-2

B)0-4

C)0-9

D)1-4

E) 1-9

How can you use algebra to solve this problem without working backwards by plugging in the answer choices?

We are told that the majority of points are scored by 7 point baskets.

For this to happen the total points scored by 7 points baskets need to be greater than \(\frac{39+34}{2}\) or \(36.5\)

This is only possible for a minimum of six 7 point baskets or 42 points(shot by 7-point baskets)

Maximum 4 point baskets by winning teamWe need a minimum of 7 point baskets. This will enable the winning team to have the maximum

number of 4 point baskets. The maximum 7 point baskets the losing team can have is 4(giving it

28 of the total 34 points). Now, we have 14 points(two 7 point baskets) of the 39 points. For the

remaining 25 points, we can have 3x + 4y = 25. Testing values - y = 1,x = 7 |

y = 4,x = 3.

Minimum 4 point baskets by winning teamWe need a maximum of 7 point baskets. This will enable the winning team to have the minimum

number of 4 point baskets. The minimum 7 point baskets the losing team can have is 2(giving it

14 of the total 34 points). Now, we can have 28 points(four 7 point baskets) of the 39 points. For

the remaining 7 points, we have one 3-point basket and one 4-point basket.

Let's try three 7-point baskets for both the team. That way 21 points of 39 points for the winning

team will be via 7-point baskets. The remaining 18 points can be scored by six 3-point baskets,

making the total number of 4-point baskets ZERO(which is the minimum 4-point baskets)

Attachment:

Point.JPG [ 33.99 KiB | Viewed 1591 times ]
Therefore, the total range of the 4-point baskets possible made by the winning team is

0-4(Option B)
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