In a game of Numberball, teams can score in increments of 3, 4, or 7

Question

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

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

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?

pushpitkc · Accepted Answer

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 team We 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 team We 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) Point.JPG Therefore, the total range of the 4-point baskets possible made by the winning team is 0-4(Option B)

chetan2u · Answer

Majority of (39+34=73) are scored from 7 pointers..
So >73/2 ~>36.5 are scored by 7 pointers...
42 is next greater multiple of 7, so AT LEAST 42 out of 73 are through 7 points..

Let's find the range of winning team ...
1) Max points by 4...
Now the team could have scored at the max 28, which is 4*7 and remaining 6 could be 2*3 pointers..
34=7*4+3*2
So remaining points from 7-pointers that is 42-28=14 must be from the winning team
7x+4y+3z=39.....
We have to maximize y...
If we take X as minimum possible that is 2, 14+4y+3z=39
4y+3z=25.......
 y=\frac{25-3z}{4} 
What is the least value of z such that 25-3z is multiple of 4...
It is 3, so y=(25-3*3)/4=16/4=4

2) min points
39..
7x+4y+3z=39
Now max value of 7 is 5..
7*5+4y+3z=39......4y+3z=4, so y is 1
But if we take 7 pointers as 4
7*4+4y+3z=39.....4y+3z=39-28=11, so y=2 and z=1
But we are looking for min value of y
7*3+4y+3z=39.....4y+3z=18..so y can be 0 when z is 6

So range is 0 to 4

B

Foreheadson · Answer

Majority of points must be scored with 7 pointers. and on your second graph number of 7 pointers is 6 and number of 3 pointers is 9. How is 9 bigger tham 6. Oh wait, does the problem mean that not the number of 7 pointers, but actually the total score by 7 pointers was the biggest? If this is the logic then ok, but the problem is written in ambigous language, which needs very very close attention

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Fdambro294 · Answer

(Step 1). Start with the constraint that the majority of the TOTAL POINTS must be 7 pointers

(39 + 34) = 73 total points

50% of these points ———> 36.5

Thus, if we have 5 of the (7point) baskets, we wouldn’t have a majority of the total points.

Thus, we need a MIN of 6 of the (7point) baskets by the two teams together

Winning team has 39

3a + 4b + 7c = 39

Losing team has 34

3x + 4y + 7z = 34

(Step 2) since the MIN number of 4 point baskets is either None or 1, first let’s see if 0 (4point) baskets is possible

We need to fulfill the condition that at least 6 scores must be (7point) scores

The most the losing team can have is 4 of these (z = 4)

7z = 7(4) = 28 ———> leaves us with 34 - 28 = 6, a multiple of 3

y = 2 ———> 3y = 3(2) = 6

Thus, we can have the losing team score 4 of the 6 minimum (7 point) scores that we need

Winning team would thus need to contribute at least 2 of these (7 point) scores. We can try that first.

3a + 4b + 7(2) = 39

3a + 4b = 25

Can we have b = 0? No, 25 is not a multiple of 3

Next, what if c = 3 of the (7 point baskets)

39 - 7(3) = 39 - 21 = 18 points would remain

3a + 4b = 18

B CAN equal 0, because 18 is a multiple of 3

Final tally:

Winning team: 3 of the (7point) scores and 6 of the (3point) scores. ———> total = 21 + 18 = 39

Losing team: 4 of the (7point) scores and 2 of the (3point) scores ———> total = 28 + 6 = 34

And the majority of the total points scored by both sides is from (7point) scores

21 from winning team and 28 from losing team = 49 points ————> which is greater than 50% of the 73 total points scored

Eliminate D and E

(Step 3) can the winning team score 9 of the (4point) scores such that C is the correct answer?

From above, we already know that the winning team must score at minimum 2 of the (7point) scores

3a + 4b + 7(2) = 39

3a + 4b = 25

We can not make a = 0 since 25 is not a multiple of 4

a = 1? 22 remains

a = 2? 19 remains

a = 3? 16 remains ——-> which IS divisible by 4

In which case we can have 4 of the scores be (4points)

Thus we can eliminate answer A.

If we can not find a scenario in which 9 4 point baskets are scored, then the answer must be B

Since the winning team must, at the very least, cover 2 of the (7point) scores, that means there is only 25 points to distribute among (3point) and (4point) scores

3a + 4b = 25

In such a case, if b = 9, we would be way over the allowable points scored by the winning team. Therefore, 9 can NOT be the MAX

Since we found 4 worked, the answer must be B

B

0 of the (4point) scores is possible by the winning team

4 of the (4point) scores is possible by the subbing team

These are the MIN and MAX

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Pragun1710 · Answer

"majority of the total points were scored by way of 7 point baskets". By both teams? By a single team?

No way this comes on the actual exam. Too lengthy and ambiguous.

Adarsh_24 · Answer

When majority is by 7 pointers. Cant it be the case that 7:4:3 are in the ratio 40:30:30 percents
Here it may work, but it doesnt always mean more than half rght?

Vivek1707 · Answer

*** The key language point to note here is that the majority of  TOTAL  is scored by 7 points (not just the winning team).

Now, (39+36)/2 = 36.5 so number of 7s have to be minimum 6
So minimum 42 points coming in from 7 pointers ---> that means 6 units of 7

So to find the range we need to check how to maximize the number of 4s
Firstly, 6 units of 7 cannot all go the losing team since their total is 34
So minimum number of 7 pointers for winning team would be 2. Since remaining 4 units of 7 pointers can fit for the losing team i.e. 7 x 4 = 28 +3 + 3 = 34. So we can have minimum of 2 units of 7 pointers for the winning team

Now check the max and minimum value of the number of units of 4

7 7 | 4 4 4 4 4| = 34 --> but since we need 39 the remaining 5 cannot come from 3 so 5 units of 4  doesn't work

7 7 | 4 4 4 4 | 3 3 3| = 39 i.e. 4 units of 7  works . So max number of units of 4 can be 4

Check if number of units of 4 can be made to 0

If we take 7 7 7 which is 21 and remaining is 18 which can come from 3 3 3 3 --->  this works

Just to ensure that we are  not violating any conditions  ---> we should check whether  3 units of 7 works for the losing team  3 units of 7 would be 21 and since we need 13 more we will need 1 unit of 4 and 3 units of 3 so that would total up to 34 i.e  works. 
So range of terms for 4 would be 0 - 4

GMATINATOR9000 · Answer

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

Show Spoiler

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

A) 0-2
B)0-4
C)0-9
D)1-4
E) 1-9

In a game of Numberball, teams can score in increments of 3, 4, or 7

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In a game of Numberball, teams can score in increments of 3, 4, or 7

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