Question

In a certain game, each player scores either 2 points or 5 points. If n players score 2 points and m players score 5 points, and the total number of points scored is 50, what is the least possible positive difference between n and m?

• A. 5
• B. 3
• C. 1
• D. 7

Answer 1

The Correct Option is B

Let's analyze the problem:
● Let n be the number of players who score 2 points. 
● Let m be the number of players who score 5 points. 
We are given that the total number of points scored is 50. So, we can create the equation: 
2n + 5m=50 
Now, we want to find the least possible positive difference between n and m. To do this, we need to minimize n while maximizing m. We should start with the smallest possible value for n, which is 1, and then increment it while keeping the equation valid. 
Starting with n = 1: 
2(1) + 5m = 50 
2 + 5m = 50 
5m = 50-2
5m = 48m 
 \(=\frac{48}{5m}=\) 9.6
Since the number of players must be whole numbers, we can't have 9.6 players scoring 5 points. Therefore, we should increase n to 2: 
2(2) + 5m = 50 
4 + 5m = 50
5m = 50-4
5m = 46m 
\(=\frac{46}{5m}=\) 9.2 
Again, m cannot be a fraction. So, let's increase n to 3: 
2(3) + 5m = 50 
6 + 5m = 50
5m = 50-6
5m = 44m 
\(=\frac{44}{5m}=\) 8.8 
Still not a whole number. Let's try n = 4: 
2(4) + 5m = 50 
8 + 5m = 50
5m = 50-8
5m = 42m 
\(=\frac{42}{5m}=\) 8.4 
Again, m is not a whole number. Let's try n = 5: 
2(5) + 5m = 50 
10 + 5m = 50
5m = 50-10
5m = 40m 
\(\frac{40}{5m}=\)= 8 
Now, we have a whole number for m. So, with n = 5 and m = 8, 
we have: n- m=5-8
=-3
The least possible positive difference between n and m is 3.
The correct option is (B): 3