In a certain game, each player scores either 2 points or 5 points. If

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. 1
B. 3
C. 5
D. 7
E. 9

Kudos for a correct solution.

A. 1
B. 3
C. 5
D. 7
E. 9

hsbinfy · Answer

is it B?

multiple of 2 always even so we need multiples of 5 ending with 0.

i see only one case in which the difference can be minimizd

5*8 +2*5=50
8-5=3

Harley1980 · Answer

We have equation 2n + 5m = 50
We have factor 2 in first number and we have factor 5 in second number.
LCM(2, 5) = 10
So we can try some numbers and we should start from 5 because it will be less list than for 2 
2 * 5 = 10 and n should be equal 20
4 * 5 = 20 and n should be equal 15 
6 * 5 = 30 and n should be equal 10
8 * 5 = 40 and n should be equal 5
10 * 5 = 50 and n should be equal 0

third variant give us the mininal difference 
n - m = 10 - 6 = 4

And there is some mistake in my way of thinking because we don't have such answer )

If we change the task and will seek for difference between m and n
than minimal result will be 8 - 5 = 3
And answer B

Harley1980 · Answer

You've made final calculation as m - n 
But question asks about n - m

AmoyV · Answer

2n+5m=50

Now m will always be even.

m 2 4 6 8
n 20 15 10 5

Lowest |n-m|=3

Answer: B

Bunuel · Answer

KAPLAN OFFICIAL SOLUTION:

The quickest solution is to pick numbers for n and m. Since n = 1 and m = 1 would amount to 7 points, and since we want to minimize the difference between n and m, and since 50/7 is just a bit more than 7, we'll start with values near 7. The key is to discover what values for n, when multiplied by 2 points, will leave a multiple of 5 as the remaining points. The solution turns out to be 5 for n (10 points), which allows for 8 for m (40 points). That's a total of 50 points, and the positive difference between the two values is only 3.

Answer: B.

Konstantin1983 · Answer

Yes i agree. Does it matter if the question asks about n-m or m-n? If about n-m then answer should be 4. If about m-n then 3. Please correct if i am wrong

manishkhare · Answer

If we create an equation then it will be 2n+5m=50 . It implies n=(50-5m)/2 so m is a multiple of 2 .
Similarly m=(50-2n)/5 so n is a multiple of 5 .Minimum difference can be the case when n= 5 and m=2 so difference is 5-2=3. 
Note: Number of players is a integer ,so this approach will work fine.

Harley1980 · Answer

Hello  Konstantin1983 
When we have such formulation "least possible positive difference between n and m" it means both variants  m - n  and  n - m 
And yes, you are right: if question asks exactly about  n - m  than answer will be  4

Konstantin1983 · Answer

Hello Harley1980!
Thanks for reply. Valuable info. I didn't know that. It will help me. Дякую!=))

Harley1980 · Answer

Hello Konstantin1983 . I've found another task that exploit this trick with wording "difference between X and Y". Think it'll be interesting for you ) a-certain-salesman-s-yearly-income-is-determined-by-a-base-126533.html

Konstantin1983 · Answer

Hello Konstantin1983 . I've found another task that exploit this trick with wording "difference between X and Y". Think it'll be interesting for you ) a-certain-salesman-s-yearly-income-is-determined-by-a-base-126533.html Hello Harley1980! Thanks! Really good question! Absolute Value is a trick here

kanusha · Answer

Each Game Players 
 2 n

5 m
Total Points 50
Case m and n are 
1 m - n = 5-2 = 3
m = 0 , n = 1 ; difference 2
m = 1, n = 0 ; difference 5

B

ScottTargetTestPrep · Answer

We can create the equation:

2n + 5m = 50

Since 2n and 50 are even, then 5m must also be even. Since 5 is not even, m must be even. Therefore, m could be 0, 2, 4, 6, 8, 10.

Since we want the least possible difference between n and m, let’s let m = 6, and we have:

2n + 5(6) = 50

2n = 20

n = 10

We see the difference between n and m is 4.

If m = 8, then we have:

2n + 5(8) = 50

2n = 10

n = 5

We see the difference between n and m is 3.

If m = 10, then we have:

2n + 5(10) = 50

2n = 0

n = 0

We see the difference between n and m is 10.

Thus, the smallest possible difference between n and m is 3 (when n = 5 and m = 8).

Answer: B

Kinshook · Answer

(5+2)*7=49
If m = 8; n=5;n-m=3
If m=6;n=10; n-m=4

IMO B

Posted from my mobile device

Ananabanana · Answer

With "x" is the difference between n and m, we have: 2n+5m=50 <-> 2n+5(n+x)=50 <-> 7n+5x=50 <-> n=(50-5x)/7 Now the task becomes: what is x to make n an integer (because n is the number of players who score 2 points, n must be an integer) if x=1: n= 45/7 --> out if x=3: n= (50-15)/7 = 5 (integer) --> correct! Hope this helps

HWPO · Answer

I did it a bit differently:

2n + 5m = 50
=> 2n = 50 - 5m
=> 2n = 5 (10-m).

Now... m MUST be positive, so really the largest m can be is 9 but then the equation is not possible.
So we go to the next number, 8, and then n = 5.
Then we go to 7; the equation is not possible..
Then we go to 6; n = 10 ...
You can technically STOP here because the difference between 8 and 5 (3) is smaller than the difference between 10 and 6 (4).

Bixy34 · Answer

Hi  Bunuel  I solved this quite quickly but wasted a lot of time figuring out if there is an even smaller positive difference case. How do we figure out that there are no more potential cases and stop wasting time? Any suggestions here?

Bunuel · Answer

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. 1
B. 3
C. 5
D. 7
E. 9

Kudos for a correct solution.

A. 1
B. 3
C. 5
D. 7
E. 9

A quick way to know when to stop is to rewrite the equation.

Method 1:

We have 2n + 5m = 50

Since 5m and 50 are multiples of 5, 2n must also be a multiple of 5. Thus n must be a multiple of 5. Let n = 5k.

Then:

2(5k) + 5m = 50
2k + m = 10
m = 10 - 2k

So:

|n - m| = |5k - (10 - 2k)| = |7k - 10|

The closest multiple of 7 to 10 is 7, so k = 1. Thus n = 5 and m = 8, and the difference is 3.

Method 2:

We have 2n + 5m = 50

Since 2n and 50 are even, 5m must be even. Thus m must be even. Let m = 2r.

Then:

2n + 5(2r) = 50
n + 5r = 25
n = 25 - 5r

So:

|n - m| = |25 - 5r - 2r| = |25 - 7r|

The closest multiple of 7 to 25 is 28, so r = 4. Thus m = 8 and n = 5, and the difference is 3.

Answer: B.

TheAdmissionHub · Answer

We can set up the following equation:  2n + 5m = 50

So:  n = \frac{50-5m}{2}   →  n = 25-\frac{5}{2}m  →  m  has to be  even .

To minimize the positive difference  |n−m| , we have to maximize  m   and minimize  n .

If   m = 2, n = 20  →  |n−m|   = 18

If   m = 4, n = 15  →  |n−m|   = 9

If   m = 6, n = 10  →  |n−m|   = 4

If   m = 8, n = 5   →   |n−m|   = 3

If   m = 10, n = 0   →  |n−m|   = 0  (  not positive  )

The least possible positive difference is equal to  3 .
 
  Answer:B

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