M18-05

Question

A tennis ball factory packages its newly produced balls in either large boxes, which hold 25 balls each, or small boxes, containing 17 balls each. If there are 94 freshly made tennis balls to be stored, and no boxes can be partially filled, what is the minimum number of balls that will remain unboxed?

A. 0
B. 1
C. 2
D. 3
E. 4

Show Answer

Official Answer

B

A. 0
B. 1
C. 2
D. 3
E. 4

Bunuel · Answer

Official Solution:

A tennis ball factory packages its newly produced balls in either large boxes, which hold 25 balls each, or small boxes, containing 17 balls each. If there are 94 freshly made tennis balls to be stored, and no boxes can be partially filled, what is the minimum number of balls that will remain unboxed?

A. 0
B. 1
C. 2
D. 3
E. 4

Show Answer

Official Answer

B

A. 0
B. 1
C. 2
D. 3
E. 4

Is it possible to place all 94 balls with no balls remaining unboxed?
 
• 94 is neither a multiple of 25 nor a multiple of 17.
 
• If we use one 25-ball box, we'll be left with 94 - 1(25) = 69 balls. 69 is not a multiple of 17. Hence, this case is not possible.
 
• If we use two 25-ball boxes, we'll be left with 94 - 2(25) = 44 balls. 44 is not a multiple of 17. Hence, this case is not possible either.
 
• If we use three 25-ball boxes, we'll be left with 94 - 3(25) = 19 balls. 19 is not a multiple of 17. Hence, this case is not possible either.
 
Since none of the possible combinations of 17-ball and 25-ball boxes can store all 94 balls without any remaining unboxed, it is not possible to place all 94 balls with no balls left unboxed.
 
Is it possible to place all 93 balls with no balls remaining unboxed?
 
• 93 is neither a multiple of 25 nor a multiple of 17.
 
• If we use one 25-ball box, we'll be left with 93 - 1(25) = 68 balls. 68 is a multiple of 17 (68 = 17 * 4). Hence, this case is possible.
 
• In this case, we would need one 25-ball box and four 17-ball boxes to store all 93 balls without any remaining unboxed.
 
Therefore, the minimum number of balls that will remain unboxed when storing 94 freshly made tennis balls is 1.
 
Essentially, we need to find the non-negative integers x and y such that the sum  25x + 17y  is closest to, but not greater than, 94. It turns out that when  x = 1  and  y = 4 , the sum equals 93, which is 1 less than 94. Therefore, the minimum number of balls that will remain unboxed is 1.

Answer: B

mbadude2017 · Answer

Honestly, just listing multiples of 25 (0, 25, 50, 75) and 17 (0, 17, 34, 51, 68, 85) and eyeballing which led to the best combo worked well. "25+68 looks good. What's the total? 93. Can we do better...nope,", took 45 secs.

jusjmkol740 · Answer

For these kind of Questions, one of the better ways to Organize the data is somewhat like this:

A) Form an Equation: Here, 25x + 17y = 94 (ideally it's less than equal to 94) for your own understanding. Thus, 17y = 94 - 25x ....
B) Then take the bigger no (Here, 25) and write the possible values of 25x upto or less than the required no: Here these will be 0, 25, 50, 75
C) Then subtract these from the required number (i.e. 94): Here these nos will be 94, 69, 44, 19
D) Now take the smaller no (Here, 17) and check the possible values of 17y nearest to the results of step C: Here these will be (85, 68, 34, 17)
E) Subtract the results of Step D from those of Step C to find the Leftover: Here the leftovers are (9, 1, 10, 2)

While it might look complicated, but if you put it in a Table format (attached file) this is Easy,

Thus, least leftover = 1 (Ans: B)

spetznaz · Answer

A good tricky question .... Well explained above !

Bunuel · Answer

I have edited the question and the solution by adding more details to enhance its clarity. I hope it is now easier to understand.

mellapeche · Answer

Without using formulas (and the way I found the fastest): 
 
 25*1=25 + 69 (69 / 17 = 4x, 1 remainder) 
 25*2=50 + 44 (44/17 = 2x, 10 remainder) 
 25*3=75 + 19 (19/17 = 1x, 2 remainder).

Thus the answer is 1 box of 25 and 4 boxes of 17, with 1 tennis ball left over.

MegB07 · Answer

Hi, I used the following approach, please tell where I went wrong:

94/25 give the remainder as 19 then 19/17 gives the remainder as 2, hence 2 balls will remain unboxed.

Bunuel · Answer

Your method doesn’t work here because you treated it as if all balls must first go into 25-ball boxes, and then whatever is left must go into 17-ball boxes. But the problem allows  any  mix of 25-ball and 17-ball boxes.

When you force it into “all 25’s, then 17’s,” you miss valid combinations. For example, 1 large box (25) plus 4 small boxes (4*17 = 68) gives 93, leaving just 1 ball unboxed. That’s better than your remainder 2.

So the mistake is assuming a fixed order of packing. The right approach is to check combinations of 25x + 17y <= 94 and see which gets closest to 94. That’s why the correct minimum leftover is 1, not 2.

agrasan · Answer

I like the solution - it’s helpful.

M18-05

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