A bag contains at least one marble of four different colors.

Question

A bag contains at least one marble of four different colors. The bag contains 10 more black marbles than green marbles and twice as many green marbles as blue marbles. If exactly 15 percent of the marbles in the bag are blue marbles, what is the smallest possible number of marbles in the bag?

A) 40
B) 41
C) 50
D) 60
E) 80

Show Answer

Official Answer

D

A) 40
B) 41
C) 50
D) 60
E) 80

roshaun25 · Answer

Is there an official explanation for this one? I calculated this by figuring out that the blue marbles are 15 percent, making the green marbles 30%. This means that combined, the blue and green marbles are 45%. Since there are 4 colors, the remaining 2 account for 55%. We want to minimize the amount of marbles in the bag, which means maximizing the percentage of black marbles since this also maximizes the percentage that the 10 marble difference between black and green accounts for. (i.e. you want the 10 marbles to represent a higher percentage of total marbles).

Therefore, I assigned 54% to the black marbles, leaving 1% for the unknown 4th color. This means that the black marbles, 54% of X (total marbles in the bag) minus the green marbles (which are 30% of X) is the 10 marble difference. 54x/100 - 30x/100 = 10. Solving for this gives us 24x/100 = 10, which solves to X = 41.67. Can someone explain the official solution and how this is wrong?

Dereno · Answer

I will let you know my approach.

The fixed part is blue, let’s take it a b.

Green = 2* blue = 2b

Black = Green + 10 = 2b+10

Fourth colour be X.

Total = b + 2b + 2b +10+ X = 5b+10+X

Given that : Blue makes 15% of the total.

(b/ 5b+10+X ) = (15/100)

20b = 15b + 30 + 3x

5b = 30 + 3x

5b - 3x = 30

To make the equation true, there are two possible ways

x=0 (which cannot be done, as question mentions atleast 1)

x = multiple of 5 ( which is feasible)

if we take x = 5, 5b = 45 , leaving b =9.

x= 5, b=9

Green = 2*b = 18

Black = 2b+10 = 28

Total = 9+18+28+5 =  60

Answer is 60.

Dereno · Answer

Blue = 15%

Green = 2b = 30%

Black = 30%+10

Fourth colour X .

So, Black + X should account for 100% - (15%+30%) = 55%.

30%+10 + x = 55%

10+x should be 25% of Total.

We know x can be minimum 1.

Total = 4*(10+x)

With x =1 , Total= 44

x=2, Total =48

x=3, Total = 52

X= 4, Total = 56

X=5, we get a total = 60. Which matches our option.

Hope this clarifies your doubts.

If the options had been 44,48,52,56. Then would have calculated total, then to 15% for blue and so on. The answer should be an integer.

kevincan · Answer

x blue marbles 
2x green marbles
2x + 10 black marbles 
r red marbles

15% means 3 out of every 20, so x must be a multiple of 3 and the total number of marbles a multiple of 20

Writing x = 3k for some positive integer k, we have a total of 20k marbles .

We get 15k + 10 marbles that are not red

As 20k is greater than 15k + 10, k must be at least 3
So the total must be at least 60

Dereno · Answer

Awesome. Much faster and efficient. 😎. Thanks for sharing.

Flyingdutch1307 · Answer

No nonsense approach

Blue = 15% of Total

hence, total must be a multiple of 15. Therefore, could be 15,30,45,60.....

only 60 fits here, as per the available options

kevincan · Answer

Careful ! The total must be a multiple of 20, not of 15. But I like your thinking

A bag contains at least one marble of four different colors.

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