Question

In a jar of red, green, and blue marbles, all but $6$ are red marbles, all but $8$ are green, and all but $4$ are blue. How many marbles are in the jar?

$6$

$8$

$9$

$10$

$18$

Answer: C

Concepts:system of equations

Difficulty rating: 1370

Small Hint:

Translate each “all but” statement into a sum of two colors.

Big Hint:

Adding the three equations counts every marble twice.

Video solution:

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Written solution:

Let $r,g,b$ be the number of red marbles, green marbles, and blue marbles respectively. We then know $r+g+b -r=6,$ $r+g+b-g = 8,$ $r+g+b-b = 4$ by the statements given. Adding these equations yields $3(r+g+b) -(r+g+b) = 18.$ This would mean $2(r+g+b) = 18,$ so $r+g+b = 9.$ Therefore, the sum of all of the marbles is $9.$

Thus, the answer is C .

2012 AMC 8 Problem 19