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Math Competition Problems - Problem of the Week - 18

Updated: Mar 13, 2021

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? - AMC 8 2012


This is an easy problem once you interpret the question right. So let's see how we can write the equations from the given info. Since the question talks about red, green and blue marbles, let

g - be the number of green marbles

b - be the number of blue marbles

r - be the number of red marbles

Since "all but 6 are red marbles" - the number of green and blue marbles must be 6.

So g+b=6 ---> eqn 1

"all but 8 are green" - which means the number of red and blue marbles must be 8.

So r+b=8 ---> eqn 2

Similarly, all but 4 are blue, means the number of red and blue marbles must be 4

So, r+g=4 ---> eqn 3

Now that we have our 3 equations we can solve them. While there are a variety of ways to solve them the below method is the easiest.

Adding all 3 equations together we get


Dividing the whole equation by 2 we get


So the total number of marbles in the jar should be 9. From here you can find the individual number of marbles, but that's not necessary for this question :)

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Vasudha Uddavan
Vasudha Uddavan
Oct 16, 2020

Absolutely Terry. I started online because my student's parents were busy. Even though they lived just 20 mins away, dropping the kids and picking them up meant a waste of 2 hours! So they wanted to try online and this was 5 years ago and there was no looking back, especially since I moved continents. Try the Continental Math League or Online Math League. does not let you register as an individual. does, but that's the most challenging Elementary competition I've seen.


Hi Vasudha, thank you for sharing another alternative site for me. I am more than happy to continue reading your posts about math competition. I think online class learning is becoming a trend with rapid development and it is boosted by the pandemic as well because lots of schools shut down. I would like to try Math League while keeping Beestar to have better comprehensive information for my son.


Vasudha Uddavan
Vasudha Uddavan
Oct 16, 2020

Hi Terry, that's great. I haven't worked with Beestar much, but I usually do CML (Continental Math League) which can also be taken at home and is low cost too :) Glad to know that you enjoy it. I do have a blog on a lot of other competitions that can be taken at home too. If you want a variety, do check it out


Hi, There, I agree with you that Math competitions are fun and effective for elementary school students. I like to help my son to learn and practice his math skills, and apply math tests for him to test his ability in Math. So I have found Beestar, which has a National Math Competition for kids with low cost, especially it can be taken at home. This is a great option for students to save time in study.

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