Inequalities
Problem
A bag contains 12 marbles.
There are more red than green but taking the green and blue together exceeds the number of reds.
The total number of yellow and green marbles is more than the total number of red and blue.
Find exactly how many of each colour there are in the bag.
How many different solutions can you find?
Student Solutions
Another crop of good solutions! Robbie said he and his Mum nearly lost their marbles over this one. Quite a number of you reasoned that there are the more yellow marbles than any other colour. All three solutions were found by Josh of Alameda School Ampthill (well done Josh!), by Y9 of the Mount School, York as a team effort, by James of Hethersett High School, who used a spreadsheet to good effect and, from the same school, by Rachel, Geoffrey, and Sarah and also by the Key Stage 3 Maths Club, Strabane Grammar School, N. Ireland. The most carefully reasoned argument covering all possible cases was done by Xin Ying from Tao Nan School, Singapore.
Let R, G, Y and B be the number of Red, Green, Yellow and Blue marbles respectively.
There are 4 conditions to fulfil: (1) R + G + Y + B = 12; (2) R > G; (3) (G+B) > R;
(4) (Y+G)> (R+B). We shall use logical thinking to get the answer.
From condition (3), (R+B) is at most (12 ¸ 2) - 1 = 5.
So (R+B) could be 5, 4, 3, 2, or 1.
Case 1 : (R+B)=1
Impossible as R and B are both not zeros.
Case 2 : R+B = 2
Then, R=1 and B=1. But R> G and G is not zero. So this case is not possible.
Case 3 : (R+B) = 3
Then (R+B) = 1+2 or 2 +1. But it must be that R=2 and B=1 so as to fulfil condition (2). Now, R> G tells us that G=1. But (G+B) is not greater than R. Condition (3) is violated. So this case is not possible.
Case 4 : (R+B) = 4
(a) R+B = 1+3 or 3+1. It must be that R=3 and B=1 from condition (2). Since R> G, G=2 or 1. Either case, (G+B) is not greater than R. So this case is impossible.
(b) R+B=2+2. Then R=2 and B=2. Since R> G, G=1. Then, Y=7. Upon checking, all 4 conditions are fulfilled. This case works!
Case 5 : (R+B)=5
(a) R=2 and B=3. Since R> G, G=1. So, Y=6. Upon checking, all 4 conditions are fulfilled. This case works!
(b) R=3 and B=2. Since R> G, G=2 or 1. If G=1, (G+B) is not greater than R. This case is not possible. So G=2. Then Y=5. Upon checking, all 4 conditions are fulfilled. This case works!
(c) R + B = 4 + 1 (1+4 is impossible). So R=4 and B=1. Since R> G, G = 3, 2 or 1. But either case, (G+B) is not greater than R. This case is impossible.
In short, only 3 situations are possible:
5 yellow marbles, 3 red marbles, 2 blue marbles and 2 green marbles.
7 yellow marbles, 2 red marbles, 2 blue marbles and 1 green marble.
6 yellow, 2 red, 3 blue and 1 green marbles.