Solving Marble Problems Through Mathematical Logic and Optimization

Marble problems often require clever application of mathematical principles to find the solution. One such problem involves determining the smallest number of yellow marbles in a collection of marbles that can be blue, red, green, or yellow. This article delves into the problem-solving process using mathematical logic and optimization techniques, providing a step-by-step guide to solving a similar problem.

Problem Statement

Yan has a collection of marbles, some of which are blue, some are red, some are green, and the rest are yellow. One third of Yan's marbles are blue, one fourth of them are red, and there are 6 green marbles. What is the smallest number of yellow marbles?

Step-by-Step Solution

Let n be the total number of marbles Yan has. The number of blue marbles is 1/3 × n. The number of red marbles is 1/4 × n.

The total number of marbles is the sum of the blue, red, and green marbles.

n (1/3 × n) (1/4 × n) 6

Combining the terms, we get:

n (4/12 × n) (3/12 × n) 6

n (7/12 × n) 6

Isolating n on one side, we have:

n - (7/12 × n) 6

(5/12 × n) 6

n (6 × 12/5) 72

The number of yellow marbles is the total number of marbles minus the number of blue, red, and green marbles:

Number of yellow marbles 72 - (1/3 × 72) - (1/4 × 72) - 6

Number of yellow marbles 72 - 24 - 18 - 6 24

Therefore, the smallest number of yellow marbles Yan has is 24.

Alternative Solutions

Another approach involves using algebra to express the relationships between the fractions:

T B R G Y

B 1/3 T so T 3B

R 1/4 T so T 4R

G 6.

R 2B - Y - 6 and 3R B Y 6.

And 5B 4Y 24 whence Y 5/4B - 6.

Plugging in values for B starting with 1 and increasing by 1 the smallest value of B which makes this work is B 8. Then Y 5/4(8) - 6 10 - 6 4.

1/3 4/12 are blue 1/4 3/12 are red 7/12 are blue or red so the total number must be divisible by 12.

With 6 green we’ve passed 12 already.

At 24 marbles 8 would be blue 6 would be red 6 would be green and there are 4 marbles left to be yellow.

Conclusion

Solving marble problems like this one requires a systematic approach and the application of mathematical principles. By understanding the relationships between fractions and using algebra, we can determine the smallest number of yellow marbles efficiently. This problem is not only an exercise in logic but also serves as a practical application of mathematical reasoning and optimization techniques.