Solving for the Ratio a:b:c When 2a 5b 6c

The problem of solving algebraic equations often involves expressing variables in terms of a common factor. In this case, we are given the equation 2a 5b 6c and need to determine the ratio a:b:c.

Method 1: Using the Least Common Multiple (LCM)

First, let's use the Least Common Multiple (LCM) to solve for the ratio.

Find the LCM of the coefficients 2, 5, and 6. The LCM of 2, 5, and 6 is 30. Divide each coefficient by the LCM and equate: 2a / 30 5b / 30 6c / 30 This implies: a/15 b/6 c/5 Thus, the ratio a:b:c can be determined by multiplying each proportional part by its respective factor: a:b:c 15:6:5

Following this method, we achieve the simplified ratio 15:6:5 for the variables a, b, and c.

Method 2: Equating Algebraic Expressions

Another method involves expressing the variables in terms of a common parameter

Let 2a 5b 6c k. By definition, k is a constant. Implying: a k/2 b k/5 c k/6 Now, the ratio a:b:c can be found by simplifying: a:b:c k/2 : k/5 : k/6 By multiplying each term by the least common multiple (LCM) of the denominators, which is 30: a:b:c 15:6:5

Method 3: Direct Algebraic Manipulation

A third approach involves manipulating the given equations directly.

Given: 2a 5b 6c Implying: a 5/2b c 5/6b Now, the ratio a:b:c can be expressed as: a:b:c 5/2b : b : 5/6b Simplifying the ratios: a:b:c 15:6:5

Conclusion

In conclusion, the process of solving for the ratio a:b:c when 2a 5b 6c can be achieved through various methods including using the LCM, equating algebraic expressions, and direct algebraic manipulation. All these methods lead to the same result: a:b:c 15:6:5.

Related Keywords

a:b:c ratio solving algebraic equations least common multiple (LCM)

References

All mathematical expressions and methods used in this article adhere to standard algebraic principles and can be referred to for further reading and understanding.