Solving for 2a^2b^2 / b^2 when a/b 5/2

When dealing with algebraic expressions and solving equations, understanding various methods to manipulate and simplify expressions is crucial. This article will walk you through the process of solving for 2a^2b^2 / b^2 given that a/b 5/2. We'll explore multiple methods to arrive at the correct solution. By the end, you will have a clear understanding of each step and the underlying algebraic principles at play.

Introduction to the Problem

Given the equation a/b 5/2, our goal is to find the value of the expression 2a^2b^2 / b^2. This problem requires us to apply the fundamentals of algebraic manipulation and ratio and proportion to simplify the given expression.

Method 1: Direct Substitution

Step 1: Start by recognizing that a/b 5/2. We can write this as:

a/b 5/2

Step 2: Substitute a/b 5/2 directly into the expression 2a^2b^2 / b^2.

2a^2b^2 / b^2 2(a/b)^2 * b^2 / b^2

Step 3: Simplify the expression by dividing b^2 by b^2 (which equals 1) and squaring the fraction a/b.

2(a/b)^2 * b^2 / b^2 2(5/2)^2 * 1

Step 4: Calculate the value of the expression.

2(5/2)^2 * 1 2 * (25/4) 25/2

Step 5: Multiply by 2 to get the final answer.

25/2 27/2 or 13.5

Method 2: Substitution of Equivalent Ratios

Step 1: Given a/b 5/2, we can express a and b in terms of a common variable k.

a/5 b/2 k

Step 2: From the above, a 5k and b 2k.

Step 3: Substitute these values into the expression 2a^2b^2 / b^2.

2a^2b^2 / b^2 2(5k)^2 (2k)^2 / (2k)^2

Step 4: Simplify the expression by dividing (2k)^2 by (2k)^2 (which equals 1).

2(5k)^2 (2k)^2 / (2k)^2 2 * 25k^2 * 4k^2 / 4k^2

Step 5: Further simplify the expression.

2 * 25k^2 * 4k^2 / 4k^2 2 * 25k^2 50k^2

50k^2 / 2k^2 27/2

Step 6: Final answer is 13.5.

Method 3: Substitution Using the Given Ratio

Step 1: Given a/b 5/2, express a in terms of b as a 5b/2.

Step 2: Substitute a 5b/2 into the expression 2a^2b^2 / b^2.

2a^2b^2 / b^2 2(5b/2)^2 b^2 / b^2

Step 3: Simplify the expression by squaring (5b/2).

2(5b/2)^2 b^2 / b^2 2 * 25b^2 / 4 b^2 / b^2

Step 4: Further simplify the expression to find the final value.

2 * 25b^2 / 4 b^2 / b^2 12.5 b^2 / b^2 27/2

Step 5: The final answer is 13.5.

Conclusion

We have successfully solved the problem of finding the value of 2a^2b^2 / b^2 given that a/b 5/2 using multiple methods. Each method provides a step-by-step approach to simplifying and solving the expression, reinforcing the importance of substitution, ratio manipulation, and algebraic simplification in solving such problems. Practice these methods to enhance your problem-solving skills in algebra and ratio and proportion.