Determining the Value of (2A - B) / (2AB) Using Equations

Let's explore a fascinating algebraic equation that involves two variables, A and B. The problem at hand is given by the equation: 20 of AB 50 of B. Our goal is to determine the value of the expression (2A - B) / (2AB) using the given information. We will proceed step by step to find a solution that is both clear and mathematically sound.

Step 1: Simplifying the Given Equation

The provided equation is:

0.20 of AB 0.50 of B

First, we'll simplify this equation to eliminate the decimals. Multiplying both sides by 10, we obtain:

2A - B 5B

Step 2: Isolating A

Next, we need to isolate A in the equation. Let's start by adding 2B to both sides:

2A 5B 2B

Combining like terms on the right side, we have:

2A 3B

We can then solve for A by dividing both sides by 2:

A frac{3}{2}B

Step 3: Substituting and Simplifying the Desired Expression

Now that we have A in terms of B, we can substitute this expression into the desired equation (2A - B) / (2AB) to find its value.

Starting with the numerator (2A - B), we substitute A frac{3}{2}B:

2A - B 2cdotfrac{3}{2}B - B 3B - B 2B

For the denominator (2AB), we use the expression A frac{3}{2}B:

2AB 2cdotfrac{3}{2}Bcdot B 3B^2

Thus, the expression becomes:

frac{2A - B}{2AB} frac{2B}{3B^2} frac{2}{3B}

Simplifying further, we get:

frac{2}{3B} frac{2}{3cdotfrac{3}{2}B} frac{2}{frac{9}{2}B} frac{2cdot2}{9B} frac{4}{9B}

However, since B is a common factor in the numerator and the denominator when evaluated, the simplified form results in:

frac{2}{3B} frac{2}{3cdotfrac{3}{2}B} frac{2}{frac{9}{2}B} frac{2cdot2}{9B} frac{4}{9B} frac{2}{3} frac{1}{2}

Hence, the value of the expression (2A - B) / (2AB) is boxed{frac{1}{2}}.

Conclusion

The detailed steps demonstrate how to simplify and solve algebraic equations to find the desired expression's value. This process involves manipulating and substituting variables to reach the final solution. The key to solving such problems lies in following each step carefully and maintaining consistency throughout the calculation.