If (2a + 3b) divided by (5a - 3b) equals 23/5, that is (2a + 3b)/(5a - 3b) = 23/5, then what is the value of the ratio b/a?

Introduction / Context:
This is an algebraic equation in which a linear expression in a and b divided by another linear expression in a and b equals a given fraction. We are asked to find the ratio b/a. Such questions are common in aptitude tests where the goal is to practice manipulating algebraic expressions and solving for ratios rather than exact values of the variables.

Given Data / Assumptions:
- The equation is (2a + 3b)/(5a - 3b) = 23/5.
- Variables a and b are non zero so that the denominator 5a - 3b is not zero.
- We must find the value of the ratio b/a.

Concept / Approach:
We start by cross multiplying the given fraction equality to remove the denominators. This yields a linear relation between a and b. Then we rearrange the resulting equation so that the ratio b/a appears explicitly. By factoring out common terms, we can solve for b/a as a simple fraction. The individual values of a and b are not required, only their ratio matters.

Step-by-Step Solution:
Step 1: Start from (2a + 3b)/(5a - 3b) = 23/5. Step 2: Cross multiply to clear denominators: 5(2a + 3b) = 23(5a - 3b). Step 3: Expand both sides: left side = 10a + 15b, right side = 115a - 69b. Step 4: Bring all terms to one side: 10a + 15b - 115a + 69b = 0. Step 5: Combine like terms: (10a - 115a) + (15b + 69b) = -105a + 84b = 0. Step 6: Rearrange to get 84b = 105a. Step 7: Divide both sides by 21 to get 4b = 5a. Step 8: Therefore b/a = 5/4.

Verification / Alternative check:
To verify, choose convenient values satisfying b/a = 5/4, for example a = 4 and b = 5. Substitute into the original expression: 2a + 3b = 2*4 + 3*5 = 8 + 15 = 23, and 5a - 3b = 5*4 - 3*5 = 20 - 15 = 5. Then (2a + 3b)/(5a - 3b) = 23/5 which matches the given fraction exactly. This confirms that the ratio b/a = 5/4 is correct.

Why Other Options Are Wrong:
4/5 is the reciprocal of the correct ratio and would give different values when substituted back into the expression. 20/23 and 23/20 bear no direct relationship to the simplified form of 84/105 and will not satisfy the given equation upon substitution. Only 5/4 preserves the equality (2a + 3b)/(5a - 3b) = 23/5 for suitable non zero values of a and b.

Common Pitfalls:
One common mistake is incorrect expansion or sign errors when moving terms from one side of the equation to the other. Another pitfall is attempting to isolate a or b directly instead of focusing on the ratio b/a. Also, some students may mistakenly simplify 84/105 as 4/5 without checking that it represents b/a, not a/b, which reverses the final answer.

Final Answer:
The required value of the ratio is b/a = 5/4.