In aptitude (algebraic simplification), if the ratio P / Q is equal to 7 for non-zero real numbers P and Q with P not equal to Q, what is the simplified value of the expression (P + Q) / (P - Q)?

Introduction / Context:This aptitude question tests basic algebraic simplification using a given ratio between two variables. Such questions are extremely common in competitive exams because they check whether a student can translate a ratio like P / Q = 7 into equivalent fractional relationships and then manipulate expressions like (P + Q) / (P - Q). The key skill here is to recognise that when a ratio is given, one variable can be expressed in terms of the other, which makes the expression much easier to simplify. No actual numeric values for P and Q are required; only the ratio matters.

Given Data / Assumptions:- P / Q = 7, where P and Q are non-zero real numbers.- P is not equal to Q, so the denominator (P - Q) is non-zero.- We need the simplified value of (P + Q) / (P - Q).- Standard algebraic manipulation rules apply.

Concept / Approach:The main idea is to convert the ratio P / Q = 7 into a relationship between P and Q. From P / Q = 7, we can write P = 7Q. Substituting this into the expression (P + Q) / (P - Q) turns the entire expression into something involving only Q. After substitution, we simplify numerator and denominator separately and then cancel common factors. Since Q is non-zero, we can safely divide both numerator and denominator by Q.

Step-by-Step Solution:Step 1: From P / Q = 7, write P = 7Q.Step 2: Substitute in the numerator: P + Q = 7Q + Q = 8Q.Step 3: Substitute in the denominator: P - Q = 7Q - Q = 6Q.Step 4: So (P + Q) / (P - Q) = (8Q) / (6Q).Step 5: Since Q is non-zero, cancel Q to get 8 / 6.Step 6: Simplify 8 / 6 by dividing numerator and denominator by 2 to obtain 4 / 3.

Verification / Alternative check:We can verify by choosing actual numbers that satisfy P / Q = 7. For example, let Q = 1, then P = 7. Compute (P + Q) / (P - Q) = (7 + 1) / (7 - 1) = 8 / 6 = 4 / 3, which agrees with the simplified result. Any other choice like Q = 2, P = 14 leads to (14 + 2) / (14 - 2) = 16 / 12 = 4 / 3 again, confirming the expression is consistently equal to 4 / 3.

Why Other Options Are Wrong:- 5/3: This would come from a different incorrect manipulation of the ratio or from an arithmetic mistake when simplifying 8 / 6.- 3/5: This inverts the correct ratio and has no algebraic justification from the given condition.- 7/4: This incorrectly mixes the given ratio 7 with the expression but does not follow from correct substitution.- 6/5: This is a random-looking fraction that does not arise from any valid simplification of (P + Q) / (P - Q).

Common Pitfalls:Students sometimes forget to express P in terms of Q, or they substitute incorrectly, such as writing P = Q / 7 instead of P = 7Q. Another common error is to stop at 8 / 6 and not simplify the fraction, which is mathematically acceptable but not usually the final required form in aptitude questions. Mixing up (P + Q) / (P - Q) with (P - Q) / (P + Q) can also produce wrong answers that look reasonable but are incorrect.

Final Answer:The simplified value of (P + Q) / (P - Q) is 4/3.