In ratio and proportion, the two equal ratios a : b and c : d are both given to be 1 : 6. Using this information, find the exact value of the expression (a^2 + c^2) / (b^2 + d^2).

Introduction / Context:
This question checks understanding of ratios and how squaring the terms in proportional relationships affects expressions built from them. By specifying that a : b = c : d = 1 : 6, the problem sets up two similar ratios and then asks for the value of a fractional expression involving squares of these quantities. This is a standard style of question in aptitude tests under the topic of ratio and proportion.

Given Data / Assumptions:

• The ratios a : b and c : d are equal and both equal to 1 : 6.
• So a / b = 1 / 6 and c / d = 1 / 6.
• We need to evaluate (a^2 + c^2) / (b^2 + d^2).
• All quantities a, b, c, and d are assumed to be non zero real numbers.

Concept / Approach:
Since a : b = 1 : 6, we can express a and b in terms of a common positive factor, say k, as a = 1 * k and b = 6 * k. Similarly, since c : d = 1 : 6, we can express c and d as c = 1 * m and d = 6 * m for some positive factor m. Substituting these forms into the expression (a^2 + c^2) / (b^2 + d^2) allows us to factor out common terms and see that the result is independent of k and m. This is a typical method when working with ratios: replace each term by a scaled version of the simplest ratio.

Step-by-Step Solution:
1) From a : b = 1 : 6, write a = k and b = 6k for some non zero real k. 2) From c : d = 1 : 6, write c = m and d = 6m for some non zero real m. 3) Compute a^2 + c^2 = k^2 + m^2. 4) Compute b^2 + d^2 = (6k)^2 + (6m)^2 = 36k^2 + 36m^2. 5) Factor 36 from the denominator: b^2 + d^2 = 36(k^2 + m^2). 6) Form the required expression: (a^2 + c^2) / (b^2 + d^2) = (k^2 + m^2) / [36(k^2 + m^2)]. 7) Since k^2 + m^2 is common to numerator and denominator and is non zero, it cancels out, leaving 1 / 36.

Verification / Alternative check:
Choose simple numerical values that satisfy the given ratio to verify the result. For example, take k = 1 and m = 2. Then a = 1, b = 6, c = 2, and d = 12. Compute a^2 + c^2 = 1^2 + 2^2 = 1 + 4 = 5. Compute b^2 + d^2 = 6^2 + 12^2 = 36 + 144 = 180. The ratio (a^2 + c^2) / (b^2 + d^2) = 5 / 180 = 1 / 36. This matches the algebraic result and confirms that the choice of k and m does not affect the final value.

Why Other Options Are Wrong:
Any value other than 1/36 would imply that the ratio changes with the particular choice of a, b, c, and d, which contradicts the proportional structure. Option b (1/60), option c (1/360), option d (1/600), and option e (1/12) can be ruled out either by the algebraic cancellation or by testing a simple numeric example as above. Only option a matches the consistent ratio obtained from both methods.

Common Pitfalls:
Some learners incorrectly assume that if a : b = 1 : 6, then a^2 : b^2 must be 1 : 36 and mistakenly attempt to combine ratios without carefully forming the full expression (a^2 + c^2) / (b^2 + d^2). Others forget to introduce separate scaling factors for the two equal ratios and instead use the same factor for both, which is not necessary but also not harmful if handled correctly. The safest approach is to express each pair with its own factor and then see which terms cancel.

Final Answer:
The expression (a^2 + c^2) / (b^2 + d^2) evaluates to the constant fraction 1/36.