Find the fraction that bears the same ratio to 1/27 as 3/7 does to 5/9, that is, x : (1/27) = (3/7) : (5/9).

Introduction / Context:
This question tests the ability to work with ratios of fractions. We are asked to find a fraction x such that the ratio of x to 1/27 is the same as the ratio of 3/7 to 5/9. This means we must handle ratios of fractions and use proportional reasoning to find x. Such problems are common in aptitude tests and help strengthen comfort with fractional arithmetic and cross multiplication.

Given Data / Assumptions:
• Target fraction = x. • Reference fraction = 1/27. • Required that x : (1/27) = (3/7) : (5/9). • All fractions are positive.

Concept / Approach:
Ratios of fractions can be handled by writing them as division expressions. The ratio (3/7) : (5/9) can be written as (3/7) / (5/9). Similarly, x : (1/27) is x / (1/27). The condition that the two ratios are equal can then be written as x / (1/27) = (3/7) / (5/9). We compute the right-hand side and then solve for x by multiplying both sides by 1/27. The key tools are fraction division and cross multiplication.

Step-by-Step Solution:
Step 1: Write the given condition: x / (1/27) = (3/7) / (5/9). Step 2: Simplify the right-hand side. Division of fractions means (3/7) / (5/9) = (3/7) * (9/5). Step 3: Multiply numerators and denominators: (3 * 9) / (7 * 5) = 27 / 35. Step 4: So we now have x / (1/27) = 27 / 35. Step 5: Multiply both sides by 1/27 to solve for x: x = (1/27) * (27 / 35). Step 6: Simplify (1/27) * (27 / 35). The 27 in numerator and 27 in denominator cancel, giving x = 1 / 35.

Verification / Alternative Check:
We can verify by recomputing both ratios with x = 1/35. First compute x : (1/27) = (1/35) / (1/27) = (1/35) * (27/1) = 27 / 35. Next compute (3/7) : (5/9) as before, which we already simplified to 27 / 35. Since both ratios are equal, the chosen value x = 1/35 satisfies the given condition and is therefore correct.

Why Other Options Are Wrong:
• 5/9 is far larger than 1/35 and would give x / (1/27) = (5/9) / (1/27) = 15, which does not match 27 / 35. • 45/7 and 7/45 are also incorrect; substituting either in place of x leads to ratios that do not equal 27 / 35.

Common Pitfalls:
A common error is to confuse ratio with difference and attempt x − 1/27 instead of x / (1/27). Another mistake is performing the division of fractions incorrectly, for example multiplying instead of inverting the second fraction. Careful handling of fraction division and clear writing of each step helps avoid these mistakes.

Final Answer:
The required fraction that bears the same ratio to 1/27 as 3/7 does to 5/9 is 1/35.