A fraction bears the same ratio to 1/27 as 3/7 bears to 5/9. Find the fraction.

Introduction / Context:
We are given a proportion involving a fraction x: x : (1/27) = (3/7) : (5/9). Solving for x uses the property that a : b = c : d implies a/b = c/d, hence a = b * (c/d).

Given Data / Assumptions:

• x / (1/27) = (3/7) / (5/9)
• All values are positive rational numbers.

Concept / Approach:
Compute the right-hand ratio exactly, then multiply by 1/27 to get x. Be careful when dividing by a fraction: (3/7)/(5/9) = (3/7)*(9/5).

Step-by-Step Solution:
(3/7) / (5/9) = (3/7) * (9/5) = 27/35x = (1/27) * (27/35) = 1/35

Verification / Alternative check:
Compute the left ratio with x = 1/35: (1/35)/(1/27) = 27/35, matching the right side.

Why Other Options Are Wrong:
7/45 and 5/21 are unrelated ratios; 45/7 is far too large; 3/35 is close but not equal to the required 1/35.

Common Pitfalls:
Inverting the division incorrectly; canceling 27s prematurely; arithmetic mistakes with complex fractions.

Final Answer:
1 / 35