Evaluate the quotient of recurring decimals Compute (0.142857) ÷ (0.285714) exactly and choose the correct value.

Introduction / Context:Many repeating decimals correspond to simple fractions. Recognizing 0.142857 and 0.285714 as classic recurring forms linked to sevenths makes the division straightforward without long computation.

Given Data / Assumptions:

• 0.142857 = 1/7 (recurring cycle of 142857).
• 0.285714 = 2/7 (the same cycle, shifted).
• We must compute (1/7) ÷ (2/7).

Concept / Approach:Division of fractions a/b ÷ c/d equals (a/b) * (d/c). Here, (1/7) ÷ (2/7) = (1/7) * (7/2) = 1/2. Identifying the recurring-decimal–fraction mapping saves time.

Step-by-Step Solution:0.142857 = 1/7; 0.285714 = 2/7.Compute the quotient: (1/7) ÷ (2/7) = (1/7) * (7/2) = 1/2.

Verification / Alternative check:Decimal check: 0.142857 ÷ 0.285714 ≈ 0.5. Since both are precise recurring forms of sevenths, the exact result is 1/2.

Why Other Options Are Wrong:1/3 and 2 do not match the straightforward fraction division; 10 is much too large; 3/2 inverts the intended ratio.

Common Pitfalls:Forgetting that both decimals are exact fractional forms of sevenths, leading to unnecessary long division.

Final Answer:1/2