Which fraction is equivalent to 0.625?

The fraction 5/8 is equivalent to the decimal 0.625 when the fraction is converted to decimal form through division of numerator by denominator (5 ÷ 8 = 0.625) or when the decimal is converted to fraction form by recognizing 0.625 as 625 thousandths then simplifying 625/1000 by dividing both numerator and denominator by their greatest common divisor of 125 to yield 5/8.

A. 5/8

This fraction precisely equals 0.625 through multiple verification pathways demonstrating equivalence mastery. Division verification: 5 ÷ 8 = 0.625 exactly, as 8 into 5.000 yields 0.6 (4.8 remainder 0.2), then 0.02 (0.16 remainder 0.04), then 0.005 (0.04 exactly) terminating after three decimal places. Decimal-to-fraction conversion: 0.625 = 625/1000; GCD of 625 and 1000 is 125; 625 ÷ 125 = 5; 1000 ÷ 125 = 8; reduced form 5/8. Percentage verification: 0.625 × 100% = 62.5%; 5 ÷ 8 = 0.625 × 100% = 62.5%, confirming equivalence. Memory aid recognition: 5/8 is a standard benchmark fraction with well-known decimal equivalent 0.625 frequently encountered in measurement systems (⅝ inch = 0.625 inch). This solution demonstrates comprehensive understanding of rational number equivalence across representations essential for measurement conversion, recipe scaling, financial calculations, and scientific notation where seamless translation between fractional and decimal forms enables flexible problem-solving approaches adapted to contextual requirements.

B. 3/5

This fraction equals 0.6 exactly (3 ÷ 5 = 0.6), falling 0.025 short of 0.625 a difference representing 25 thousandths or precisely 4% of the target value. The error likely originates from approximating 0.625 as 0.6 then selecting the nearest benchmark fraction, revealing reliance on estimation rather than precise equivalence determination. Students might recognize that 0.625 lies between 0.6 (3/5) and 0.666... (2/3) but incorrectly select the lower bound without executing exact conversion. Another plausible pathway involves misreading the decimal as 0.600 or truncating after the tenths place. The 0.025 discrepancy, while seemingly small, represents a meaningful difference in precision-critical contexts such as engineering tolerances (where 0.025 inch = 1/40 inch affects mechanical fit), pharmaceutical dosing (2.5% concentration error), or financial calculations (2.5% interest miscalculation compounds significantly over time). This distractor effectively identifies students who lack fluency with non-terminating versus terminating decimal patterns or who confuse adjacent benchmark fractions without executing verification procedures.

C. 2/3

This fraction equals approximately 0.666... (repeating), exceeding 0.625 by approximately 0.0417 a difference of about 6.7% relative to the target value. The error likely stems from recognizing 0.625 falls between 0.6 and 0.666... then incorrectly selecting the upper bound, or from misapplying fraction conversion procedures (attempting 625/1000 simplification but dividing by incorrect GCD to yield 2/3). Students might execute 625 ÷ 312.5 = 2 and 1000 ÷ 333.33 ≈ 3 through flawed proportional reasoning. The repeating decimal nature of 2/3 (0.666...) fundamentally distinguishes it from terminating decimals like 0.625 a critical conceptual difference rooted in denominator prime factorization (denominators with only 2s and 5s yield terminating decimals; denominators with other prime factors yield repeating decimals). This distractor proves particularly instructive for diagnosing whether students recognize terminating versus repeating decimal patterns and understand their relationship to fraction denominator composition a foundational number theory concept essential for advanced rational number work, decimal expansion prediction, and computational precision awareness.

D. 7/12

This fraction equals approximately 0.5833... (repeating), falling short of 0.625 by approximately 0.0417 symmetrically opposite to option C's error magnitude but in the negative direction. Calculation: 7 ÷ 12 = 0.58333... The error likely originates from miscalculating 625/1000 simplification possibly dividing numerator and denominator by 89.2857... (incorrect GCD approximation) to yield 7/12, or from confusing 0.625 with 7/12's actual value through memory error. Another plausible pathway involves solving 0.625 = x/12 → x = 7.5 then incorrectly rounding down to 7 rather than recognizing the non-integer numerator indicates non-equivalence. The denominator 12 contains prime factor 3 (12 = 2² × 3), guaranteeing a repeating decimal expansion immediately signaling non-equivalence with terminating decimal 0.625 to students fluent in decimal termination principles. This distractor effectively identifies learners who lack systematic fraction-decimal conversion procedures and instead rely on guesswork or incomplete memory of benchmark equivalences without verification capabilities.

Conclusion

The equivalence between 0.625 and 5/8 emerges through rigorous conversion procedures in both directions (fraction-to-decimal and decimal-to-fraction) validated through multiple independent methods ensuring result integrity. This problem reinforces critical rational number literacy essential across mathematical domains: recognizing terminating decimal patterns based on denominator prime factorization (only 2s and 5s yield termination), executing systematic simplification using greatest common divisors, verifying equivalences through bidirectional conversion, and memorizing key benchmark fractions with their decimal equivalents for computational efficiency. Mastery of these integrated competencies proves indispensable for measurement systems (imperial fraction-decimal conversions), financial calculations (interest rate representations), scientific notation (significant figure management), and culinary arts (recipe scaling) where seamless translation between representations enables flexible problem-solving adapted to contextual constraints. The distractors strategically target common error patterns including approximation to adjacent benchmarks, confusion between terminating and repeating decimals, and incomplete simplification procedures highlighting the necessity of verification protocols rather than reliance on estimation or memory alone when precise equivalence determination is required.