A school has 15 teachers and 20 teaching assistants. They have 200 students. What is the ratio of faculty to students?

The ratio of faculty to students equals 7:40 when the total faculty count (15 teachers plus 20 teaching assistants equals 35 faculty members) is compared to the 200 students, then simplified by dividing both terms by their greatest common divisor of 5 to produce the reduced ratio 7:40.

A. 3:20

This ratio would require 30 faculty members for 200 students (since 200 ÷ 20 × 3 = 30), but the actual faculty count is 35 five members higher than required for this proportion. This error likely emerges from omitting teaching assistants from faculty calculations (using only 15 teachers: 15:200 simplifies to 3:40, not 3:20) or miscalculating the simplification process by dividing 35 and 200 by incorrect common factors. Students selecting this option may have divided 15 teachers by 5 to get 3 but incorrectly divided 200 students by 10 to get 20, demonstrating inconsistent application of simplification procedures. The ratio 3:20 actually represents a faculty-to-student proportion of 15%, whereas the correct ratio represents 17.5% a meaningful difference in educational resource allocation contexts where faculty density impacts instructional quality.

B. 4:17

This ratio produces a non-integer faculty count when applied to 200 students (200 ÷ 17 × 4 ≈ 47.06 faculty members), immediately signaling its incompatibility with the discrete whole-number faculty count of 35. This option likely distracts students who confuse faculty-to-student ratio with student-to-faculty ratio (200:35 simplifies to 40:7, not 17:4) or who execute erroneous cross-multiplication during simplification attempts. The presence of prime number 17 in the denominator creates computational complexity that might trap students attempting mental simplification without systematic GCD identification. This ratio represents approximately 23.5% faculty representation substantially higher than the actual 17.5% making it educationally useful for identifying ratio directionality confusion and simplification errors.

C. 3:2

This ratio would require 300 faculty members for 200 students (200 ÷ 2 × 3 = 300), representing a faculty surplus rather than the actual faculty deficit relative to student population. This extreme disproportion immediately signals implausibility in educational contexts where student-to-faculty ratios typically exceed 10:1 even in elite institutions. The error likely originates from completely reversing the ratio components (students:falulty = 200:35 ≈ 40:7, then misreading as 7:40 and further distorting to 3:2 through unstructured approximation) or from misinterpreting the question as asking for student-to-faculty rather than faculty-to-student proportion. This option serves as an extreme distractor testing whether students recognize realistic magnitude relationships in institutional ratios and maintain consistent ratio ordering throughout problem-solving processes.

D. 7:40

This ratio correctly represents the faculty-to-student relationship through precise calculation and proper simplification. Total faculty = 15 teachers + 20 teaching assistants = 35 members. Initial ratio = 35:200. Greatest common divisor of 35 and 200 is 5. Dividing both terms by 5 yields 7:40 the fully reduced form where 7 and 40 share no common factors beyond 1. Verification: 7 ÷ 40 = 0.175 or 17.5% faculty representation; 35 ÷ 200 = 0.175 confirming equivalence. This ratio reflects realistic educational staffing patterns where approximately one faculty member serves every 5.7 students a plausible configuration for specialized instructional settings with teaching assistants supplementing certified teachers. The solution demonstrates mastery of ratio construction (correctly identifying component quantities), ratio simplification (systematic GCD application), and ratio interpretation (understanding proportional relationships in real-world contexts).

Conclusion

The 7:40 ratio emerges through accurate faculty aggregation followed by mathematically rigorous simplification using greatest common divisor methodology. This problem reinforces critical proportional reasoning skills essential across disciplines: correctly identifying quantities comprising ratio components (recognizing both teachers and teaching assistants constitute faculty), maintaining consistent ratio ordering (faculty:students not students:faculty), executing systematic simplification rather than approximation, and interpreting reduced ratios within realistic contextual frameworks. Mastery of these integrated competencies proves indispensable for data analysis, resource allocation modeling, and comparative institutional assessment where ratio literacy enables sophisticated quantitative interpretation. The common error of omitting teaching assistants from faculty calculations highlights the importance of precise terminology interpretation a skill extending beyond mathematics into legal, medical, and technical domains where definitional precision determines analytical accuracy.