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07 Aug 2026, 05:17
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At a volunteer clinic, every volunteer is assigned exactly one of four roles: trainee, assistant, nurse, or coordinator. The ratio of trainees to assistants is 3 : 4. The ratio of trainees to nurses is 3/4 of the ratio of trainees to assistants and the ratio of assistants to coordinators is twice the ratio of trainees to assistants. If the combined number of trainees and coordinators is less than the smallest three-digit positive integer with distinct digits, what is the maximum possible combined number of assistants and nurses?
A. 112
B. 128
C. 140
D. 168
E. 196
A. 112
B. 128
C. 140
D. 168
E. 196
07 Aug 2026, 05:17
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Official Solution:
At a volunteer clinic, every volunteer is assigned exactly one of four roles: trainee, assistant, nurse, or coordinator. The ratio of trainees to assistants is 3 : 4. The ratio of trainees to nurses is 3/4 of the ratio of trainees to assistants and the ratio of assistants to coordinators is twice the ratio of trainees to assistants. If the combined number of trainees and coordinators is less than the smallest three-digit positive integer with distinct digits, what is the maximum possible combined number of assistants and nurses?
A. 112
B. 128
C. 140
D. 168
E. 196
Let:
\(T\) = trainees
\(A\) = assistants
\(N\) = nurses
\(C\) = coordinators
First, work out all the ratios.
The ratio of trainees to assistants is:
\(T:A = 3:4\)
The ratio of trainees to nurses is \(\frac{3}{4}\) of the ratio of trainees to assistants, thus:
\(\frac{T}{N} = \frac{3}{4} * \frac{3}{4} = \frac{9}{16}\)
The ratio of assistants to coordinators is twice the ratio of trainees to assistants, thus:
\(\frac{A}{C} = 2 * \frac{3}{4} = \frac{3}{2}\)
So:
\(T:A = 3:4\)
\(T:N = 9:16\)
\(A:C = 3:2\)
Now adjust the ratios so that the matching variables have the same values. Multiply the first ratio by 3 and the third ratio by 4:
\(T:A = 9:12\)
\(T:N = 9:16\)
\(A:C = 12:8\)
So:
\(T:A:N:C = 9:12:16:8\)
The smallest three-digit positive integer with distinct digits is 102. Therefore, the combined number of trainees and coordinators is less than 102:
\(T + C = 9x + 8x < 102\)
\(17x < 102\)
\(x < 6\)
The greatest possible value of \(x\) is 5.
So, the maximum possible combined number of assistants and nurses is:
\(A + N = 12x + 16x = 28x = 28 * 5 = 140\)
Answer: C
At a volunteer clinic, every volunteer is assigned exactly one of four roles: trainee, assistant, nurse, or coordinator. The ratio of trainees to assistants is 3 : 4. The ratio of trainees to nurses is 3/4 of the ratio of trainees to assistants and the ratio of assistants to coordinators is twice the ratio of trainees to assistants. If the combined number of trainees and coordinators is less than the smallest three-digit positive integer with distinct digits, what is the maximum possible combined number of assistants and nurses?
A. 112
B. 128
C. 140
D. 168
E. 196
Let:
\(T\) = trainees
\(A\) = assistants
\(N\) = nurses
\(C\) = coordinators
First, work out all the ratios.
The ratio of trainees to assistants is:
\(T:A = 3:4\)
The ratio of trainees to nurses is \(\frac{3}{4}\) of the ratio of trainees to assistants, thus:
\(\frac{T}{N} = \frac{3}{4} * \frac{3}{4} = \frac{9}{16}\)
The ratio of assistants to coordinators is twice the ratio of trainees to assistants, thus:
\(\frac{A}{C} = 2 * \frac{3}{4} = \frac{3}{2}\)
So:
\(T:A = 3:4\)
\(T:N = 9:16\)
\(A:C = 3:2\)
Now adjust the ratios so that the matching variables have the same values. Multiply the first ratio by 3 and the third ratio by 4:
\(T:A = 9:12\)
\(T:N = 9:16\)
\(A:C = 12:8\)
So:
\(T:A:N:C = 9:12:16:8\)
The smallest three-digit positive integer with distinct digits is 102. Therefore, the combined number of trainees and coordinators is less than 102:
\(T + C = 9x + 8x < 102\)
\(17x < 102\)
\(x < 6\)
The greatest possible value of \(x\) is 5.
So, the maximum possible combined number of assistants and nurses is:
\(A + N = 12x + 16x = 28x = 28 * 5 = 140\)
Answer: C
10 Aug 2026, 11:40
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I didn't say simplify my ratio, so my ratio was 18:24:32:16 then sum was 34x<102 , i got x as 2 and marked 112 as the answer which is wrong. Where did i go wromg?
