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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


 


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GMAT Club 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 3/4 of the ratio of trainees to assistants, thus:

T/N = 3/4 * 3/4 = 9/16

The ratio of assistants to coordinators is twice the ratio of trainees to assistants, thus:

A/C = 2 * 3/4 = 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.
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T:A=3:4
T:N=3/4(3:4)=9:16
A:C=2(3:4)=3:2
T+C<102

combine ratios: T:A:N:C=9:12:16:8
T+C=17x<102
x=5
A+N= (12+16)x
28*5=140
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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?

Let the number of trainees, assistants, nurses and coordinators be t, a, n & c respectively

t/a = 3/4
t/n = 3/4*t/a = 3/4 * 3/4 = 9/16
a/c = 2*t/a = 2*3/4 = 3/2

t: a : n : c = 3 : 4 : 16/3 : 8/3 = 9: 12: 16: 8
t = 9k; a = 12k; n = 16k; c = 8k

t + c = 9k + 8k = 17k < 102; k < 6

Maximum possible combined number of assistants and nurses = 12k + 16k = 28k = 28*5 = 140

IMO C
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Given:
t:a = 3/4
t:n = 3/4(3/4) = 9/16
a:c = 2*3/4 = 3/2

using common multiple m
t =9m, a=12m, n=16m & c= 8m
most important smallest 3- digit with distinct digits is 102 not 123
given: t+c <102
(9+8)m => 17m <102
m<6

then, a+n = 12m+16m = 28*5 = 140
Answer: C
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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

total four roles T:A:N:C
ratio given is

T/A = 3x/4x

ratio of T/N is 3x/4x of T/A

3/4 * 3/4 = 9/16 = T/N

we can get N = 16x/3 ; x can be multiple of 3
x=3y
T= 9y
A= 12y
N= 16y

A/C = 2 * T/A
12y/C = 2 * 9y/12y
C= 8y

given that T+C <100
least value will be
17y<100
y can be 5

A + N
12*5 + 16*5 = 60 + 80 ,140

OPTION C is correct ; 140
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trainee : assistant =3 : 4

trainee : nurse =3.3 : 4.4 = 9 :16

assistant : coordinator =3: 2

hence coordinator:assistant:trainee:nurse = 8:12:9:16

now , trainee + coordinator =9+8 =17
assistant+ nurse=12+16=28

we know that ,smallest 3 digit positive integer = 102

total possible number of assistant and nurse = 102 .28 /17 =140


ans is 140
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T/ A = 9/12

T/N = 9/16

A/ C = 3/2

Combining = T/N/A/C = 9/16/12/8

Given 17x < 102

x = 5

28 * 5 = 140
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T:A= 3:4
T:N=(3/4)(T:A) , 4A=3N or T:N=9:16
A:C=2(T:A) , A:C=3:2
T:A:N:C =9:12:16:8
Or T= 9m; A=12m; N=16m; C=8m
T+C<102
9m+8m<102
17m <102
m< 102/17 ; m<6
Maximum possible value of m is 5.
Max possible value of A+N = 12m+16m=28 m =28(5)=140

C. 140
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Given T:A = 3:4 = T/A = 3/4
T/N = 3/4 (T/A) = 9/16 = 9:16
A/C = 2 (T/A) = 3/2 = 3:2

For T = 9, Multiply T:A with 3 = 9:12
therefore T:N:A = 9:16:12
A:C = 3:2 Multiply with 4 = 12:8

therefore T:N:A:C = 9:16:12:8
Let T = 9x
N = 16x
A = 12x
C = 8x

T+C < 102 = 17x<102 -> x<6
so let us consider x = 5 for max possible people

5 (16+12) = 5 * 28 = 140
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Answer C

Let trainees T= 3x and assistants A= 4x

trainees: nurses=3/4 *3/4= 9/16 so N= 16/9T = 16x/3

hence x must be a multiple of 3.
Let x=3k

then T= 9k, A=12k, N=16k

Also, assistans: cooridantors=2*3/4=3/2 so C=8k

T+C=9k+8k=17k

17k<102 ( because smallest integer with distinct digits)
k<=5

Maximum: 12k+16k=28k= 28(5)=140
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What i did in this question is I wrote all the ratios given and then adjusted all ratio to give appropriate ratio of t:a:n:c

As given: t/a = 3/4
t/n = 3/4 * 3/4; t/n = 9/16
a/c = 2 * 3/4; a/c = 3/2

Now to get the ratio of t,a,n,c, I will convert all the number to satisfy all the ratios:

If t/n = 9/16 then i need to multiply ratio of t/a by 3/3
and NEW t/a = 9/12
to get value of a as 12 we need to multiply a/c by 4/4
NEW a/c = 12/8

now we have = t:a:n:c :: 9:12:16:8

as we were told t+a<102 (smallest 3 digit +ve integer with different digits) then
t+c = 9 + 8 = 17
and 17*6 = 102 - 17 * 5 rejected so the maximum possible value of t+c = 17*5
now multiply a+n by 5 to get maximum value

this would be (12+16) * 5 = 140

IMO C
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T:A = 3:4
T:N = 3/4 (T:A) = 3/4(3/4) = 9:16
A:C = 2(T:A) = 2(3:4) = 6:4 = 3:2

T:A:N:C = 9:12:16:8

Lowest distinct 3 digit number is 102
T+C < 102
9k+8k < 102 = 17k < 102 = k < 6
This means that max k has to be 5 as volunteers can't be decimals

Max (A+N) = 12k+16k = 12(5) + 16(5) = 60 + 80 = 140 (C)
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You take T:A = 3:4 ; T/A =3/4
Then it is given T/N = 3/4 * T/A
So T/N = 9:16
then it is given A/C = 2 (T/A)
So A/C = 3/2

T : A : N : C
3 : 4
9 : 16
3 : 2

Final Ratio
T : A : N : C
9 : 12 : 16 : 8

Then it is given T + C < 102 (Smallest 3 Digit number with distinct digits)
So 9x+8x < 102
17x < 102
x < 6

so for maximum take the highest integer value of x which is 5

we need to find maximum number of A + N
THERFORE 12x +16x = 28x = 28 * 5 = 140

so Ans C
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i am going with option c.
combined ratio - T:A:N:C =9:12:16:8
T+C = 9+8= 17 parts
T+C<102 - 17K<102 = K<6
max integer - 5, A+N = (12=16)K = 28x5 = 140
a incorrect - uses k = 4, but k=5 is still allowed so it is not the max
b incorrect - 128 not a multiple of 28 so it cannot be equal to A+N = 28K
d incorrect - 28x6 but T+C= 17x6 = 102 which is not less than 102
e incorrect - 28x7 giving T+C = 17x7 = 119>102
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Answer: C) 140
T:A = 3:4
The ratio of T:N is 3/4 of the ratio of T:A so 3/4(3/4)=9/16
The ratio of A:C is 2 times the ratio of T:A so 3/4(2)=6/4

T:A = 3:4 = 9:12
T:N = 9:16
A:C = 6:4 = 12:8
T:A:N:C = 9x:12x:16x:8x

We know that 9x + 8x < 102, x < 6, x=5
12(5)+16(5) = 140
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T:A = 3:4
T:N = 3/4 * 3*4 = 9:16
A:C = 2*3/4 = 3:2

multiply T:A by 3 so T:A:N = 9:12:16
now multiply A:C by 4 so T:A:N:C = 9:12:16:8 = 45
Let total volunteers be x
T+C ratio: (17/45)x

smallest 3-digit positive integer with distinct digits = 102 hence (17/45)x < 102 ==> x < 270

What is (A+N)max = ?

A+N ratio: (28/45)x

we know x is positive integer & x < 270. the highest multiple of 45 less than 270 is 225.
28/45 * 225 = 140
C
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Let's synthesize the given ratios to find a single, unified ratio for all four roles: Trainees (\(T\)), Assistants (\(A\)), Nurses (\(N\)), and Coordinators (\(C\)).

We are given the base ratio: \(\frac{T}{A} = \frac{3}{4}\)

From this, we can derive the other two relationships:

(1) \(\frac{T}{N} = \frac{3}{4} \times (\frac{3}{4}) = \frac{9}{16}\)
(2) \(\frac{A}{C} = 2 \times (\frac{3}{4}) = \frac{3}{2}\)

To merge these into one continuous ratio (\(T : A : N : C\)), we need common terms. Let's scale \(T : A\) from \(3 : 4\) to \(9 : 12\) so it perfectly matches the \(9\) in the \(T : N\) ratio.
Scaling \(A : C\) by 4 gives us \(12 : 8\), neatly matching our newly scaled \(A\) value.

This reveals our master unified ratio: \(T : A : N : C = 9 : 12 : 16 : 8\)
This means the actual number of people in each role can be represented as \(9x, 12x, 16x,\) and \(8x\) (where \(x\) is a positive integer).

Now for the constraint: The combined number of trainees and coordinators (\(9x + 8x = 17x\)) must be strictly less than the smallest three-digit positive integer with distinct digits.
The smallest 3-digit number is 100 (digits are not distinct), followed by 101 (digits are not distinct). The smallest one with fully distinct digits is 102 (using 1, 0, and 2).

Setting up the inequality:
\(17x < 102\)
\(x < 6\)

Since \(x\) must be a positive integer, the maximum possible value for \(x\) is 5. *(Note: If you fall for the trap of treating the inequality as "less than or equal to", you will incorrectly get x = 6 and pick option D!)*

Finally, we maximize the combined number of assistants and nurses (\(A + N\)):
\(12x + 16x = 28x\)
Maximum value = \(28 \times 5 =\) 140.

Correct Answer: C
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Firstly I converted all the ratio and proportions into one ratio of all as below:
T:A = 3:4
And T:N = 3/4x 3/4 =9/16 = 9:16
Now, A:C= 2 X 3/4 = 3/2 =3:2
Therefore,
T:A:N= 9:12:16 (I made trainee same by multiplying 3)
for cordinators:
A:C = 3:2
Since A is 12 as per our proportion, just multiply by 4 to bring it to that.
Hence,
T:A:N:C = 9:12:16:8

Secondly,
smallest 3 digist distinct number = 102
so, T+C < 102
Means, 17 parts of total < 102
Add a constant x
17x<102
so x= 5

Finally,
(12+16)x5 =140
Hence, C
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