GMAT Club Olympics 2024 (Day 6): At the Olympics, each trainer in the

Question

At the Olympics, each trainer in the French team trains the same number of the team's sportsmen as each trainer in the Italian team trains their team's sportsmen. If each sportsmen is trained by exactly one trainer at the Olympics, what is the ratio of sportsmen in the French team to those in the Italian team?

(1) The French team has 50 more sportsmen than the Italian team.
(2) The French team has 10 more trainers than the Italian team.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

Bunuel · Accepted Answer

GMAT Club Official Explanation:

At the Olympics, each trainer in the French team trains the same number of the team's sportsmen as each trainer in the Italian team trains their team's sportsmen. If each sportsmen is trained by exactly one trainer at the Olympics, what is the ratio of sportsmen in the French team to those in the Italian team?

We are given that  \frac{S_f}{T_f} = \frac{S_i}{T_i}  and are asked to find the value of  \frac{S_f}{S_i} .

(1) The French team has 50 more sportsmen than the Italian team.

This implies that  S_f = S_i + 50 , which is clearly insufficient to get the desired ratio.

(2) The French team has 10 more trainers than the Italian team.

This implies that  T_f = T_i + 10 , which is also clearly insufficient to get the desired ratio.

(1)+(2) Substituting information from the statements into  \frac{S_f}{T_f} = \frac{S_i}{T_i}  gives  \frac{S_i + 50}{T_i + 10} = \frac{S_i}{T_i} , which when simplified gives  \frac{S_i}{T_i}=\frac{5}{1} . Hence, we have that  \frac{S_f}{T_f}=\frac{S_i + 50}{T_i + 10} = \frac{S_i}{T_i}=\frac{5}{1} . This is also insufficient to get  \frac{S_f}{S_i} . For example:
 
If  S_i=5  and  T_i=1 , then  S_f=55  and  T_f=11 , making  \frac{S_f}{S_i}=\frac{55}{5}=11 .

If  S_i=10  and  T_i=2 , then  S_f=60  and  T_f=12 , making  \frac{S_f}{S_i}=\frac{60}{10}=6 .

...

Not sufficient.

Answer: E.

divi22 · Answer

Let the no of sportsmen trained by each trainer = x;
Let no. of trainer in French team = Tf;
Let no. of trainer in Italian team = Ti;
And we have to find Tf*x/Ti*x = Tf/Ti (as each sportsmen is trained by exactly one trainer);

According to Statement 1: 
Tf*x = 50 + Ti*x; 
Tf/Ti = 1 + 50/(Ti*x); Which depends on no. of trainers in the Italian team and no of sportsmen trained by each trainer.
So alone, it's not sufficient.

According to Statement 2: 
Tf = 10 + Ti; 
Tf/Ti = 1 + 10/Ti; Which depends on no. of trainers in the Italian team. So alone, it's not sufficient too.

According to Statement 1 & Statement 2: 
Tf*x = 50 + Ti*x;
Tf = 10 + Ti; 
=> (10+Ti)*x = 50 + Ti*x;
=> x = 5; 
Still, Tf/Ti = 1 + 10/Ti; Which depends on no. of trainers in the Italian team. So together, it's not sufficient too.

Ans: E

rg270903 · Answer

At the Olympics, each trainer in the French team trains the same number of the team's sportsmen as each trainer in the Italian team trains their team's sportsmen. If each sportsmen is trained by exactly one trainer at the Olympics, what is the ratio of sportsmen in the French team to those in the Italian team?

(1) The French team has 50 more sportsmen than the Italian team.
(2) The French team has 10 more trainers than the Italian team.

The question states that the ratio of trainers to sportsmen for the 2 teams are equal.
If x is number of trainers and y the number of sportsmen on France team, and X is number of trainers and Y the number of sportsmen on Italy team, then x/y = X/Y

We need to find ratio of sportsmen in the French team to those in the Italian team, which is x:X = x/X

Statement (1) -
The French team has 50 more sportsmen than the Italian team.
Therefore, x = X + 50
The ratio x/X = (X +50) /X

Statement (2) -
The French team has 10 more trainers than the Italian team.
Therefore, y = Y + 10

x/y = X/Y
Cross multiply,
we get xY = Xy
Y(X+50) = X(Y+10)
XY + 50 Y = XY +10 X
50 Y = 10 X
X = 5Y

However, I am not able to reach to a ratio without variables, therefore, my answer is   E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

meet459 · Answer

At the Olympics, each trainer in the French team trains the same number of the team's sportsmen as each trainer in the Italian team trains their team's sportsmen. If each sportsmen is trained by exactly one trainer at the Olympics, what is the ratio of sportsmen in the French team to those in the Italian team?

(1) The French team has 50 more sportsmen than the Italian team.
(2) The French team has 10 more trainers than the Italian team.

Let us assume that 
F = number of sportmen in French team 
I = number of sportmen in Italian team
f = number of trainers in French team
i = number of trainers in Italian team

From the question, 
F/f = I/i

and we need to find F/I or f/i (Since both are same)

Statement (1): F = I + 50, this is clearly  INSUFF 
Statement (2); f = i + 10, this is clearly  INSUFF

Let us consider both options, then we have

(I + 50) / I  = (i + 10) / i

We have I/i = 5 (which is the count each trainer teaches the sportsmen in the team), we still don't know the ratio of F/I
 Hence the answer will be E

Kinshook · Answer

Given: At the Olympics, each trainer in the French team trains the same number of the team's sportsmen as each trainer in the Italian team trains their team's sportsmen.
Asked: If each sportsmen is trained by exactly one trainer at the Olympics, what is the ratio of sportsmen in the French team to those in the Italian team?

Let the sportsmen in French team & sportsmen in Italian team be f & i respectively
Let the trainers be k times the sportsmen which is same for both French and Italian teams.

(1) The French team has 50 more sportsmen than the Italian team.
f = i + 50
The ratio of sportsmen in the French team to those in the Italian team =  \frac{f}{i} =  \frac{i + 50}{i} 
Case 1: i = 20; f/i = 70/20 = 7/2
Case 2: i = 50; f/i = 100/50 = 2
NOT SUFFICIENT

(2) The French team has 10 more trainers than the Italian team.
f/k - i/k = 10
f = i + 10k
The ratio of sportsmen in the French team to those in the Italian team =  \frac{f}{i} =  \frac{i + 10k}{i} 
Case 1: i = 20; k = 5; f/i = 70/20 = 7/2
Case 2: i = 50; k = 5; f/i = 100/50 = 2
NOT SUFFICIENT

(1) + (2) 
(1) The French team has 50 more sportsmen than the Italian team.
f = i + 50
(2) The French team has 10 more trainers than the Italian team.
f/k - i/k = 10
f - i = 10k = 50; k = 5
f = i + 50
Case 1: i = 20; f/i = 70/20 = 7/2
Case 2: i = 50; f/i = 100/50 = 2
NOT SUFFICIENT

IMO E

rsrobin864 · Answer

Let Sf= sportsmen in the French team
Si=sportsmen in the Italian team
Tf=trainer in the French team
Ti=trainer in the Italian team

Given -- Sf/Tf= Si/Ti

Find Sf/Si

1-- Given Sf=50+ Si

So Sf/Si=(50+Si)/Si

So Sf/Si=50/Si+1

Not suff as we need value of Si

2-- Tf=10+Ti

Sf/Si= Tf/Ti= 10+Ti/Ti

So Sf/Si= 10/Ti+1

Not suff as we need value of Ti

1+2-- Not suff as we cant get the value

Ans is E

HarshaBujji · Answer

Let #French Trainers = f; 
#Sportsmen = kf
Similarly, #Italian trainers = i;
#Sportsment = ki

Looking for kf/ki => f/i;

(1) The French team has 50 more sportsmen than the Italian team.
kf = ki + 50;
We cannot find f/i??

(2) The French team has 10 more trainers than the Italian team.
f = i+10;
We cannot find f/i.

Even after using both the equations, we cannot find f/i;

k =  5 from both equations. We cannot get what's required.

Hence  IMO E

pranjalshah · Answer

Answer: E

Let no of trainers for the French and Italian teams be Tf and Ti respectively, and the no of sportspersons be F and I.

Let each trainer train x sportspersons. To find, F/I.

(1) The French team has 50 more sportsmen than the Italian team.

Given, F = I + 50.

F/I = (I+50)/I = ? (insufficient)

(2) The French team has 10 more trainers than the Italian team.

Given, Tf = Ti + 10

We know, F = Tf*x and I = Ti*x

Therefore, F/I = Tf/Ti = (Ti+10)/Ti = ? (insufficient)

(1) and (2) together:

Tf = Ti + 10

F = I + 50 => x*Tf = x*Ti + 50, comparing this with the above equation, we can say that x = 5.

But these 2 equations don't give us the values for I or Ti. Therefore, insufficient.

Lizaza · Answer

At the Olympics, each trainer in the French team trains the same number of the team's sportsmen as each trainer in the Italian team trains their team's sportsmen. If each sportsmen is trained by exactly one trainer at the Olympics, what is the ratio of sportsmen in the French team to those in the Italian team?

(1) The French team has 50 more sportsmen than the Italian team.
(2) The French team has 10 more trainers than the Italian team.

A. Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient to answer the question asked.
B. Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient to answer the question asked.
C. BOTH statements (1) and (2) TOGETHER are sufficient to answer the question asked, but NEITHER statement ALONE is sufficient to answer the question asked.
D. EACH statement ALONE is sufficient to answer the question asked.
E. Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

If each coach trains a sportsmen, then the number of sportsmen for Italy  i  and France  f  is:
 i = ax 
 f = ay, 
where x and y is the number of coaches.

The question is:  f:i = ?

f = i+50 
So we need to calculate  (i+50) : i = ? 
 Insufficient by itself.

y = x + 10 
so  f:i = a(x+10) : ax = (x+10) : x 
 Insufficient by itself.

+   
 f = i+50 
 y = x + 10 
So,  i+50 = a(x+10) = ax + 10a 
and   i = ax 
Subtractng the second one,  50 = 10a  and  a = 5 
However, even inserting it in the equations,  we can't get  to the final value of the ratio.
Basically, we have 4 variables and 2 equations only.

The correct answer is E.

aviraj1703 · Answer

Let,
Sportsmen in French team = Sf
Sportsmen in Italian team = Si
Trainers in French team = Tf
Trainers in Italian team = Ti

According to question, each trainer trains same number of sportsmen in Freanch team as Italian team.
And each sportsmen is trained by only 1 trainer.
That means  Sf > Tf and Si > Ti , also if Tf and Ti trains same number of sportsmen.
Let say x number of sportsmen are being trained by Ti and Tf then

Sf/Tf = Si/Ti ...(i)

1)  Sf = Si + 50, this does not provide any information about Tf and Ti.
So, we can eliminate A, D.

2)  Tf = Ti + 10, this does not provide any information about Sf and Si.
So, eliminate B.

Now combining both we get...

Sf = Si + 50 ...(ii)
Tf = Ti + 10 ...(iii)

Solving these 2 we can only get relation between Sf, Tf and Si, Ti. But we can not determine answer as this will not provide any specific answer.

Hence, eliminate C.

Answer: E.  Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

DG1989 · Answer

At the Olympics, each trainer in the French team trains the same number of the team's sportsmen as each trainer in the Italian team trains their team's sportsmen. If each sportsman is trained by exactly one trainer at the Olympics, what is the ratio of sportsmen in the French team to those in the Italian team?

(1) The French team has 50 more sportsmen than the Italian team.
(2) The French team has 10 more trainers than the Italian team.

Solution:  Let's assume
X as the number of sportsmen in the French team
Y as the number of sportsmen in the Italian team
A as the number of trainers in the French team
B the number of trainers in the Italian team

We need to find  \frac{X}{Y}

As each trainer in the French team trains the same number of the team's sportsmen as each trainer in the Italian team trains their team's sportsmen.
Thus,  \frac{X}{A}  =  \frac{Y}{B}  ------------ (1)

Statement 1:    The French team has 50 more sportsmen than the Italian team 
This means, X = Y + 50
or,  \frac{X}{Y}  = 1 +  \frac{50}{Y}  ------------ (2)
Since we don't know the value of Y
We cannot determine the  \frac{X}{Y} 
  INSUFFICIENT

Statement 2:    The French team has 10 more trainers than the Italian team 
This means, A = B + 10
Substituting the value of A in (1)
 \frac{X}{(B+10)}  =  \frac{Y}{B} 
 \frac{X}{Y}  = 1 +  \frac{10}{B}  ------------ (3)
Since we don't know the value of B
We cannot determine the  \frac{X}{Y} 
  INSUFFICIENT

Combining Statements 1 & 2
X = Y + 50
A = B + 10
From (2) and (3)
1 +  \frac{50}{Y}  = 1 +  \frac{10}{B} 
Y = 5B
Substituting the value of Y in (3)

\frac{X}{Y}  = 1 +  \frac{10}{B} 
 \frac{X}{5B}  = 1 +  \frac{10}{B} 
X = 5B + 50
Thus,  \frac{X}{Y}  =  \frac{(5B + 50)}{5B} 
Since we don't know the value of B
We cannot determine the  \frac{X}{Y} 
  INSUFFICIENT

The correct answer is E

kavyavishnoi02 · Answer

Let's analyze the situation:

- Each trainer trains the same number of sportsmen in both teams.
- Each sportsmen is trained by exactly one trainer.

We want to find the ratio of sportsmen in the French team to those in the Italian team.

Using statement (1) alone:

- French team has 50 more sportsmen than the Italian team.
- We don't know the number of trainers or the number of sportsmen per trainer, so we can't find the ratio.

Using statement (2) alone:

- French team has 10 more trainers than the Italian team.
- We don't know the number of sportsmen or the number of sportsmen per trainer, so we can't find the ratio.

Using both statements together:

- French team has 50 more sportsmen than the Italian team.
- French team has 10 more trainers than the Italian team.

Let's assume the Italian team has x sportsmen and y trainers. Then, the French team has x+50 sportsmen and y+10 trainers.

Since each trainer trains the same number of sportsmen, we can set up an equation:

(x+50) / (y+10) = x / y

Now, this equation has multiple solutions, and without additional information, we can't determine the unique values of x and y.

Therefore, even with both statements, we need additional data to solve the problem, making the  correct answer E .

d_patel · Answer

Question stem tells us that,
1. Each trainer in the French team trains the same number of the team's sportsmen as each trainer in the Italian team trains their team's sportsmen.
2. Each sportsmen is trained by exactly one trainer at the Olympics

If we assume,
Number of french trainers =  F_T 
Number of french sportsmen =  F_S 
Number of italian trainers =  I_T 
Number of italian sportsmen =  I_S

Then we are given that  \frac{F_T}{I_T}  =  \frac{F_S}{I_S}

And we need to find  \frac{F_S}{I_S}

Statement-1 
The French team has 50 more sportsmen than the Italian team.

So,  F_S  =  I_S + 50

\frac{F_S}{I_S}  =  \frac{(I_S + 50) }{ I_S}

We do not know value of  I_S  so ratio could be anything as it will change with value of  I_S .

Statement-1 is not sufficient.

Statement-2 
The French team has 10 more trainers than the Italian team.

So,  F_T  =  I_T + 10

\frac{F_S}{I_S}  =  \frac{(I_T + 10) }{ I_T}

We do not know value of  I_T  so ratio could be anything as it will change with value of  I_T .

Statement-2 is not sufficient.

Combining Statement-1 and Statement-2,  
 \frac{(I_S + 50) }{ I_S}  =  \frac{(I_T + 10) }{ I_T}

(I_T)(I_S + 50) = (I_S)(I_T + 10)

I_T*I_S + 50I_T = I_T * I_S + 10I_S

\frac{I_S }{ I_T} = 5

This does give us relation between  I_S  and  I_T .

But even if we subsitute it in ratio equation, there will always be one unknown variable. So, we cannot get unqiue value for raito. If we had found relation between  F_S  and  I_S  like in this ratio format than we would have found unique value.

Final answer  -  E 
Statements (1) and (2) TOGETHER are NOT sufficient to answer the question asked, and additional data specific to the problem are needed.

captain0612 · Answer

Choice E

Given:
1. Two teams French and Italian
2. Trainers at each team train the same number of sportsmen
3. Each sportsmen is trained by exactly 1 trainer  (no overlaps)

Question: Ratio of sportsmen at French team to those at Italian team

Let, f and i be the number of trainers for the 2 teams French and Italian respectively.

Let Ft and It be the number of sportsmen at the 2 teams French and Italian respectively.

Let x be the number of sportsmen each trainer trains.

=> Ft = f * x and It = i * x

Now, Ratio of sportsmen = (f)(x)/(i)(x) = f/i
= ratio of trainers at respective teams

Statement 1:

French team has  50  more sportsmen than Italian team

Ft = It + 50

f * x = i * x + 50

From, the above 2 equations, we cannot find the ratio of sportmen.  Insufficient

Statement 2:

French team has  10  more trainers than Italian team

f = i + 10

Total sportsmen = (number of trainers) * x (number of sportsmen each trainer trains)

Total sportsmen at French team = f * x = (i + 10) * x
Total sportsmen at Italian team = i * x

Ratio of sportsmen = (i + 10) * x / i * x = (i + 10)/i
Depending on the value of i (trainers at Italian team), ratio changes

from the above again, we cannot find the ratio f/i. Hence  Insufficient

Statement 1 and Statement 2 :

Ft = It + 50

f = i + 10

Total sportsmen at French team = f * x = (i + 10) * x
Total sportsmen at Italian team = i * x

=> (i + 10) * x = i * x + 50
=> i * x + 10 * x = i * x + 50
=> 10 * x = 50 ( i * x is common on both sides )

x = 5. So, each trainer trains exactly 5 sportsmen.

Again, for the total number of sportsment at either team, we need to know the number of trainers at least at 1 team.

Hence, even with both statements we cannot find the ratio E

pianogirl · Answer

We are looking for the ratio of sportsmen in the French team to those in the Italian team: 
 R = \frac{FS}{IS}  
We also know that the number of French sportsmen FS to the number of French trainer FT is equal to that for the Italian team
 \frac{FS}{FT} = \frac{IS}{IT}   =>  \frac{FS}{IS} = \frac{FT}{IT}

(1) The French team has 50 more sportsmen than the Italian team. 
Which means  FS = IS + 50  so  R = 1+\frac{50}{IS} 
We don’t know the number of Italian sportsmen so
 the statement (1) is not sufficient

(2) The French team has 10 more trainers than the Italian team. 
FT= IT +10 =>  R = 1+\frac{10}{IT} 
We don’t know the number of Italian trainers so  the statement (2) is not sufficient

Combining the two statements gives us the ratio of sportsmen to trainers for each team 
 \frac{FS}{FT} = \frac{IS}{IT} = \frac{50}{10} =5

We still don’t have the ratio between teams therefore the right answer is  E

Posted from my mobile device

paradise1234 · Answer

At the Olympics, each trainer in the French team trains the same number of the team's sportsmen as each trainer in the Italian team trains their team's sportsmen. If each sportsmen is trained by exactly one trainer at the Olympics, what is the ratio of sportsmen in the French team to those in the Italian team?

Trainer French = Tf
Sportsmen French = Sf
Trainer Italian = Ti
Sportsmen Italian = Si

Tf:Sf = Ti:Si (given)
Need: Sf:Si

(1) The French team has 50 more sportsmen than the Italian team. 
Sf = Si + 50
Sf:Si will change according to the value of Si
I is insufficient

(2) The French team has 10 more trainers than the Italian team. 
Tf = Ti + 10
Tf:Ti will change according to the value of Ti
II is insufficient

Combining
Sf = Si + 50
Tf = Ti + 10
 \frac{(Ti + 10)}{(Si + 50)} = \frac{(Ti)}{(Si)} 
We cant get any value.

Ans E

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