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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.
(1) The French team has 50 more sportsmen than the Italian team.
(2) The French team has 10 more trainers than the Italian team.
29 Jul 2024, 08:17
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Official Solution:
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
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
