The chess club and the ballet club of Nahk University were founded at the same time, and they each accept new members at a constant rate.

Question

The chess club accepts
\[5\] new members each week, and it had
\[37\] members after
\[6\] weeks.
The number of members in the ballet club as a function of time (in weeks) is given by the following table of values:
\[\text{Time}\] (weeks)
\[\text{Members}\]
\[3\]
\[28\]
\[10\]
\[70\]
\[17\]
\[112\]
Which club accepts more members each week?
Choose 1 answer:
Choose 1 answer:
(Choice A) The chess club
A
The chess club
(Choice B) The ballet club
B
The ballet club
(Choice C) They both accept the same number of new members each week
C
They both accept the same number of new members each week
Which club had more founding members?
Choose 1 answer:
Choose 1 answer:
(Choice A) The chess club
A
The chess club
(Choice B) The ballet club
B
The ballet club
(Choice C) They both had the same number of founding members
C
They both had the same number of founding members

Answer 1

To determine which club accepts more members each week, we should find the membership growth of the ballet club from the provided table values.

1. Chess Club:

   • The chess club has 5 new members each week.
   • After 6 weeks, it has 37 members.
   • The founding members (M_0) can be calculated as: \[ M_0 + 5 \times 6 = 37 \quad \Rightarrow \quad M_0 + 30 = 37 \quad \Rightarrow \quad M_0 = 37 - 30 = 7 \]
   • Thus, the chess club had 7 founding members.

2. Ballet Club:

   • We need to find the number of members at various weeks and determine the rates.
     • At week 3: 28 members
     • At week 10: 70 members
     • At week 17: 112 members
   • We can find the number of new members accepted each week by calculating the increase in membership.

   From week 3 to week 10: \[ 70 - 28 = 42 \quad \text{members over } (10 - 3) = 7 \text{ weeks} \] \[ \text{Rate for this period} = \frac{42}{7} = 6 \quad \text{members per week} \]

   From week 10 to week 17: \[ 112 - 70 = 42 \quad \text{members over } (17 - 10) = 7 \text{ weeks} \] \[ \text{Rate for this period} = \frac{42}{7} = 6 \quad \text{members per week} \]

   It appears that members are accepted at a constant rate of 6 members per week.

3. Comparing both clubs:

   • Chess club: 5 members per week
   • Ballet club: 6 members per week

Conclusion:

• Which club accepts more members each week?
  (Choice B) The ballet club

Next, we can determine which club had more founding members:

• Chess club founding members: 7
• Ballet club founding members need to be calculated using the first point in the table (week 3): \[ M_0 + 6 \times 3 = 28 \quad \Rightarrow \quad M_0 + 18 = 28 \quad \Rightarrow \quad M_0 = 28 - 18 = 10 \]
• Therefore, the ballet club had 10 founding members.

Conclusion:

• Which club had more founding members?
  (Choice B) The ballet club