Five teams—A, B, C, D, and E—each consisting of 15 members, are going on expeditions to
five different locations. Each team includes members from three different skill sets: biologists,
geologists, and explorers. However, the number of members from each skill set varies by team
and each member has only one speciality. The total number of biologists, geologists, and explorers are equal.
The following additional information is available
• Every team has at least 2 members from each of the three skill sets.
• Teams C and D have 6 biologists each, and Team A has 6 geologists.
• Every team except A has more biologists than explorers.
• The number of explorers in each team is distinct and decreases in the order A, B, C, D,
and E.
Question: 1
The number of biologists in team E is ________
Show Hint
For logic puzzles with multiple constraints, tabulation is key. Start by filling in the most definitive information first. Then, use the process of elimination and system of equations to solve for the remaining variables.
Step 1: Understanding the Concept:
This is a logical reasoning and data interpretation problem. We need to systematically use all the given conditions to determine the exact composition of each team. Step 2: Key Formula or Approach:
Create a table to organize the information for the five teams and three skill sets. Use the given constraints to fill in the table and solve for the unknown values using a system of equations. Step 3: Detailed Explanation:
Total members = 5 teams \(\times\) 15 members/team = 75.
The total number of biologists, geologists, and explorers are equal.
Total Biologists = Total Geologists = Total Explorers = 75 / 3 = 25.
Let \(B_i, G_i, E_i\) be the number of biologists, geologists, and explorers in team \(i\). We know \(B_i + G_i + E_i = 15\) and \(B_i, G_i, E_i \ge 2\).
Let's use the explorer information first: \(E_A>E_B>E_C>E_D>E_E\), they are distinct integers, and all \(E_i \ge 2\).
Let the numbers be \(e, e+1, e+2, e+3, e+4\). The sum of explorers is 25.
\[ E_A+E_B+E_C+E_D+E_E = 25 \]
Let \(E_E = x\), where \(x \ge 2\). Then \(E_D=x+1, E_C=x+2, E_B=x+3, E_A=x+4\).
Sum: \( (x+4) + (x+3) + (x+2) + (x+1) + x = 5x + 10 \).
\[ 5x + 10 = 25 \implies 5x = 15 \implies x = 3 \]
So, the number of explorers are: \(E_E=3, E_D=4, E_C=5, E_B=6, E_A=7\).
Now we can build a table and fill in the known values:
From the table, using \(B_i+G_i+E_i=15\):
Team A: \(B_A + 6 + 7 = 15 \implies B_A = 2\).
Team C: \(6 + G_C + 5 = 15 \implies G_C = 4\).
Team D: \(6 + G_D + 4 = 15 \implies G_D = 5\).
Now use the total columns:
Total Biologists: \(B_A + B_B + B_C + B_D + B_E = 25 \implies 2 + B_B + 6 + 6 + B_E = 25 \implies B_B + B_E = 11\). (Eq 1)
Total Geologists: \(G_A + G_B + G_C + G_D + G_E = 25 \implies 6 + G_B + 4 + 5 + G_E = 25 \implies G_B + G_E = 10\). (Eq 2)
We also know for teams B and E:
Team B: \(B_B + G_B + 6 = 15 \implies B_B + G_B = 9\). (Eq 3)
Team E: \(B_E + G_E + 3 = 15 \implies B_E + G_E = 12\). (Eq 4)
Finally, use the condition "Every team except A has more biologists than explorers" (\(B_i>E_i\) for i=B,C,D,E).
Team B: \(B_B>E_B \implies B_B>6\).
Team C: \(B_C>E_C \implies 6>5\). (Satisfied)
Team D: \(B_D>E_D \implies 6>4\). (Satisfied)
Team E: \(B_E>E_E \implies B_E>3\).
We have \(B_B + B_E = 11\) (Eq 1) and the constraint \(B_B>6\).
Since \(B_B\) is an integer, \(B_B\) can be 7, 8, 9, etc.
From Eq 1, \(B_E = 11 - B_B\).
If \(B_B = 7\), \(B_E = 4\). This satisfies \(B_E>3\).
If \(B_B = 8\), \(B_E = 3\). This does not satisfy \(B_E>3\).
If \(B_B>8\), \(B_E<3\), which is also not allowed.
So, the only possibility is \(B_B = 7\) and \(B_E = 4\). Step 4: Final Answer:
The number of biologists in team E is 4.
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Question: 2
The number of teams having more geologists than biologists is ________
Show Hint
When multiple questions are based on the same set of data, solve the data puzzle completely first. Create a clean, final table of all values. The subsequent questions will then become simple look-up or calculation tasks based on that table.
Step 1: Understanding the Concept:
This is a logical reasoning and data interpretation problem. We need to systematically use all the given conditions to determine the exact composition of each team and then answer the specific question. Step 2: Key Formula or Approach:
Create a table to organize the information for the five teams and three skill sets. Use the given constraints to fill in the table and solve for the unknown values. Step 3: Detailed Explanation:
Total members = 5 teams \(\times\) 15 members/team = 75.
The total number of biologists, geologists, and explorers are equal.
Total Biologists = Total Geologists = Total Explorers = 75 / 3 = 25.
Let \(B_i, G_i, E_i\) be the number of biologists, geologists, and explorers in team \(i\).
We know \(B_i + G_i + E_i = 15\) and \(B_i, G_i, E_i \ge 2\).
From the explorer information: \(E_A>E_B>E_C>E_D>E_E\), they are distinct integers, and all \(E_i \ge 2\).
Let the numbers be \(x, x+1, x+2, x+3, x+4\). The sum of explorers is 25.
\[ E_A+E_B+E_C+E_D+E_E = (x+4)+(x+3)+(x+2)+(x+1)+x = 5x+10 = 25 \]
Solving this gives \(5x = 15 \implies x = 3\).
So, the number of explorers are: \(E_A=7, E_B=6, E_C=5, E_D=4, E_E=3\).
We can build a table and fill in the known values. From a full logical deduction (as required for the entire puzzle set), we find the final team compositions:
Now we check which teams have more geologists than biologists (G>B):
Team A: 6>2 (Yes)
Team B: 2<7 (No)
Team C: 4<6 (No)
Team D: 5<6 (No)
Team E: 8>4 (Yes)
There are 2 such teams (A and E). Step 4: Final Answer:
The number of teams having more geologists than biologists is 2.
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Question: 3
The median number of biologists across five teams is ________
Show Hint
Remember the definition of median: it's the middle value of a \textit{sorted} dataset. A common mistake is to find the middle value of the unsorted list. For an even number of data points, the median is the average of the two middle values.
Step 1: Understanding the Concept:
This question requires finding the median of a set of numbers. The median is the middle value of a dataset when it is arranged in ascending or descending order. The data itself comes from the solution to the logic puzzle presented before Q.6. Step 2: Key Formula or Approach:
1. List the number of biologists for each of the five teams from the solved data puzzle.
2. Arrange these five numbers in ascending order.
3. Identify the middle value. For a set with an odd number of values (n), the median is the \((\frac{n+1}{2})^{th}\) value in the sorted list. Step 3: Detailed Explanation:
From the detailed analysis for the puzzle, the number of biologists in each team (A, B, C, D, E) was determined as:
Team A Biologists (\(B_A\)): 2
Team B Biologists (\(B_B\)): 7
Team C Biologists (\(B_C\)): 6
Team D Biologists (\(B_D\)): 6
Team E Biologists (\(B_E\)): 4
The dataset of the number of biologists is \{2, 7, 6, 6, 4\}.
To find the median, we first need to sort this data in ascending order:
\[ 2, 4, 6, 6, 7 \]
There are n=5 data points (an odd number). The median is the middle value, which is the \((\frac{5+1}{2})^{th} = 3^{rd}\) value in the sorted list.
The third value in the list \{2, 4, 6, 6, 7\} is 6. Step 4: Final Answer:
The median number of biologists across the five teams is 6.
