• 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.
The number of biologists in team E is ________
Show Hint
Correct Answer: 4
Solution and Explanation
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:
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.
The number of teams having more geologists than biologists is ________
Show Hint
Correct Answer: 2
Solution and Explanation
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:
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.
The median number of biologists across five teams is ________
Show Hint
Correct Answer: 6
Solution and Explanation
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.
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CONVERSATION ANALYSIS: Read the following transcript and choose the answer that is closest to each of the questions that are based on the transcript.
Lucia Rahilly (Global Editorial Director, The McKinsey Podcast): Today we’re talking about the next big arenas of competition, about the industries that will matter most in the global business landscape, which you describe as arenas of competition. What do we mean when we use this term?
Chris Bradley (Director, McKinsey Global Institute): If I go back and look at the top ten companies in 2005, they were in traditional industries such as oil and gas, retail, industrials, and pharmaceuticals. The average company was worth about $250 billion. If I advance the clock forward to 2020, nine in ten of those companies have been replaced, and by companies that are eight times bigger than the old guards.
And this new batch of companies comes from these new arenas or competitive sectors. In fact, they’re so different that we have a nickname for them. If you’re a fan of Harry Potter, it’s wizards versus muggles.
Arena industries are wizardish; we found that there’s a set of industries that play by very different set of economic rules and get very different results, while the rest, the muggles (even though they run the world, finance the world, and energize the world), play by a more traditional set of economic rules.
Lucia Rahilly: Could we put a finer point on what is novel or different about the lens that you applied to determine what’s a wizard and what’s a muggle?
Chris Bradley: Wizards are defined by growth and dynamism. We looked at where value is flowing and the places where value is moving. And where is the value flowing? What we see is that this set of wizards, which represent about ten percent of industries, hog 45 percent of the growth in market cap. But there’s another dimension or axis too, which is dynamism. That is measured by a new metric we’ve come up with called the ”shuffle rate.” How much does the bottom move to the top? It turns out that in this set of wizardish industries, or arenas, the shuffle rate is much higher than it is in the traditional industry.
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Chris Bradley: The economic profit, which is the profit you make minus the cost for the capital you employ is in the wizard industries. It’s where R&D happens; they’re two times more R&D intensive. They’re big stars, the nebulae, where new business is born.- IPMAT - 2025
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