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.
The number of biologists in team E is
Enter your answer to attempt
The Setup: We are dealing with a logical puzzle disguised as a matrix grid. We have 5 teams of 15 members each, meaning there are 75 total members. Since the three roles (Biologists, Geologists, Explorers) have equal total numbers, we just divide 75 by 3. That means exactly 25 Biologists, 25 Geologists, and 25 Explorers across the entire board. Time to build the matrix and min-max the stats.
Step 1:The Explorer Distribution (The Meta)
The problem states that the number of Explorers in each team strictly decreases in the order A > B > C > D > E. We also know they must sum up to 25, and every single team must have at least 2 of *each* role.
Let's find the maximum number of Explorers team A can have. Team A already has 6 Geologists (given) and must have at least 2 Biologists.
Max_ExplorersA=15−6−2=7
If Team A has a maximum of 7 Explorers, we need 5 strictly decreasing integers starting from ≤7 that sum to 25. There is literally only one valid sequence in the entire universe that satisfies this:
7+6+5+4+3=25
Lock it in:
Team A = 7, Team B = 6, Team C = 5, Team D = 4, Team E = 3.
Step 2:The Biologist Constraints (Vibe Check)
Now let's look at the Biologists (B). We know the total is 25.
The prompt feeds us free intel: BC=6 and BD=6.
This means the remaining Biologists for teams A, B, and E are:
25−6−6=13
So, BA+BB+BE=13.
Now we apply the heavy constraint: "Every team except A has more biologists than explorers."
* For Team B: BB>EB→BB>6→BB≥7
* For Team E: BE>EE→BE>3→BE≥4
* For Team A (the exception): It just needs to meet the baseline minimum of BA≥2.
Step 3:Solving the Biologist Equation
We have our minimum thresholds: BA≥2, BB≥7, and BE≥4.
Let's add these bare minimums together:
2+7+4=13
Since the sum of the absolute minimums equals exactly the total number of remaining Biologists (13), there is zero wiggle room. The math is mathing perfectly. None of these values can be higher, or the sum would break 13.
Therefore, BA=2, BB=7, and BE=4.
Step 4:The Audit (Double Check Protocol)
Let's run a full matrix audit to ensure we aren't throwing the game. We'll fill in the Geologists (G) using the formula G=15−B−E to make sure no team breaks the rules.
* Team A: E=7, B=2, G=6. (Sum = 15, all ≥2. Works. G=6 matches the prompt.)
* Team B: E=6, B=7, G=2. (Sum = 15, all ≥2, B>E. Works.)
* Team C: E=5, B=6, G=4. (Sum = 15, all ≥2, B>E. Works.)
* Team D: E=4, B=6, G=5. (Sum = 15, all ≥2, B>E. Works.)
* Team E: E=3, B=4, G=8. (Sum = 15, all ≥2, B>E. Works.)
Now, let's verify total Geologists:
6+2+4+5+8=25
Perfectly balanced, as all things should be. The board is fully validated and legally solved.
Final Answer: 4
Eight teams take part in a tournament where each team plays against every other team exactly once. In a particular year, one team got suspended after playing 3 matches, due to a disciplinary issue. The organizers decide to proceed, nonetheless, with the remaining matches. The total number of matches that were played in the tournament that year is
Enter your answer to attempt
The Setup:
We are dealing with a round-robin tournament combinatorics problem. The base logic requires calculating the total number of matches in a perfect bracket, and then subtracting the matches that got canceled because a team got hit with the ban hammer. I have set up a stat box to extract the core data before we run the calculations.
Step 1:Extracting the Meta (Stat Box)
Let's map out exactly what is happening in this bracket before the suspension ruins the schedule.
Tournament Stat
Value
Math Logic
Total Teams
8
Base roster (n=8)
Total Matches (No Bans)
28
8C2 combination formula
Intended Matches per Team
7
n−1 (You play everyone except yourself)
Matches Played by Suspended Team
3
Stated in prompt
Step 2:The Base Timeline
In a flawless, perfectly run round-robin tournament, every team plays every other team exactly once. The formula to find total unique handshakes (or matches) is nC2:
8C2=28×78C2=28
So, the organizers originally scheduled exactly 28 matches.
Step 3:Calculating the Dropped Matches
One team gets toxic and gets suspended.
Every single team in this tournament was scheduled to play 7 matches.
This suspended team managed to play exactly 3 matches before getting kicked.
To find the number of canceled matches, subtract what they played from what they were supposed to play:
7−3=4
Exactly 4 scheduled matches literally evaporated from the bracket.
Step 4:The Final Tally
Subtract the canceled matches from the total base schedule to find out how many matches actually happened.
28−4=24
There were 24 matches officially played that year.
Step 5:The Audit (Double Check Protocol)
Running it back using an alternative method to guarantee we aren't throwing.
Let's look at the tournament from the perspective of the 7 teams that *didn't* get banned.
If those 7 teams just played a private tournament among themselves, the total matches would be:
7C2=27×6=21
Now, we just add the matches that the banned team played against 3 of these valid teams before getting kicked:
21+3=24
Both methods independently output exactly 24. The logic is completely locked in and validated. No cap.
Final Answer: 24
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.
The number of teams having more geologists than biologists is
Enter your answer to attempt
The Setup:
We are literally back in the exact same lobby as Question 2. This is a follow-up question utilizing the same logical matrix we already decoded. Since we previously theory-crafted the exact distribution of all 75 members across the grid, we can just pull up our cached data and run a quick stat check to find out who has higher Geologist DPS than Biologist DPS. Free points, honestly.
Step 1:Retrieving the Master Matrix (From Memory)
Let's pull the fully solved grid we established earlier. Remember, every team has exactly 15 members, and each class (Explorer, Biologist, Geologist) totals exactly 25 across all teams.
Team
Explorers (E)
Biologists (B)
Geologists (G)
A
7
2
6
B
6
7
2
C
5
6
4
D
4
6
5
E
3
4
8
Step 2:The Stat Check
The prompt asks for the number of teams where Geologists > Biologists (G>B). Let's run down the roster and vibe check every single team:
* Team A:G=6, B=2. (6>2) →Valid (W)
* Team B:G=2, B=7. (2<7) → Invalid (L)
* Team C:G=4, B=6. (4<6) → Invalid (L)
* Team D:G=5, B=6. (5<6) → Invalid (L)
* Team E:G=8, B=4. (8>4) →Valid (W)
Only Team A and Team E meet the specific criteria.
Step 3:The Audit (Double Check Protocol)
Let's do a quick sanity check to ensure our base matrix didn't glitch.
Are there exactly 25 of each role?
Explorers: 7+6+5+4+3=25 (Check).
Biologists: 2+7+6+6+4=25 (Check).
Geologists: 6+2+4+5+8=25 (Check).
Do all teams have 15 members? Yes.
Does every team have ≥2 in each role? Yes.
Do C and D have 6 Biologists? Yes.
Does A have 6 Geologists? Yes.
Does every team except A have B>E?
Team B (7>6), Team C (6>5), Team D (6>4), Team E (4>3). Yes.
Does E strictly decrease from A to E? 7>6>5>4>3. Yes.
The matrix is perfectly locked in. The final count of teams with G>B is strictly 2.
Final Answer: 2
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.
The median number of biologists across five teams is
Enter your answer to attempt
The Setup:
We are dropping back into the team arrangement lobby for round 3. Since we already theory-crafted and completely solved the master matrix in Question 2, we just need to query our Biologist data and run a basic median calculation. Free elo.
Step 1:Retrieving the Biologist Stats
Let's pull the exact number of Biologists (B) for each team from our previously validated master grid:
* Team A: 2
* Team B: 7
* Team C: 6 (Given in prompt)
* Team D: 6 (Given in prompt)
* Team E: 4Step 2:Sorting the Array
To find the median, we can't just pick the middle team. We have to sort the Biologist values in ascending order to find the true statistical middle (the 50th percentile).
Unsorted array: {2,7,6,6,4}
Sorted array: {2,4,6,6,7}Step 3:Finding the Median
Since there are exactly 5 teams (an odd number), the median is literally just the dead-center value—the 3rd number in our sorted array.
Looking at our sorted list {2,4,6,6,7}, the middle value is 6.
Step 4:The Audit (Double Check Protocol)
Let's run a quick sanity check to make sure the data hasn't been corrupted.
Did they sum to 25? 2+4+6+6+7=25. (Yes).
Did every team except A have more Biologists than Explorers?
Team B: 7>6. Team C: 6>5. Team D: 6>4. Team E: 4>3. (Yes).
Are we picking the 3rd index of a 5-item sorted list? Yes.
The math is flawless. The logic is strictly locked in.
Final Answer: 6
The table given below provides the details of monthly sales (in lakhs of rupees) and the value of products returned by the customers (as a percentage of sales) of an e-commerce company for three product categories for the year 2024. Net sales (in lakhs of rupees) is defined as the difference between sales (in lakhs of rupees) and the value of products returned (in lakhs of rupees).
Month
Sales: Apparel
Sales: Footwear
Sales: Electronics
Returns: Apparel
Returns: Footwear
Returns: Electronics
January
262
104
289
13%
7%
2%
February
279
113
387
16%
9%
3%
March
236
121
283
20%
7%
2%
April
258
58
325
16%
8%
1%
May
249
69
359
12%
6%
4%
June
230
111
321
19%
5%
3%
July
244
119
341
17%
9%
4%
August
252
60
336
16%
6%
2%
September
288
118
355
10%
9%
5%
October
222
108
383
15%
8%
2%
November
228
93
282
14%
9%
4%
December
221
86
268
18%
10%
1%
Which month had highest percentage decline in monthly sales as compared to previous month for the Apparel category?
AJune
BMarch
CDecember
DOctober
Pick an option to attempt
We don't need to calculate exact percentages - we can compare the fractions directly to find the largest one.
From the Apparel sales data, let's identify consecutive months where sales decreased:
Month Transition
Sales Change (Lakhs)
Change Type
Analysis
January → February
262→279
Increase (+17)
-
February → March
279→236
Decrease (-43)
Significant decrease - Consider
March → April
236→258
Increase (+22)
-
April → May
258→249
Decrease (-9)
Ignore - Minimal decrease while in the 250s range
May → June
249→230
Decrease (-19)
Ignore - Same reason as May
June → July
230→244
Increase (+14)
-
July → August
244→252
Increase (+8)
-
August → September
252→288
Increase (+36)
-
September → October
288→222
Decrease (-66)
Much higher decrease - Consider with March
October → November
222→228
Increase (+6)
-
November → December
228→221
Decrease (-7)
Ignore - Same reason as May
% Decline=Previous Month SalesPrevious Month Sales−Current Month Sales×100%
**February → March:**
% Decline=279279−236×100%=27943×100%=15.41%
**September → October:**
% Decline=288288−222×100%=28866×100%=22.92%
Comparing via fractions → March: 27943 and October: 28866
* October has a numerator (66) that is about 1.5 times larger than March's numerator (43).
* October has a denominator (288) that is only about 1.03 times larger than March's denominator (279).
Since the numerator grows much faster than the denominator, October's fraction 28866 is larger than March's fraction 27943.
October had the highest percentage decline in Apparel sales, dropping from 288 lakhs in September to 222 lakhs in October.
Final Answer: October
The table given below provides the details of monthly sales (in lakhs of rupees) and the value of products returned by the customers (as a percentage of sales) of an e-commerce company for three product categories for the year 2024. Net sales (in lakhs of rupees) is defined as the difference between sales (in lakhs of rupees) and the value of products returned (in lakhs of rupees).
Month
Sales: Apparel
Sales: Footwear
Sales: Electronics
Returns: Apparel
Returns: Footwear
Returns: Electronics
January
262
104
289
13%
7%
2%
February
279
113
387
16%
9%
3%
March
236
121
283
20%
7%
2%
April
258
58
325
16%
8%
1%
May
249
69
359
12%
6%
4%
June
230
111
321
19%
5%
3%
July
244
119
341
17%
9%
4%
August
252
60
336
16%
6%
2%
September
288
118
355
10%
9%
5%
October
222
108
383
15%
8%
2%
November
228
93
282
14%
9%
4%
December
221
86
268
18%
10%
1%
For which categories did the value of the products returned (as a percentage of sales) increase for three consecutive months?
AOnly Electronics
BOnly Apparel
CBoth Apparel and Footwear
DOnly Footwear
Pick an option to attempt
The Setup: We need to identify which product categories had the value of products returned (as a percentage of sales) increase for three consecutive months. The concept here is to track each category's return percentage month by month and look for patterns where the percentage consistently rises over a three-month span (meaning the current month's return percentage is strictly higher than the previous month's for three transitions).
Step 1: Analyze the Apparel category.
For Apparel, we see continuous increases from January to February to March:
* January: 13%
* February: 16%
* March: 20%
This gives us three consecutive months of increasing percentages (13%→16%→20%).
Step 2: Analyze the Footwear category.
For Footwear, we see continuous increases from October to November to December:
* October: 8%
* November: 9%
* December: 10%
This gives us three consecutive months of increasing percentages (8%→9%→10%).
Step 3: Analyze the Electronics category.
Scanning the Electronics return percentages, we do not find any sequence where the values strictly increase for three consecutive months.
Step 4: Conclusion.
The key insight is that we are looking for a pattern where each month's return percentage is strictly greater than the previous month's percentage for three consecutive months. Both the Apparel and Footwear categories satisfy this condition.
Final Answer: Both Apparel and Footwear
The table given below provides the details of monthly sales (in lakhs of rupees) and the value of products returned by the customers (as a percentage of sales) of an e-commerce company for three product categories for the year 2024. Net sales (in lakhs of rupees) is defined as the difference between sales (in lakhs of rupees) and the value of products returned (in lakhs of rupees).
Month
Sales: Apparel
Sales: Footwear
Sales: Electronics
Returns: Apparel
Returns: Footwear
Returns: Electronics
January
262
104
289
13%
7%
2%
February
279
113
387
16%
9%
3%
March
236
121
283
20%
7%
2%
April
258
58
325
16%
8%
1%
May
249
69
359
12%
6%
4%
June
230
111
321
19%
5%
3%
July
244
119
341
17%
9%
4%
August
252
60
336
16%
6%
2%
September
288
118
355
10%
9%
5%
October
222
108
383
15%
8%
2%
November
228
93
282
14%
9%
4%
December
221
86
268
18%
10%
1%
By what percentage the net sales for June increased as compared to May in the Footwear category?
A62.58 percent
B7.21 percent
C18.97 percent
D60.87 percent
Pick an option to attempt
Net Sales=Sales−Value of Products Returned
The value of products returned is given as a percentage of sales, so we need to convert this to actual rupees first.
For May in the Footwear category:
* Sales=69 lakhs
* Return percentage=6% of sales
* Value of products returned=1006×69=4.14 lakhs
* Net sales for May=69−4.14=64.86 lakhsFor June in the Footwear category:
* Sales=111 lakhs
* Return percentage=5% of sales
* Value of products returned=1005×111=5.55 lakhs
* Net sales for June=111−5.55=105.45 lakhsTo find the percentage increase from May to June:% Increase=May net salesJune net sales−May net sales×100%=64.86105.45−64.86×100%=64.8640.59×100%=62.58%Final Answer: 62.58 percent
The table given below provides the details of monthly sales (in lakhs of rupees) and the value of products returned by the customers (as a percentage of sales) of an e-commerce company for three product categories for the year 2024. Net sales (in lakhs of rupees) is defined as the difference between sales (in lakhs of rupees) and the value of products returned (in lakhs of rupees).
Month
Sales: Apparel
Sales: Footwear
Sales: Electronics
Returns: Apparel
Returns: Footwear
Returns: Electronics
January
262
104
289
13%
7%
2%
February
279
113
387
16%
9%
3%
March
236
121
283
20%
7%
2%
April
258
58
325
16%
8%
1%
May
249
69
359
12%
6%
4%
June
230
111
321
19%
5%
3%
July
244
119
341
17%
9%
4%
August
252
60
336
16%
6%
2%
September
288
118
355
10%
9%
5%
October
222
108
383
15%
8%
2%
November
228
93
282
14%
9%
4%
December
221
86
268
18%
10%
1%
Among the following four months, for which month the contribution of the Apparel category in the total monthly sales was the highest?
AJanuary
BApril
CDecember
DAugust
Pick an option to attempt
The Setup: The contribution of Apparel to Total Sales is calculated as the ratio of Apparel sales to the total sales of all categories for that specific month, converted to a percentage.
Apparel contribution=Total sales of all categoriesApparel sales×100%Step 1: Calculate the total sales for each given month.
We find the total sales by adding the sales from all three categories (Apparel + Footwear + Electronics):
* January:262+104+289=655
* April:258+58+325=641
* August:252+60+336=648
* December:221+86+268=575Step 2: Calculate the exact contribution percentage for each month.
* January:655262×100%=40.00%
* April:641258×100%≈40.25%
* August:648252×100%≈38.89%
* December:575221×100%≈38.43%Step 3: Compare and Conclude.
Comparing the percentages, April has the highest contribution (40.25%).
*Conceptual insight:* Even though January had higher absolute Apparel sales (262 vs 258), April had proportionally lower sales from other categories, making Apparel's share of the total pie larger.
Final Answer: April
The table given below provides the details of monthly sales (in lakhs of rupees) and the value of products returned by the customers (as a percentage of sales) of an e-commerce company for three product categories for the year 2024. Net sales (in lakhs of rupees) is defined as the difference between sales (in lakhs of rupees) and the value of products returned (in lakhs of rupees).
Month
Sales: Apparel
Sales: Footwear
Sales: Electronics
Returns: Apparel
Returns: Footwear
Returns: Electronics
January
262
104
289
13%
7%
2%
February
279
113
387
16%
9%
3%
March
236
121
283
20%
7%
2%
April
258
58
325
16%
8%
1%
May
249
69
359
12%
6%
4%
June
230
111
321
19%
5%
3%
July
244
119
341
17%
9%
4%
August
252
60
336
16%
6%
2%
September
288
118
355
10%
9%
5%
October
222
108
383
15%
8%
2%
November
228
93
282
14%
9%
4%
December
221
86
268
18%
10%
1%
Among the following four months, for which month the value of the Footwear returned (in lakhs of rupees) was the highest?
ASeptember
BJuly
CJune
DMarch
Pick an option to attempt
The Setup: The value of products returned is given as a percentage of sales, so we need to calculate the actual value in lakhs of rupees using the formula:
Value of products returned=100Return percentage×SalesStep 1: Calculate the exact return value for each given month.
Extracting the Footwear sales and return percentage data from the table:
* September:1009×118=10.62 lakhs
* July:1009×119=10.71 lakhs
* June:1005×111=5.55 lakhs
* March:1007×121=8.47 lakhsStep 2: Compare and Conclude.
Comparing the calculated values, July has the highest absolute value of products returned (10.71 lakhs).
*Conceptual insight:* The absolute value of returns depends on both the return percentage and the total sales volume. July combines a high return percentage (9%) with substantial sales (119 lakhs), creating the largest absolute return amount.
* September → percentage is the same (9%) but the sales are lower (118).
* June → sales are around the same but the percentage is almost half (5%).
* March → percentage is 2% lower (7%) and sales are only 2 lakhs more than July, which makes it definitively lower.
Final Answer: July