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The workshop after the move
Divide and conquer
Make the difficult whole manageable—and keep what makes it a whole.
A new room, moving boxes and several things to put right. The team starts by deciding what will help.
Mina and Arjun want their workshop usable again. A storage corner cuts into the floor. Their model rover needs a trail. Some jobs can happen together; others must wait.
Mina “I keep seeing all the things we have to do at once.”
CKRobo “What would make the first problem easier to think about?”
CKSTEM Thinking CompassStuck? Find your next move.
You don’t need the whole route yet. Start with one useful move, then notice what it tells you.
What do you know for certain? What are you trying to find?
After checking your steps against every clue, choose your own check:
Ready to explore.
The direction reflects your self-check. Use the lesson’s answer and reasoning to check further when you are ready.
A room that bends
The floor starts as an 8 m by 6 m rectangle, with a 3 m by 2 m storage corner removed from the top right. Every edge follows the 1 m grid. Each whole mat is 1 m by 1 m.
Outside: 8 m × 6 mStorage removed: 3 m × 2 m
Each little square is 1 m by 1 m. The storage corner is outside the mat-covered floor.
One square is 1 m by 1 m. Only the remaining floor needs mats.
How many mats cover the remaining floor exactly, without cutting, gaps or overlaps? Explain how you know the whole floor is covered.
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How many mats cover the remaining floor exactly, without cutting, gaps or overlaps? Explain how you know the whole floor is covered.
A room that bends
The floor starts as an 8 m by 6 m rectangle, with a 3 m by 2 m storage corner removed from the top right. Every edge follows the 1 m grid. Each whole mat is 1 m by 1 m.
Outside: 8 m × 6 mStorage removed: 3 m × 2 m
Each little square is 1 m by 1 m. The storage corner is outside the mat-covered floor.
One square is 1 m by 1 m. Only the remaining floor needs mats.
Your answer
Enter the mat count. Also explain a division that covers the entire floor once and show why no cutting is needed.
These quantities fit. Explain your choice of parts, how they recombine and why every original rule is met; reasoning stays unscored.
Break it into steps
Step 1 of 3
How many one-square-metre mats would cover the whole outer rectangle?
Eight times six is forty-eight.
Step 2 of 3
How many square metres are removed?
Three times two is six.
Step 3 of 3
How many whole mats cover the actual remaining floor?
The actual floor has forty-two square metres and forty-two one-square-metre mats fit.
You have worked through these help steps. Return to Your answer when you are ready, and check every original clue.
Show the answer and reasoning: cover the floor
Answer:42 whole mats cover the floor exactly.
Arjun “I could count every little square. Or I could make rectangles I can count more quickly.”
CKRobo “How will you check that your pieces still make this floor?”
Mina “I’ll trace their edges. Nothing can be missing or counted twice.”
Outside: 8 m × 6 mStorage removed: 3 m × 2 m
A Left: 5 m × 6 m
B Right: 3 m × 4 m
These two regions cover every floor square once.
A line from the inside corner makes a 5 m by 6 m rectangle and a 3 m by 4 m rectangle. They meet along an edge; their interiors do not overlap.
\[5\times6+3\times4=30+12=42.\]
Mina tries another useful division:
Outside: 8 m × 6 mStorage removed: 3 m × 2 m
A Bottom: 8 m × 4 m
B Top-left: 5 m × 2 m
A different valid division describes the same whole.
\[8\times4+5\times2=32+10=42.\]
Check the original floor: the outside has \(8\times6=48\) grid squares. The removed storage corner has \(3\times2=6\), leaving \(48-6=42\). The diagrams also show how 42 whole square mats actually fit.
Optional mathematics: the familiar pieces
Area
Area measures covered space in square units. One mat covers \(1\text{ m}^2\).
Area of a rectangle
Equal rows let us multiply length by width. The strategy was choosing familiar rectangles that rejoin to make the awkward floor.
Divide and conquer
Understand what the whole problem asks. Choose smaller questions that help. Solve them, then reconnect their answers and check every original condition.
The pieces need not be equal or physically separate. They might be regions, jobs, parts of a route or different conditions on one choice. The useful split depends on the goal.
If you are stuck, try one manageable part and ask what it tells you about the whole. Do not keep making smaller pieces after they stop helping.
Putting answers back together takes thought too. You may need to add, remove an overlap, wait for both jobs, combine choices or check which conditions hold together.
Two rectangles, one total
For the same floor, Mina counts the bottom strip, 8 m by 4 m, and the left strip, 5 m by 6 m. She adds their counts and gets 62 mats.
Outside: 8 m × 6 mStorage removed: 3 m × 2 m
Solid outline: bottom strip, 8 m × 4 m
Dashed outline: left strip, 5 m × 6 m
The solid and dashed outlines show the two proposed strips.
Is 62 correct? If not, identify exactly what was counted twice and repair the total.
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Is 62 correct? If not, identify exactly what was counted twice and repair the total.
Two rectangles, one total
For the same floor, Mina counts the bottom strip, 8 m by 4 m, and the left strip, 5 m by 6 m. She adds their counts and gets 62 mats.
Outside: 8 m × 6 mStorage removed: 3 m × 2 m
Solid outline: bottom strip, 8 m × 4 m
Dashed outline: left strip, 5 m × 6 m
The solid and dashed outlines show the two proposed strips.
Your answer
Enter the repaired total. Identify exactly the shared region and decide whether sixty-two was correct.
These quantities fit. Explain your choice of parts, how they recombine and why every original rule is met; reasoning stays unscored.
Break it into steps
Step 1 of 2
What area lies in both counted strips?
Their common five-by-four part has area twenty.
Step 2 of 2
What is the repaired mat total?
Sixty-two minus twenty is forty-two.
You have worked through these help steps. Return to Your answer when you are ready, and check every original clue.
Show the answer and reasoning: the overlapping strips
Answer:No. The bottom-left 5 m by 4 m region was counted twice. Remove its 20 squares once: 42 mats.
Mina “Both strips are useful rectangles, but they share some floor.”
Arjun “We need those shared squares once. Our sum included them twice.”
Outside: 8 m × 6 mStorage removed: 3 m × 2 m
Bottom strip: 8 m × 4 m
Left strip: 5 m × 6 m
Hatched region: 5 m × 4 m, counted twice.
The hatch marks the squares belonging to both proposed strips.
\[32+30-20=42.\]
Removing one copy leaves every floor square counted once. This agrees with the two divisions and the outside-minus-storage check. Recognising useful parts was only the beginning; their relationship mattered.
Two jobs, one finish
Mina’s preparation takes 12 minutes; Arjun’s takes 18 minutes. They have separate tools and may both start at minute 0. A mat crew needs 7 minutes, starting only when both preparations are complete. A 4-minute check follows the crew’s work. All times are fixed, and there are no other jobs.
Mina prepares
12 minutes
Arjun prepares
18 minutes
Mat crew
7 minutes
Waits for both preparations
Final check
4 minutes
Waits for the crew
Separate tools are available. The later work obeys the stated waiting rules.
What is the earliest minute when the checked room can be ready? Give a schedule and explain why it cannot finish earlier.
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What is the earliest minute when the checked room can be ready? Give a schedule and explain why it cannot finish earlier.
Two jobs, one finish
Mina’s preparation takes 12 minutes; Arjun’s takes 18 minutes. They have separate tools and may both start at minute 0. A mat crew needs 7 minutes, starting only when both preparations are complete. A 4-minute check follows the crew’s work. All times are fixed, and there are no other jobs.
Mina prepares
12 minutes
Arjun prepares
18 minutes
Mat crew
7 minutes
Waits for both preparations
Final check
4 minutes
Waits for the crew
Separate tools are available. The later work obeys the stated waiting rules.
Your answer
Enter the earliest finish, then give an attaining schedule and explain why an earlier finish is impossible.
These quantities fit. Explain your choice of parts, how they recombine and why every original rule is met; reasoning stays unscored.
Break it into steps
Step 1 of 3
What is the earliest minute both preparations can be complete?
The later finish is eighteen.
Step 2 of 3
What is the earliest minute the crew can finish?
Eighteen plus seven is twenty-five.
Step 3 of 3
What is the earliest minute the checked room is ready?
Twenty-five plus four is twenty-nine.
You have worked through these help steps. Return to Your answer when you are ready, and check every original clue.
Show the answer and reasoning: the earliest finish
Answer: The room can be ready at minute 29, and no earlier.
Arjun “Adding every job gives forty-one minutes.”
CKRobo “Does that total describe what the people and tools must do?”
Mina “Our preparations can overlap. The crew waits for the later one, not for their sum.”
Forward schedule
Mina prepares · minutes 0–12
Arjun prepares · minutes 0–18
Mat crew · minutes 18–25
Check · minutes 25–29
Every bar uses the same 0–29 minute scale. The crew starts when both preparations are finished.
Mina finishes at 12 and Arjun at 18. The crew works from 18 to 25, then the check runs from 25 to 29.
\[18+7+4=29\text{ minutes}.\]
Why this is earliest: the 18-minute preparation cannot finish before 18. The crew and check must follow it and need another 11 minutes. The schedule attains 29 and meets every dependency.
Routes for a model rover
Arjun makes a one-way trail for a small wheeled model. Mina wants to know how many different complete routes they can try.
The map shows every allowed connection. A route starts at Start and ends at Finish. Follow the arrows only; do not travel backwards or add paths. A route is different if it uses a different labelled branch.
Follow the arrows from Start to Finish. There are no other connections.
How many complete routes are there? Identify every route so you can check that none is missed.
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How many complete routes are there? Identify every route so you can check that none is missed.
Routes for a model rover
Arjun makes a one-way trail for a small wheeled model. Mina wants to know how many different complete routes they can try.
The map shows every allowed connection. A route starts at Start and ends at Finish. Follow the arrows only; do not travel backwards or add paths. A route is different if it uses a different labelled branch.
Follow the arrows from Start to Finish. There are no other connections.
Your answer
Enter the count. You must also identify every complete route and explain why recombination has no missing or repeated route.
These quantities fit. Explain your choice of parts, how they recombine and why every original rule is met; reasoning stays unscored.
Break it into steps
Step 1 of 3
How many labelled branches can reach the middle gate?
There are three first branches.
Step 2 of 3
How many labelled branches continue from the gate to Finish?
There are two continuations for each first branch.
Step 3 of 3
How many complete directed routes are there?
Three times two gives six complete routes.
You have worked through these help steps. Return to Your answer when you are ready, and check every original clue.
Show the answer and reasoning: the complete routes
Answer: There are 6 routes: L1–R1, L1–R2, L2–R1, L2–R2, L3–R1 and L3–R2.
Mina “Every route reaches Gate. I can look at the journey to Gate and the journey from Gate separately.”
CKRobo “How do those smaller answers fit together?”
Arjun “Each first part can use either second part. We need a route from each side.”
Every complete route: one left branch and one right branch
First branch
Then R1
Then R2
L1
L1–R1
L1–R2
L2
L2–R1
L2–R2
L3
L3–R1
L3–R2
Each row fixes its first branch; both permitted continuations are included.
\[3\times2=6.\]
Every complete route has exactly one left label and one right label. Every displayed pairing follows the original arrows. That establishes both completeness and no duplicates.
Adding \(3+2\) counts separate route pieces, not complete journeys. A direct list of the six full routes is valid too; choosing Gate as a joining point explains why the multiplication works.
Ready by noon
For another room, Mina needs 12 minutes of preparation and Arjun needs 18 minutes. They have separate tools and may work at the same time. They do not have to start together. The mat crew needs 7 minutes after both are ready, followed by a 4-minute check. All times are fixed; no other jobs or delays occur.
The checked room must be ready by 12:00 noon. Each child wants to start as late as possible.
Mina prepares
12 minutes
Arjun prepares
18 minutes
Mat crew
7 minutes
Waits for both preparations
Final check
4 minutes
Waits for the crew
Checked room ready by 12:00
The finish is fixed; the two people do not have to start together.
What is the latest start time for each child? Give a complete working schedule and explain why neither child can start later.
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What is the latest start time for each child? Give a complete working schedule and explain why neither child can start later.
Ready by noon
For another room, Mina needs 12 minutes of preparation and Arjun needs 18 minutes. They have separate tools and may work at the same time. They do not have to start together. The mat crew needs 7 minutes after both are ready, followed by a 4-minute check. All times are fixed; no other jobs or delays occur.
The checked room must be ready by 12:00 noon. Each child wants to start as late as possible.
Mina prepares
12 minutes
Arjun prepares
18 minutes
Mat crew
7 minutes
Waits for both preparations
Final check
4 minutes
Waits for the crew
Checked room ready by 12:00
The finish is fixed; the two people do not have to start together.
Your answer
Enter each person’s independent latest start. Also give the shared crew/check schedule and prove that neither start can be later.
These quantities fit. Explain your choice of parts, how they recombine and why every original rule is met; reasoning stays unscored.
Break it into steps
Step 1 of 4
What is the latest clock time the final check must begin?
The check must start by 11:56.
Step 2 of 4
What is the latest clock time the crew must begin?
The crew must start by 11:49.
Step 3 of 4
What is Mina’s latest start?
11:49 minus twelve minutes is 11:37.
Step 4 of 4
What is Arjun’s latest start?
11:49 minus eighteen minutes is 11:31.
You have worked through these help steps. Return to Your answer when you are ready, and check every original clue.
A question to help me begin
What would a complete answer need to tell each person?
Show the answer and reasoning: each latest start
Answer: Mina can start at 11:37; Arjun at 11:31. The crew works 11:49–11:56, then the check runs 11:56–12:00.
Divide and conquer
We need each person’s latest start, not one common start. Separate the shared ending from the two preparation branches. Both branches must rejoin before the crew can begin, so neither can be checked in isolation from that joining time.
Arjun “Starting us both at eleven thirty-one would work.”
CKRobo “Does that settle what each person was asking?”
Mina “It makes me start before I need to. I’ll trace my own preparation back from the time we both have to be ready.”
Work backwards
The noon deadline is known, while the starts are unknown. Go back through the shared ending first: undo the 4-minute check, then the 7-minute crew job. Both preparations must be complete by 11:49. Now go back 12 minutes on Mina’s branch and 18 on Arjun’s.
Work backwards from noon
Checked room ready
12:00
Back 4 minutes ↓
Check must start
11:56
Back 7 minutes ↓
Both preparations ready
11:49
Mina latest start
Back 12 minutes
11:37
Arjun latest start
Back 18 minutes
11:31
Follow the shared ending backwards, then follow each person’s branch. Teal labels show the backward reasoning.
Forward check:
Forward schedule
Mina prepares · 11:37–11:49
Arjun prepares · 11:31–11:49
Mat crew · 11:49–11:56
Check · 11:56–12:00
The amber forward check uses the original durations and reaches the required finish.
Mina’s 12 minutes and Arjun’s 18 both end at 11:49. The crew’s 7 minutes end at 11:56; the check’s 4 end at noon. All original dependencies hold.
Why neither can start later: Mina still needs her own \(12+7+4=23\) minutes before noon; Arjun needs \(18+7+4=29\). A later start on either branch would push the shared ending past noon. One common start is a valid plan, but is not the latest start for each person.
The order they can bring home
The team needs mats for another project. A low price is tempting, but an order is useful only if the whole plan works.
The project needs exactly 81 identical mats. Buy from one supplier only, in whole sealed packs, with no extra mats. Zero packs of a type is allowed. Both suppliers have enough stock.
Supplier A
6-mat pack: $15
8-mat pack: $22
Supplier B
5-mat pack: $14
8-mat pack: $24
All packs contain the same type of mat. Either pack size occupies one cart slot.
These are the complete prices. The cart has 12 slots; either pack size uses one slot. There is one trip, and packs cannot be split.
Find the least-cost order that satisfies every rule. Give its supplier, pack quantities and cost, and explain why no cheaper feasible order is missed.
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Find the least-cost order that satisfies every rule. Give its supplier, pack quantities and cost, and explain why no cheaper feasible order is missed.
The order they can bring home
The team needs mats for another project. A low price is tempting, but an order is useful only if the whole plan works.
The project needs exactly 81 identical mats. Buy from one supplier only, in whole sealed packs, with no extra mats. Zero packs of a type is allowed. Both suppliers have enough stock.
Supplier A
6-mat pack: $15
8-mat pack: $22
Supplier B
5-mat pack: $14
8-mat pack: $24
All packs contain the same type of mat. Either pack size occupies one cart slot.
These are the complete prices. The cart has 12 slots; either pack size uses one slot. There is one trip, and packs cannot be split.
Your answer
Choose the supplier and enter both pack quantities and the lowest cost. Check exactly 81 mats, the 12-slot cart in one trip, and every complete price comparison.
These quantities fit. Explain your choice of parts, how they recombine and why every original rule is met; reasoning stays unscored.
Break it into steps
Step 1 of 7
What is the remainder when any Supplier A order’s mat count is divided by 2?
Every Supplier A total is even. Since 81 is odd, none of those orders can fill the exact demand.
Step 2 of 7
At most how many five-mat packs fit without going over 81 mats? Ignore exact filling and cart space for this bound.
Sixteen five-mat packs use eighty mats; seventeen use eighty-five. Sixteen is only a ceiling, not a complete order.
Step 3 of 7
At most how many of Supplier B’s eight-mat packs fit without going over 81 mats? Ignore the other constraints for this bound.
Ten eight-mat packs use eighty mats; eleven use eighty-eight. Test bounded whole pack combinations against the cart and exact-inventory rules.
Step 4 of 7
Which supplier provides the least-cost feasible order?
Supplier B supplies the feasible exact order.
Step 5 of 7
How many smaller packs does the least-cost feasible order use?
The feasible winning order uses five of its supplier’s smaller packs.
Step 6 of 7
How many larger packs does that order use?
Seven larger packs and five smaller packs use all twelve slots and make exactly 81 mats.
Step 7 of 7
What is the least cost of an order satisfying every rule?
Supplier B with five five-mat and seven eight-mat packs costs two hundred and thirty-eight dollars.
You have worked through these help steps. Return to Your answer when you are ready, and check every original clue.
Show the answer and complete reasoning: the collectible order
Answer: Buy from Supplier B: five 5-mat packs and seven 8-mat packs, costing $238. They supply exactly 81 mats and use all 12 cart slots. No cheaper order meets every rule.
Arjun “I want to compare the pack prices first.”
Mina “Price matters. Let’s compare whole orders we can collect. I’ll start with the exact mat count.”
CKRobo “What will you need to keep from that first part?”
Arjun “The whole possible orders—their pack quantities. A single cheap pack won’t tell us enough.”
Divide and conquer
The count, collection and price conditions must hold together. We first keep the exact-count condition to find possible complete orders. We retain their pack quantities, then recombine them with the cart limit and total cost. Picking a cheap pack alone could miss the whole decision.
Use even and odd
There is no point pricing a supplier that cannot make the exact count. Every A pack contains an even number of mats, so every A order is even. It cannot make the odd total 81. For B, eight-packs always add an even number of mats. The five-packs must make an odd total, so there must be an odd number of five-packs.
A: 6 mats
A: 8 mats
Demand: 81 mats = 40 pairs and 1 unpaired
Every dot is one mat. A’s complete pairs cannot make a total with one unpaired mat.
Even and odd numbers help reject a whole supplier and restrict the remaining candidates. An odd count of five-packs is necessary, but does not yet make an exact order.
Make an organized list
We still need every exact B order, so no cheaper feasible choice is missed. There can be at most 16 five-packs in 81 mats. List every allowed odd count from 1 to 15, then test whether the remaining mats form a nonnegative whole number of eight-packs.
Every possible odd count of B’s five-packs
5-mat packs
Mats left
8-mat packs
1
76
Not a multiple of 8
3
66
Not a multiple of 8
5
56
7 packs; exact
7
46
Not a multiple of 8
9
36
Not a multiple of 8
11
26
Not a multiple of 8
13
16
2 packs; exact
15
6
Not a multiple of 8
An odd five-pack count is necessary. The remainder must still make whole eight-packs.
Only two complete orders survive. Now return to the rest of the original problem:
5 small + 7 large
25 + 56 = 81 mats
$238
12 cart slots
Fits the one trip
13 small + 2 large
65 + 16 = 81 mats
$230
15 cart slots
Cannot fit the one trip
Both meet the mat count; only one meets the cart rule. Compare complete orders.
Arjun “Thirteen small and two large packs cost less—but need fifteen slots.”
Mina “They cannot come home in our one trip. The other complete order costs more, but meets every condition.”
Full check: \(5\times5+7\times8=81\) mats; \(5+7=12\) slots; \(5\times\$14+7\times\$24=\$238\). All packs come from B, are whole and are available. A is impossible; the complete B list contains only one order fitting the cart. That proves $238 is the least feasible cost.
Another valid start: begin with the cart
Supplier A is still impossible: all its packs contain an even number of mats, while 81 is odd.
For Supplier B, ten packs could hold at most 80 mats, while the cart holds at most twelve packs. So an exact order must use 11 or 12 packs.
Eleven eight-packs hold 88 mats. Replacing an eight-pack with a five-pack removes 3 mats each time. Reaching 81 would require removing 7, which cannot be made from groups of 3.
Twelve eight-packs hold 96 mats. We must remove 15, so five replacements give five small packs and seven large packs. This meets the cart rule and gives the same unique feasible B order.
This route combines the count and cart conditions sooner and needs only two cases. The earlier route keeps both exact orders, which makes it easier to see why the cheaper one fails collection. Both are valid; choose a split that helps.
Your turn
Choose useful parts for these new situations. Each answer stays closed until you want to check.
One book, two reading lists
A bird reading list has 18 distinct books; a river list has 14 distinct books. Exactly 5 books appear on both lists. Neither list repeats a book. The collection includes precisely the books appearing on either list, with one copy of each.
Mina says the collection needs 32 books. Is that correct? Find the number of distinct books and explain how each shared book is counted.
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Mina says the collection needs 32 books. Is that correct? Find the number of distinct books and explain how each shared book is counted.
One book, two reading lists
A bird reading list has 18 distinct books; a river list has 14 distinct books. Exactly 5 books appear on both lists. Neither list repeats a book. The collection includes precisely the books appearing on either list, with one copy of each.
Your answer
Enter the distinct count. Also evaluate Mina’s claim and explain how a shared title is counted once.
These quantities fit. Explain your choice of parts, how they recombine and why every original rule is met; reasoning stays unscored.
Break it into steps
Step 1 of 2
What is the sum of the two list lengths?
Eighteen plus fourteen is thirty-two list entries.
Step 2 of 2
How many distinct books are in the collection?
Remove one copy of each shared book: thirty-two minus five is twenty-seven.
You have worked through these help steps. Return to Your answer when you are ready, and check every original clue.
Show the answer and reasoning: the book collection
Answer:No. The collection needs 27 distinct books. Each shared book is included once.
The sum \(18+14=32\) includes each of the five shared books twice. Remove one copy of each: \(32-5=27\). Alternatively, count disjoint parts: 13 bird-only, 5 shared and 9 river-only. Their sum is 27, and no book is missing or repeated.
One tool changes the plan
Mina’s preparation takes 12 minutes; Arjun’s takes 18 minutes. This time each preparation needs the same single vacuum continuously. They cannot share it at the same time. Both people and the vacuum are available at minute 0. A 7-minute crew job waits for both preparations, then a 4-minute check follows. Times are fixed; there are no other jobs.
Can the checked room still be ready at minute 29? Find the earliest finish, give a schedule and explain why earlier is impossible.
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Can the checked room still be ready at minute 29? Find the earliest finish, give a schedule and explain why earlier is impossible.
One tool changes the plan
Mina’s preparation takes 12 minutes; Arjun’s takes 18 minutes. This time each preparation needs the same single vacuum continuously. They cannot share it at the same time. Both people and the vacuum are available at minute 0. A 7-minute crew job waits for both preparations, then a 4-minute check follows. Times are fixed; there are no other jobs.
Your answer
Enter the earliest finish. Give a legal schedule and explain why the old parallel plan no longer applies.
These quantities fit. Explain your choice of parts, how they recombine and why every original rule is met; reasoning stays unscored.
Break it into steps
Step 1 of 2
How many minutes must the shared vacuum spend on both preparations?
Twelve plus eighteen is thirty minutes of non-overlapping vacuum work.
Step 2 of 2
What is the earliest checked-room finish?
Thirty plus seven plus four is forty-one.
You have worked through these help steps. Return to Your answer when you are ready, and check every original clue.
Show the answer and reasoning: the shared tool
Answer:No. The earliest finish is minute 41.
Forward schedule
Mina prepares · minutes 0–12
Arjun prepares · minutes 12–30
Mat crew · minutes 30–37
Check · minutes 37–41
The same vacuum cannot serve both preparations at once. This is one of two valid preparation orders.
For example, Mina prepares from 0 to 12, then Arjun from 12 to 30. The crew works 30–37, then the check 37–41. Reversing the two preparations works too.
The single vacuum must supply \(12+18=30\) minutes before both preparations are finished. Another \(7+4=11\) minutes must follow. Therefore no schedule can finish before 41; the displayed one attains it. Separating the jobs did not make their shared resource disappear.
Four pairings, one time limit
A park has these complete choices: from Start to Gate, route A takes 4 minutes or B takes 7 minutes. From Gate to Finish, route C takes 4 minutes or D takes 2 minutes. There are no other paths, reversals or delays at Gate. A complete journey uses one first route followed by one second route.
Start to Gate
A: 4 minutes
B: 7 minutes
Gate to Finish
C: 4 minutes
D: 2 minutes
These are the only path times; Gate adds no delay.
The whole journey must take at most 10 minutes.
Find every permitted complete journey and its time. Does having two choices for each part mean four journeys meet the time limit? Explain.
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Find every permitted complete journey and its time. Does having two choices for each part mean four journeys meet the time limit? Explain.
Four pairings, one time limit
A park has these complete choices: from Start to Gate, route A takes 4 minutes or B takes 7 minutes. From Gate to Finish, route C takes 4 minutes or D takes 2 minutes. There are no other paths, reversals or delays at Gate. A complete journey uses one first route followed by one second route.
Start to Gate
A: 4 minutes
B: 7 minutes
Gate to Finish
C: 4 minutes
D: 2 minutes
These are the only path times; Gate adds no delay.
The whole journey must take at most 10 minutes.
Your answer
Start with the blank row. Add a row for each permitted journey and its total minutes; remove any unwanted row.
Every permitted journey and its time are present. Explain why the four given pairings cover all possibilities and why each rejected pairing fails.
Break it into steps
Help step 1
What is the total time for A followed by C?
Four plus four is eight minutes.
Help step 2
What is the total time for B followed by C?
Seven plus four is eleven, so this pairing is too slow.
You have worked through these help steps. Return to Your answer when you are ready, and check every original clue.
Show the answer and reasoning: the complete journey times
Answer:Three meet the limit: A–C, 8 minutes; A–D, 6; B–D, 9. No, not all four pairings meet it.
Every complete journey checked
Journey
Minutes
Fits time limit?
A–C
4 + 4 = 8
Meets 10-minute limit
A–D
4 + 2 = 6
Meets 10-minute limit
B–C
7 + 4 = 11
Too long
B–D
7 + 2 = 9
Meets 10-minute limit
The path times, not just the number of paths, determine which pairings work.
B–C needs \(7+4=11\) minutes, so it fails. The two-by-two table checks every possible pairing exactly once. Counting parts was not enough: their times had to travel with them into the complete-journey check.
Check that the parts solve the whole problem
Recognise the cue: the whole feels awkward, while useful smaller questions look manageable. Choose the pieces, keep the information needed where they meet, and check the complete request.
Casework helps when the possibilities split into alternatives to cover. Divide and conquer also helps with pieces that must coexist: floor regions, linked jobs and conditions on one order.
When stuck, try a useful part. Ask what it settles, what remains and how the answers must reconnect. More pieces are not automatically a better plan.
The workshop becomes usable because the children keep returning to the same goal. Knowing each smaller answer is only useful when the whole works.
Mina “I keep seeing all the things we have to do at once.”
CKRobo “What would make the first problem easier to think about?”
Arjun “I could count every little square. Or I could make rectangles I can count more quickly.”