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The trip to the sound exhibition
Modular reduction
Keep the information that matters for your next decision.
The team has welcome supplies to carry and a journey to plan. CKSprout listens while Mina and Rowan think it through.
Mina and Rowan are helping at a community sound exhibition. Their coach is CKSprout. They want the display ready before visitors arrive, with a quiet corner where people can take a break.
The places, counts and timetables in this story are invented. Each problem gives the information needed for its own decision.
Mina “There are so many numbers on our records. Do we need every part of every number?”
CKSprout “What are you trying to find out first?”
Rowan “We should decide that before throwing any information away.”
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 small display for a large count
Before the trip, the team checks the ticket record at the supply desk.
23 groups will receive 47 tickets each. There are also 5 spare tickets. All tickets are counted once.
A counter displays only the units digit of the total number of tickets.
Which digit should it display? Explain why you can find that digit without calculating the complete total first.
Tickets for the groups
23 groups
47 tickets each
Spare tickets
5 more tickets
Counter display?
Units digit only
The display needs the final digit, rather than every digit of the total.A question to help me begin
Which part of the total will the display actually show? What happens when you add a complete group of ten?
Try with the CKSTEM Thinking Compass
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Which digit should it display? Explain why you can find that digit without calculating the complete total first.
A small display for a large count
Before the trip, the team checks the ticket record at the supply desk.
23 groups will receive 47 tickets each. There are also 5 spare tickets. All tickets are counted once.
A counter displays only the units digit of the total number of tickets.
Tickets for the groups
23 groups
47 tickets each
Spare tickets
5 more tickets
Counter display?
Units digit only
The display needs the final digit, rather than every digit of the total.
Your answer
Enter the digit. Explain why removing complete tens preserves exactly the information this display needs.
These quantities fit. Check the digit. Explain why removing complete tens preserves exactly the information this display needs.
Break it into steps
Step 1 of 2
What do the combined extra ones total before reducing again?
Three times seven plus five is twenty-six. It contains two new complete tens, so those must be removed too.
Step 2 of 2
What is your answer for Display digit?
Display digit: 6. Compare this quantity with the complete worked reasoning; the number alone does not check your argument.
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 display
Answer: It should display 6. Complete tens do not change the units digit.
Check the units digit
Replace 23 by its 3 extra ones and 47 by its 7 extra ones. Multiplying any discarded complete ten by a whole number still gives complete tens. Adding the 5 spares gives:
\[3\times7+5=26\]\[26=2\times10+6\]
3 × 7 + 5 = 26 extra ones
One complete ten
Another complete ten
Six extra ones
Final display: 6
Regroup the combined extras: twenty-six ones make two complete tens and six extra ones. Only the six determines the units digit.
The first leftover, 26, still contains two complete tens. Remove those too: the final units digit is 6.
Full check: \(23\times47+5=1086\), which ends in 6. Full multiplication is valid too. We kept less information because this particular display asks for only one digit.
Modular reduction
For a question about a remainder, remove complete groups and keep the extras. You can add, subtract or multiply using numbers with the same remainders, provided the group size stays the same. Remove any new complete groups afterwards.
The mathematical notation \(a\equiv b\pmod m\) means that \(a\) and \(b\) leave the same remainder when divided by the positive whole number \(m\). It does not mean their full values are equal.
A final remainder is at least 0 and smaller than the group size. A units digit is the remainder by 10. Other questions may need groups of 8, 14 or another size.
Before using the shortcut, ask what the question needs. A remainder may be enough to find a position or an unfinished group. It cannot tell you everything about the original amount.
How long until the shuttle?
The fictional Bridge Shuttle departs every 14 minutes from 09:00, including a departure at 09:00. The team is ready to board at 09:47 that morning.
A group ready at the exact departure time can board. There is space for them.
When is the next departure they can catch, and how many minutes must they wait?
Starting point
09:00
Repeat interval
Every 14 minutes
Ready to board
09:47
Next catchable departure unknown
Measure elapsed time from the service origin. The minute display by itself is not the number of elapsed minutes.Try with the CKSTEM Thinking Compass
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When is the next departure they can catch, and how many minutes must they wait?
How long until the shuttle?
The fictional Bridge Shuttle departs every 14 minutes from 09:00, including a departure at 09:00. The team is ready to board at 09:47 that morning.
A group ready at the exact departure time can board. There is space for them.
Starting point
09:00
Repeat interval
Every 14 minutes
Ready to board
09:47
Next catchable departure unknown
Measure elapsed time from the service origin. The minute display by itself is not the number of elapsed minutes.
Your answer
Enter departure and wait. Check the service origin and exact-ready boarding rule.
These quantities fit. Check departure and wait. Check the service origin and exact-ready boarding rule.
Break it into steps
Step 1 of 3
How many minutes past the last departure is 09:47?
Forty-seven is three complete periods plus five minutes. Nine minutes remain in the current period.
Step 2 of 3
What is your answer for Departure (a.m. h:mm)?
Departure (a.m. h:mm): 9:56. Compare this quantity with the complete worked reasoning; the number alone does not check your argument.
Step 3 of 3
What is your answer for Wait (minutes)?
Wait (minutes): 9. Compare this quantity with the complete worked reasoning; the number alone does not check your argument.
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 wait
Answer: The next departure is 09:56. They wait 9 minutes.
Use the service's starting point. At 09:47, 47 minutes have elapsed since 09:00:
\[47=3\times14+5\]
Three complete periods take us to 09:42. The 5 extra minutes have already passed since that departure. Another \(14-5=9\) minutes completes the next period.
09:42 · departure09:56 · departure
5 elapsed9 to wait
The whole strip is 14 minutes.
Five minutes of this period have already passed. The remaining nine minutes lead to the next departure; the two lengths use the same minute scale.
Rowan “I found a remainder of five. I nearly called that the wait.”
CKSprout “Where does that five belong in the journey?”
Mina “It is the part of the period already gone. We need the other nine minutes.”
Check: 09:56 is \(4\times14=56\) minutes after 09:00, and it is 9 minutes after 09:47. The previous departure, 09:42, has gone.
Listing a short timetable would also work. For a time after 10:00, still count elapsed minutes from 09:00; the clock's minute display alone does not give that elapsed time.
The unfinished sleeve
While they organise their supplies, Mina checks a separate badge order.
There are 27 visiting clubs with 18 people each, plus 7 volunteers. These are distinct people, each counted once; Mina and Rowan are already among the volunteers. Everyone gets one badge.
A sleeve has 8 pockets, with one badge per pocket. Fill complete sleeves first.
How many badges go in the unfinished sleeve? How many of its pockets remain empty?
People to include
27 clubs × 18 people
7 volunteers
One sleeve
Eight pockets; final fill unknown
One badge per pocket
Everyone is counted once. Complete sleeves come first; the question concerns the unfinished one.Try with the CKSTEM Thinking Compass
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How many badges go in the unfinished sleeve? How many of its pockets remain empty?
The unfinished sleeve
While they organise their supplies, Mina checks a separate badge order.
There are 27 visiting clubs with 18 people each, plus 7 volunteers. These are distinct people, each counted once; Mina and Rowan are already among the volunteers. Everyone gets one badge.
A sleeve has 8 pockets, with one badge per pocket. Fill complete sleeves first.
People to include
27 clubs × 18 people
7 volunteers
One sleeve
Eight pockets; final fill unknown
One badge per pocket
Everyone is counted once. Complete sleeves come first; the question concerns the unfinished one.
Your answer
Enter both counts. Explain why combined extras need another reduction and check the original batch.
These quantities fit. Check both counts. Explain why combined extras need another reduction and check the original batch.
Break it into steps
Step 1 of 3
What do the reduced extras total before reducing again?
Three times two plus seven is thirteen. Remove another complete eight, leaving five.
Step 2 of 3
What is your answer for Badges in unfinished sleeve?
Badges in unfinished sleeve: 5. Compare this quantity with the complete worked reasoning; the number alone does not check your argument.
Step 3 of 3
What is your answer for Empty pockets?
Empty pockets: 3. Compare this quantity with the complete worked reasoning; the number alone does not check your argument.
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 badge sleeve
Answer: The unfinished sleeve holds 5 badges and has 3 empty pockets.
We need the leftover by 8, rather than the units digit. Remove 24 of the clubs in three groups of 8 clubs. Each such group supplies a multiple of 8 badges. Three clubs remain.
In each remaining club, 16 of the 18 badges fill two complete sleeves, leaving 2 extra badges. Those extras and the volunteers' badges make \(3\times2+7=13\).
Remove 24 clubs
24 × 18 = 432 badges
54 complete sleeves
From each of the 3 clubs left
3 × 16 = 48 badges
6 complete sleeves; 2 extras per club
Combine the remaining extras
3 × 2 + 7 = 13 badges
1 more complete sleeve and 5 extras
Each discarded amount fills whole eight-pocket sleeves. Combining the extras can fill another sleeve, so the combined count must be reduced again.
Thirteen still fills one complete sleeve. Reduce again: \(13=1\times8+5\).
5 badges · 3 empty pockets
The unfinished sleeve has eight equal pockets: five hold badges and three are empty.
Mina “The extras made thirteen, so I thought the remainder was thirteen.”
CKSprout “What can you still do with those extras?”
Rowan “Fill another sleeve. Then only five remain, and three pockets are empty.”
Full check: \(27\times18+7=493\), and \(493=61\times8+5\). The unfinished sleeve has \(8-5=3\) empty pockets. Directly calculating 493 and dividing is another valid route.
The same complete-group idea works for multiplication and addition because discarded groups produce more complete groups. It does not depend on these particular badge numbers.
What did we leave behind?
Badges still use 8-pocket sleeves, with one badge per pocket. Compare orders of 17 badges and 25 badges; no badge may be omitted.
Do they leave the same number of badges in an unfinished sleeve? Does that remainder alone tell you how many sleeves to order? Find the sleeves needed for each order, then for an order of 493 badges.
Try with the CKSTEM Thinking Compass
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Do they leave the same number of badges in an unfinished sleeve? Does that remainder alone tell you how many sleeves to order? Find the sleeves needed for each order, then for an order of 493 badges.
What did we leave behind?
Badges still use 8-pocket sleeves, with one badge per pocket. Compare orders of 17 badges and 25 badges; no badge may be omitted.
Your answer
Enter the sleeves needed for each original order, including a sleeve for any remaining badges. Also answer the questions about equal leftovers and whether the remainder alone is enough.
The sleeve counts fit. Explain why the 17- and 25-badge orders have equal leftovers but need different numbers of sleeves.
Break it into steps
Step 1 of 6
How many complete eight-pocket sleeves can 17 badges fill?
Two complete sleeves hold 16 badges. One badge remains, so the order needs another sleeve.
Step 2 of 6
How many complete eight-pocket sleeves can 25 badges fill?
Three complete sleeves hold 24 badges. One badge remains. The leftover is the same as for 17, although the complete-sleeve count differs.
Step 3 of 6
How many complete eight-pocket sleeves can 493 badges fill?
61 × 8 = 488 and 62 × 8 = 496. There are 61 complete sleeves and five badges still needing pockets.
Step 4 of 6
How many sleeves must be ordered for all 17 badges?
Two full sleeves and one unfinished sleeve require 3 sleeves in all: 16 < 17 ≤ 24.
Step 5 of 6
How many sleeves must be ordered for all 25 badges?
Three full sleeves and one unfinished sleeve require 4 sleeves: 24 < 25 ≤ 32. Equal leftovers do not fix the number of sleeves to order.
Step 6 of 6
How many sleeves must be ordered for all 493 badges?
Another sleeve holds the five remaining badges, so order 62: 488 < 493 ≤ 496. Check the complete groups as well as the leftovers when explaining all three orders.
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: ordering sleeves
Answer: Both leave 1 badge, but the remainder alone is not enough to order sleeves. Seventeen badges need 3 sleeves; 25 need 4; 493 need 62.
17 badges
Full sleeve
Full sleeve
Unfinished: 1 badge
2 full + 1 unfinished = 3 sleeves
25 badges
Full sleeve
Full sleeve
Full sleeve
Unfinished: 1 badge
3 full + 1 unfinished = 4 sleeves
The same one-badge leftover accompanies different numbers of complete sleeves. Equal pockets and sleeve sizes make the comparison faithful.
Seventeen fills 2 sleeves and needs one more. Twenty-five fills 3 and needs one more. The identical leftover does not preserve the number of full sleeves.
Rowan “Both orders have one badge left, but one order needs a whole extra sleeve.”
CKSprout “What information does this new request need?”
Mina “The number of complete sleeves too. For 493, sixty-one are full and we need a sixty-second.”
Check: 2 sleeves hold only 16 badges, while 3 hold 24. Three hold fewer than 25, while 4 hold 32. For 493, 61 sleeves hold 488 and 62 hold 496. Each answer is enough, and one fewer is not.
Does sharing preserve the leftover?
An order of 14 badges and an order of 6 badges leave the same remainder when divided by 8.
Share each order equally between 2 people, with no badge split.
Do the resulting shares still have the same remainder by 8? Give both shares and explain what happened.
Try with the CKSTEM Thinking Compass
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Do the resulting shares still have the same remainder by 8? Give both shares and explain what happened.
Does sharing preserve the leftover?
An order of 14 badges and an order of 6 badges leave the same remainder when divided by 8.
Share each order equally between 2 people, with no badge split.
Your answer
Enter the shares. Also decide whether their remainders by eight are equal and explain why arbitrary division is unsafe.
These quantities fit. Check the shares. Also decide whether their remainders by eight are equal and explain why arbitrary division is unsafe.
Break it into steps
Step 1 of 3
How much do the original orders differ by?
Their difference is eight. After halving, the difference becomes four, which is no longer a complete eight-group.
Step 2 of 3
What is your answer for Half of fourteen?
Half of fourteen: 7. Compare this quantity with the complete worked reasoning; the number alone does not check your argument.
Step 3 of 3
What is your answer for Half of six?
Half of six: 3. Compare this quantity with the complete worked reasoning; the number alone does not check your argument.
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: sharing
Answer:No. The shares are 7 badges and 3 badges, whose remainders by 8 differ.
Compare amounts and their remainders by 8
Before sharing
After sharing
Original count
Left by 8
Each share
Left by 8
14
6
7
7
6
6
3
3
Both original leftovers are six. Halving the original amounts gives shares of seven and three, so unrestricted division does not preserve the same remainder.
Check: the original amounts differ by 8, a complete group. After halving, the shares differ by only 4. That is no longer a complete 8-group.
Addition and multiplication preserve same-remainder information with a fixed group size. Division needs its own justification. This example shows why we cannot simply halve matching remainders and keep the same modulus.
Today, compare the whole journey
Today the team is ready at the transport plaza at 09:47. The exhibition opens at 11:10. From the Museum Stop, walking takes 5 minutes, then setup takes 10 minutes.
Choose one of the two complete routes below. Every service runs from its listed origin, at its listed interval, that same morning. Rides and transfer times are fixed; there is space. A group ready at a departure can board it. Each transfer allowance must be completed before boarding the second service.
Bridge route
Bridge Shuttle: every 14 min from 09:00; ride 12 min.
Allow 3 min to transfer.
Clocktower Shuttle: every 45 min from 09:15; ride 22 min.
Market route
Market Shuttle: every 19 min from 09:05; ride 12 min.
Allow 3 min to transfer.
Riverside Shuttle: every 20 min from 09:00; ride 22 min.
These are the only two complete routes in the model. Each repeat interval belongs to its own service and starting point.
Which route can get the display ready by 11:10? Show its journey times. Explain why choosing the first shuttle to leave does not settle the decision.
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Which route can get the display ready by 11:10? Show its journey times. Explain why choosing the first shuttle to leave does not settle the decision.
Today, compare the whole journey
Today the team is ready at the transport plaza at 09:47. The exhibition opens at 11:10. From the Museum Stop, walking takes 5 minutes, then setup takes 10 minutes.
Choose one of the two complete routes below. Every service runs from its listed origin, at its listed interval, that same morning. Rides and transfer times are fixed; there is space. A group ready at a departure can board it. Each transfer allowance must be completed before boarding the second service.
Bridge route
Bridge Shuttle: every 14 min from 09:00; ride 12 min.
Allow 3 min to transfer.
Clocktower Shuttle: every 45 min from 09:15; ride 22 min.
Market route
Market Shuttle: every 19 min from 09:05; ride 12 min.
Allow 3 min to transfer.
Riverside Shuttle: every 20 min from 09:00; ride 22 min.
These are the only two complete routes in the model. Each repeat interval belongs to its own service and starting point.
Your answer
Enter the working route’s event times. Also name it and explain why the first shuttle to leave does not decide the whole journey.
These quantities fit. Check the working route’s event times. Also name it and explain why the first shuttle to leave does not decide the whole journey.
Break it into steps
Step 1 of 5
How many minutes must the 09:47 group wait for Bridge?
Forty-seven is three fourteens plus five, so Bridge is nine minutes away. An earlier first departure does not establish an earlier complete arrival.
Step 2 of 5
What is your answer for First shuttle (a.m.)?
First shuttle (a.m.): 10:02. Compare this quantity with the complete worked reasoning; the number alone does not check your argument.
Step 3 of 5
What is your answer for Second shuttle (a.m.)?
Second shuttle (a.m.): 10:20. Compare this quantity with the complete worked reasoning; the number alone does not check your argument.
Step 4 of 5
What is your answer for Museum (a.m.)?
Museum (a.m.): 10:42. Compare this quantity with the complete worked reasoning; the number alone does not check your argument.
Step 5 of 5
What is your answer for Ready (a.m.)?
Ready (a.m.): 10:57. Compare this quantity with the complete worked reasoning; the number alone does not check your argument.
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: today's journey
Answer: Take the Market route: depart 10:02, connect at 10:20, reach the museum at 10:42, and finish walking and setup at 10:57. The earlier Bridge departure leads to a missed opening.
Working back through the final 15 minutes gives a Museum Stop deadline of 10:55. Now use each service's own starting point to find its next departure, then follow its connection.
Bridge route
Board first shuttle
Reach interchange
Transfer complete; ready for connection
Board second shuttle
Reach Museum Stop
Final walk and setup complete
Too late for 11:10
Market route
Board first shuttle
Reach interchange
Transfer complete; ready for connection
Board second shuttle
Reach Museum Stop
Final walk and setup complete
Ready before 11:10
Follow the complete sequence, including transfers. The first departure is earlier on Bridge, but the museum-ready time is earlier on Market. These rows record events; their spacing is not a duration scale.
Bridge leaves first at 09:56. The team reaches its interchange at 10:08 and finishes transferring at 10:11. The next Clocktower departure is 10:45: it reaches the museum at 11:07, and setup finishes at 11:22.
Market starts at 09:05. At 09:47, \(42=2\times19+4\) minutes have elapsed since that origin, so its next departure is at 10:02. After the ride and transfer, the team is ready at 10:17. Riverside's next departure is 10:20.
Mina “Bridge leaves six minutes sooner. But the next connection makes its whole journey later.”
CKSprout “What should your final check include?”
Rowan “Both transfers, both service origins, and the walk and setup after we arrive.”
Full check: Market's 10:02 ride ends 10:14; the 3-minute transfer ends 10:17; the 10:20 ride ends 10:42; walking ends 10:47; setup ends 10:57, before 11:10. Bridge's earliest feasible connection is already too late. A later Bridge shuttle cannot catch an earlier Clocktower connection, so it cannot rescue that route.
Remainders locate departures. They do not compare an entire journey by themselves. Following the next connection before choosing is a useful way to look ahead.
Yesterday's deadline
Mina remembers a separate rehearsal journey from the day before. The direct service was different from today's two-route choice.
The rehearsal display had to be ready by 11:00. Setup took 7 minutes, the walk from the museum stop took 9 minutes, and the direct shuttle ride took 26 minutes.
That shuttle departed every 14 minutes from 09:00 that morning. Walking from home to its platform took 8 minutes. All durations were fixed, there was space, and being ready at the exact departure was enough to board.
What was the latest time to leave home and still have the display ready by 11:00? Check the complete journey and explain why leaving later could not work.
Finish
Display ready by 11:00
Setup 7 min
After the shuttle
Final walk 9 min
Ride 26 min
Before the shuttle
Home walk 8 min
Every 14 min from 09:00
The deadline fixes the finish. The repeating timetable restricts which departure can fit the recovered time limit.Try with the CKSTEM Thinking Compass
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What was the latest time to leave home and still have the display ready by 11:00? Check the complete journey and explain why leaving later could not work.
Yesterday's deadline
Mina remembers a separate rehearsal journey from the day before. The direct service was different from today's two-route choice.
The rehearsal display had to be ready by 11:00. Setup took 7 minutes, the walk from the museum stop took 9 minutes, and the direct shuttle ride took 26 minutes.
That shuttle departed every 14 minutes from 09:00 that morning. Walking from home to its platform took 8 minutes. All durations were fixed, there was space, and being ready at the exact departure was enough to board.
Finish
Display ready by 11:00
Setup 7 min
After the shuttle
Final walk 9 min
Ride 26 min
Before the shuttle
Home walk 8 min
Every 14 min from 09:00
The deadline fixes the finish. The repeating timetable restricts which departure can fit the recovered time limit.
Your answer
Enter the latest home departure. Explain the backward stages, timetable adjustment, full forward check and why later fails.
These quantities fit. Check the latest home departure. Explain the backward stages, timetable adjustment, full forward check and why later fails.
Break it into steps
Step 1 of 2
How many minutes past the last scheduled departure is the 10:18 limit?
Seventy-eight is five fourteens plus eight. Move back eight minutes to the last actual departure before the limit, then allow the home walk.
Step 2 of 2
What is your answer for Leave home (a.m. h:mm)?
Leave home (a.m. h:mm): 10:02. Compare this quantity with the complete worked reasoning; the number alone does not check your argument.
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 rehearsal
Answer: Leave home no later than 10:02, to catch the 10:10 shuttle. The display is ready at 10:52.
Work backwards
Subtract setup, final walking and riding from the goal. The shuttle must depart at or before 10:18.
Goal: display ready by this time
← Back 7 min: latest setup start
← Back 9 min: latest museum-stop arrival
← Back 26 min: latest possible shuttle departure
← Back 8 min: last scheduled departure at or before the limit
← Back 8 min: latest home departure
Read down from the finish to earlier times. Teal arrows work backwards; the timetable step moves to an actual departure at or before the limit.
But 10:18 is a limit, not a scheduled departure. It is 78 minutes after 09:00:
\[78=5\times14+8\]
Modular reduction identifies the part of the period already passed. Move back 8 minutes to the last allowed departure: 10:10. Then allow the 8-minute home walk.
Rowan “Yesterday we wanted a departure before a limit. Today we wanted the one after we were ready.”
CKSprout “Which goal tells you which way to move?”
Mina “Yesterday's deadline sends us back to 10:10. Going forward to 10:24 would be too late.”
Leave home
→ Walk 8 min; board shuttle
→ Ride 26 min; reach museum stop
→ Walk 9 min; reach museum
→ Set up for 7 min; ready before 11:00
Amber arrows replay the actual forward journey. Finishing at 10:52 satisfies the 11:00 deadline; a later scheduled departure would not.
Full check: leave 10:02, reach the platform 10:10, ride to 10:36, walk to 10:45, and finish setup at 10:52. Leaving later misses 10:10; the next shuttle at 10:24 would finish at 11:06. Later services cannot improve that fixed journey, so 10:02 is the latest feasible home departure.
How the tools fit: Work backwards supplies the latest possible time; Modular reduction places it on the repeating schedule. The forward replay checks the original request.
At the museum, the team finds one record that lists a leftover instead of the original order.
A separate batch contains some complete strips of 14 badges each, plus 9 spare badges. The strip count is a whole number from 1 to 12, inclusive.
Put every badge into 8-pocket sleeves, filling complete sleeves first. The unfinished sleeve contains exactly 5 badges.
Find every possible strip count. For each, give the total badges and show that complete sleeves leave exactly 5. Explain why no other allowed count works.
Unknown strip count
Whole count from 1 to 12
14 badges per strip
Additional badges
9 spares
Recorded leftover
Exactly 5 in the unfinished sleeve
The five-badge leftover is given; the original strip count is unknown. Every complete sleeve holds eight badges.A question to help me begin
One extra strip adds fourteen badges. How much of that changes the leftover in an eight-pocket sleeve? Try several consecutive strip counts, then look for a complete repeating cycle.
Try with the CKSTEM Thinking Compass
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Find every possible strip count. For each, give the total badges and show that complete sleeves leave exactly 5. Explain why no other allowed count works.
The missing packing order
At the museum, the team finds one record that lists a leftover instead of the original order.
A separate batch contains some complete strips of 14 badges each, plus 9 spare badges. The strip count is a whole number from 1 to 12, inclusive.
Put every badge into 8-pocket sleeves, filling complete sleeves first. The unfinished sleeve contains exactly 5 badges.
Unknown strip count
Whole count from 1 to 12
14 badges per strip
Additional badges
9 spares
Recorded leftover
Exactly 5 in the unfinished sleeve
The five-badge leftover is given; the original strip count is unknown. Every complete sleeve holds eight badges.
Your answer
Enter every allowed strip count with its total badges and number of complete sleeves. Add rows as needed. Explain why each leaves five badges and why no allowed count is missing.
The listed rows fit. Check each against the original batch and leftover conditions, then explain why your list covers every allowed strip count.
Break it into steps
Help step 1
After removing a complete group of eight from one strip of 14, how many badges remain?
One strip contributes six beyond a complete sleeve. Each extra strip therefore changes the leftover by six, reducing complete eights again.
Help step 2
What remainder must all the strips together leave, before adding the nine spares?
The strip contribution must leave four: four plus the spare contribution of one gives the required five.
Help step 3
What is the smallest positive number of extra strips that restores the same leftover?
The first such multiple is 4 × 14 = 56 = 7 × 8. Adding four strips leaves the unfinished-sleeve count unchanged. Check one complete four-count cycle before using that repeat.
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 missing order
Answer:2, 6 or 10 strips, giving 37, 93 or 149 badges. They fill 4, 11 or 18 complete sleeves, respectively, with exactly five left each time. No other count from one through twelve works.
Rowan “The leftover does not tell us the total. Several orders could leave five.”
CKSprout “What changes in the leftover when one more strip is added?”
Mina “Fourteen adds one complete eight and six extras. I can follow the extras until their state returns.”
Use a variable
Let \(x\) count whole strips from one through twelve. The total is \(14x+9\). The same unknown must count every fourteen-badge strip; the nine spares are added only once.
Modular reduction removes the complete eight from each strip and from the spares. The total therefore has the same remainder by eight as \(6x+1\). Follow the remainder of this expression for consecutive strip counts:
Four consecutive strip counts
Strips
Badges left by eight
1
7
2
5
3
3
4
1
Each extra strip adds six to the remainder, then complete eights are removed again. After four strips, the remainder state returns.
Casework
Four additional strips add fifty-six badges, exactly seven complete sleeves, so the leftover repeats after every four strips. Every whole strip count has exactly one of the remainders zero, one, two or three by four. These four cases cover the entire count range; they are not just four hopeful test values.
All four strip-count cases
Strip count left by four
Total badges left by eight
Leaves five?
0
1
Does not fit
1
7
Does not fit
2
5
Fits
3
3
Does not fit
Adding four strips changes the full total by fifty-six, a complete number of eight-badge groups. The four rows therefore cover every whole count, not just four examples.
The remainder-two case is the only one leaving five badges by eight. In one through twelve, that means 2, 6 and 10. Return to full totals to learn the sleeve counts:
Every fitting count from 1 to 12
Strips
Total badges
Full sleeves
Left over
2
37
4
5
6
93
11
5
10
149
18
5
Each total uses fourteen badges per strip and nine spares. These three original records all leave five; the remainder-class argument excludes every other count in the allowed range.
Check every original condition: \(37=4\times8+5\), \(93=11\times8+5\) and \(149=18\times8+5\). Each total also equals fourteen per stated strip plus nine spares. All three counts are in range. The complete four-case cycle excludes every other allowed count.
Another representation: Solving a linear congruence
Solving a linear congruence
A congruence states matching remainders. The same problem can be written:
\[14x+9\equiv5\pmod8\]\[6x\equiv4\pmod8\]
Matching remainders mean the difference is a complete multiple of eight: for an integer \(q\),
\[6x-4=8q\]\[3x-2=4q\]\[3x\equiv2\pmod4\]
We halved the integer equality, so its group size changes from eight to four. We did not cancel while keeping the old modulus. Since \(3\times3=9\) leaves one by four, multiplying the last congruence by three gives \(x\equiv2\pmod4\), the same class found by the concrete model.
How the tools fit: the variable preserves the unknown total. Reduction removes irrelevant full sleeves. Casework turns the four recurring states into a complete argument for all allowed counts. Testing all twelve full totals is valid too; the cycle reduces the work and explains why the answers are four apart.
Your turn
Try these before opening the answers. Decide what the leftover means and which information the request still needs.
Right on the departure
A shuttle departs every 14 minutes from 09:00 that morning. There is space, and being ready at the exact departure is enough to board.
Find the next catchable departure and wait for a group ready at 10:10, then for a group ready at 10:11.
Try with the CKSTEM Thinking Compass
Interactive checking needs JavaScript. The complete question and worked answer remain available.
Find the next catchable departure and wait for a group ready at 10:10, then for a group ready at 10:11.
Right on the departure
A shuttle departs every 14 minutes from 09:00 that morning. There is space, and being ready at the exact departure is enough to board.
Your answer
Enter departure and wait for both readiness times. Explain the zero-remainder boundary.
These quantities fit. Check departure and wait for both readiness times. Explain the zero-remainder boundary.
Break it into steps
Step 1 of 5
How many complete fourteen-minute periods end at 10:10?
Seventy is five fourteens. Exact-ready boarding allows this departure with no further wait.
Step 2 of 5
If the group is ready at 10:10, what is its next catchable departure?
10:10 is exactly five fourteen-minute periods after 09:00. Exact-ready boarding is allowed, so this group can catch the 10:10 departure.
Step 3 of 5
If the group is ready at 10:10, how many minutes must it wait?
The group is ready at the exact 10:10 departure, so it waits zero minutes. Equality at departure is explicitly allowed.
Step 4 of 5
If the group is ready at 10:11, what is its next catchable departure?
The 10:10 departure has already left. One further fourteen-minute period gives 10:24, the next catchable departure.
Step 5 of 5
If the group is ready at 10:11, how many minutes must it wait?
From 10:11 until the next departure at 10:24 is 13 minutes. Check this against the original departure times, rather than treating the one-minute remainder as the wait.
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 exact departure
Answer: At 10:10, catch 10:10 with 0 minutes waiting. At 10:11, catch 10:24 with 13 minutes waiting.
Check: 70 minutes from 09:00 is exactly five 14-minute periods. At 71 minutes, one minute of the next period has gone. A zero remainder does not require a full extra period when exact-ready boarding is allowed.
Are the extras finished?
A different order has 24 packs of 19 badges, plus 6 spare badges. Sleeves have 7 pockets, one badge per pocket; fill complete sleeves first.
How many badges remain after complete sleeves are filled? Is there an unfinished sleeve?
Try with the CKSTEM Thinking Compass
Interactive checking needs JavaScript. The complete question and worked answer remain available.
How many badges remain after complete sleeves are filled? Is there an unfinished sleeve?
Are the extras finished?
A different order has 24 packs of 19 badges, plus 6 spare badges. Sleeves have 7 pockets, one badge per pocket; fill complete sleeves first.
Your answer
Enter the remainder. Also decide whether any unfinished sleeve remains.
These quantities fit. Check the remainder. Also decide whether any unfinished sleeve remains.
Break it into steps
Step 1 of 2
How many reduced extra badges are there before regrouping?
Three times five plus six is twenty-one, which fills three complete seven-pocket sleeves.
Step 2 of 2
What is your answer for Badges left?
Badges left: 0. Compare this quantity with the complete worked reasoning; the number alone does not check your argument.
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: complete sleeves
Answer:0 badges remain. There is no unfinished sleeve.
By 7, use remainders 3 for 24 and 5 for 19. The extras make \(3\times5+6=21\), which fills three more complete sleeves. Reduce again to 0.
Full check: \(24\times19+6=462=66\times7\). Every badge fits a complete sleeve.
A changed leftover
A separate batch has a whole strip count from 1 to 12, with 14 badges per strip, plus 9 spares. All badges are packed in 8-pocket sleeves, filling complete sleeves first.
The record claims the unfinished sleeve contains exactly 4 badges.
Find all possible strip counts, or explain why none can work.
CKSTEM Thinking CompassPlan and check your reasoning.
Find all possible strip counts, or explain why none can work.
A changed leftover
A separate batch has a whole strip count from 1 to 12, with 14 badges per strip, plus 9 spares. All badges are packed in 8-pocket sleeves, filling complete sleeves first.
The record claims the unfinished sleeve contains exactly 4 badges.
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.
Show the answer and reasoning: the changed record
Answer:None can work. The claimed leftover is incompatible with every allowed strip count.
Fourteen times any whole count is even; adding 9 makes the total odd. Complete sleeves use an even number of badges. An odd total therefore leaves an odd leftover, so it cannot leave 4.
Check the contradiction: a total of complete 8-badge groups plus 4 would be even, while every permitted batch is odd. This rules out every count, rather than just some tested ones.
When is the remainder enough?
The display is ready. The team has checked a final digit, a waiting time, a packing remainder, complete sleeve orders and a whole journey. The question decided which information mattered.
You can discard complete groups when they cannot change the answer you need. Keep their count when the request needs a full total. Keep the starting point when the question concerns a timetable. And check the original situation after a shortcut has done its work.
The missing order connects remainders to an unknown and a complete case split. The optional algebra expresses the same argument with symbols. Earlier, Look for a pattern used Modular reduction for recurring positions; this guide develops what information a remainder preserves and what it loses.
Mina “There are so many numbers on our records. Do we need every part of every number?”
CKSprout “What are you trying to find out first?”
Rowan “We should decide that before throwing any information away.”