Problem
Farthest travel under a carry limit
Two soldiers are 160 km from base. One day's rations carries a soldier 20 km. Nobody may set out with more than 6 days' rations. Find how one of them can reach the base.
Reasoning · Grade 6-2 Optimal Design
Your answer
Watch out Handing over everything fails: the escort needs food to get back.
How to solve
Strategy Draw a Diagram — The two limits in this problem are written in different units -- one is a distance, the other a number of days of food -- so the first job is to put them on the same footing by measuring everything in days. Once the journey is 8 days long and the sack holds 6 days, the shortfall is a plain 2 days, and the whole problem becomes 'get 2 extra days of food into the walker's sack without breaking the 6-day cap'. Then a day-by-day picture does the rest: draw a line marked 0 to 8 days, put a row of ration dots under each soldier, and cross one dot off per day. Trying the handover on day 1, then day 2, then day 3 and checking each against the two rules is exactly the kind of short, bounded guess-and-check this problem invites, and listing the outcomes side by side shows that only one of them survives both rules.
1STEP 1
Measure the journey in days instead of kilometres
In days, the trip needs 8 days' rations.
160 ÷ 20 = 8 days
Dividing a total distance by a distance-per-day is the same 'how many groups of 20 fit into 160?' division used for sharing objects, and it converts the whole problem into one unit.
4.NBT.B.6Analyze The Units2STEP 2
See exactly how far short one soldier falls
Carrying six, a soldier is 2 days short.
6 × 20 = 120 km, 160 - 120 = 40 km, 8 - 6 = 2 days
Naming the shortfall as a number -- 2 days -- turns a vague 'he hasn't got enough' into a target the second soldier has to deliver.
4.MD.A.2Analyze The UnitsOne soldier alone falls short by a definite number of days, and that shortfall is what the handover has to cover.
Why?
The journey is the days walked laid end to end, so the food carried and the days needed can be set directly against each other.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
Why?
Each day costs exactly one day's food, so measuring the journey in days makes the carrying limit a plain count.
🧱A bigger unit is a fixed count of smaller unitsA bigger unit always equals the same fixed number of smaller units (1 hour = 60 minutes, 1 km = 1000 m), so a measure can be renamed between them.
3STEP 3
Realise the extra food must be handed over out in the desert
That shortfall must be handed over on the way.
Thinking of the sack as a container that empties as it is used, rather than as a fixed load, is the whole idea -- space to receive is created by walking.
4.OA.A.3Draw A Diagram4STEP 4
Try the handover on different days and check both rules
Trying each day, only day 2 works.
day 1{:} 5+1=6 ＜ 7; day 3{:} 3-3=0 to spare; day 2{:} 4+2=6=6
There are only a few days worth testing, and each test is one small addition and one small subtraction, so checking them all is quicker than trying to be clever.
4.OA.A.3Guess And Check5STEP 5
Write out the plan and follow the food day by day
Then the walker holds 6 days' rations.
2 × 20 = 40 km; 6 - 2 = 4; 4 - 2 = 2 (escort); 4 + 2 = 6 (walker)
Tracking two sacks at once is easy if you write the two numbers under each day of the picture, the way the answer key draws rows of ration dots and crosses one off per day.
4.OA.A.3Draw A Diagram6STEP 6
Check that Soldier 2 makes it all the way
Already 40 km along, he reaches the base.
6 × 20 = 120 km, 40 + 120 = 160 km
Counting the remaining days off one at a time along the drawn line is the surest way to see that the last day of food really does coincide with the last stretch of road.
4.MD.A.2Make A Systematic ListAnswer
walk together 2 days, hand over 2 days' rations, turn back
40 + 120 = 160
Every quantity stays in days of food or kilometres, and the two match up at 20 km per day throughout. All the numbers land where they should: nobody ever holds more than 6 days' rations (Soldier 2 is at exactly 6 at the handover, which the rule allows), nobody ever holds a negative amount, and each soldier's food runs out precisely as he arrives -- Soldier 1 at the village, Soldier 2 at the base. The total food eaten is a good cross-check: Soldier 1 walks 40 km out and 40 km back, which is 80 km or 4 days, and Soldier 2 walks the full 160 km, which is 8 days; that is 12 days of food in all, and 12 is exactly the 6 + 6 the two men carried out of the village, with nothing left over and nothing missing.
Takeaway
Turn the kilometres into days of food, and the escort's job becomes obvious: walk 2 days, hand over 2 days, keep 2 days to get home.
- Measure the journey in days instead of kilometres
- See exactly how far short one soldier falls
- Realise the extra food must be handed over out in the desert
- Try the handover on different days and check both rules
- Write out the plan and follow the food day by day
- Check that Soldier 2 makes it all the way