Ration Endurance and a Patrol Intercept: Two Calculations for the Quartermaster
A company of 120 personnel at a border post holds ration stock calculated to last 45 days at the authorised scale of one ration per person per day. The scale is not changed at any stage. On the morning of day 11, thirty personnel of a relieving column are attached to the post for an indefinite period. On the morning of day 21, forty of the original strength are despatched to a neighbouring post, again permanently for the period under consideration. The quartermaster must report to the commandant how long the balance stock will last, and whether it can be stretched to cover a further 30 days beyond day 21 without indenting fresh ration, since the supply column is delayed. Separately, the post runs a jeep patrol that leaves at 05:00 at an average of 4 km/h on the border track. The patrol's ordered turning point is 40 km out. At 08:00 a written message is found still lying at the post; a cyclist can leave immediately at an average of 6 km/h on the same track.
1. How long will the stock actually last, and on what day will it run out? Show the person-day working.
Read the worked answer
Work everything in person-days, because the strength changes.
Total holding = 120 x 45 = 5,400 person-days.
Days 1 to 10 (120 men): consumed 120 x 10 = 1,200. Balance = 5,400 - 1,200 = 4,200 person-days on the morning of day 11.
Days 11 to 20 (120 + 30 = 150 men): consumed 150 x 10 = 1,500. Balance = 4,200 - 1,500 = 2,700 person-days on the morning of day 21.
From day 21 (120 - 40 = 110 men): 2,700 / 110 = 24.545... days, i.e. 24 days and about 13 hours.
Total life = 10 + 10 + 24.55 = 44.55 days from day 1, so the stock runs out on day 45, shortly after midday.
Note what the shortcuts would have got wrong. Assuming 120 men throughout gives 45 days and hides the entire problem. Assuming the stock 'runs out' at day 20 plus 2,700/150 = 18 days would ignore the 40 men who left. The person-day method is the only one that survives a change of strength, and it is worth stating explicitly in the report that the answer is a fraction of a day, not a whole number, because the issue is drawn at a fixed hour.
2. By what percentage must the daily ration per head be reduced from day 21 so that the stock lasts 50 days in all? Is that a realistic measure?
Read the worked answer
Requirement: the stock must last to day 50, that is 30 more days from the morning of day 21, feeding 110 men.
Person-days demanded = 110 x 30 = 3,300. Person-days actually held = 2,700. Shortfall = 600 person-days.
Required scaling factor = 2,700 / 3,300 = 0.8182. Reduction = 1 - 0.8182 = 0.1818, i.e. about 18.2 per cent.
Check by substitution: 110 x 0.8182 = 90.0 person-equivalents per day; 90.0 x 30 = 2,700 person-days, which matches the holding exactly.
The arithmetic is clean but the measure is not. An 18 per cent cut means roughly one in five and a half ration units disappears - for a border post on physical duties that is a cut in calories, not an administrative adjustment, and it will show up in fatigue and sickness within a fortnight. The correct answer to the commandant is therefore: the arithmetic says 18.2 per cent, but the professional recommendation is a priority indent for the shortfall of 600 person-days (about 5.5 days of full ration for the present strength) and a request to advance the supply column, with the percentage cut put in writing as a contingency only if the column slips further. Quantify first, then advise - do not simply hand over a fraction.
3. Where and when can the cyclist catch the patrol, and does the ordered turning point change the answer? Advise on whether the cyclist should be sent at all.
Read the worked answer
Head start: the patrol leaves at 05:00 and the cyclist at 08:00, so the patrol is already 4 x 3 = 12 km ahead.
Relative speed: 6 - 4 = 2 km/h.
Time to close = 12 / 2 = 6 hours after 08:00, i.e. the cyclist catches the patrol at 14:00.
Meeting point = 6 x 6 = 36 km from the post (check with the patrol: 4 km/h x 9 h = 36 km, so the two agree).
Does the turning point matter? It does. The patrol reaches 40 km at 15:00, so a 36 km meeting at 14:00 happens on the outward leg and is valid. If the ordered turning point had been 30 km, the patrol would have turned at 12:30; at that moment the cyclist (08:00 start, 6 km/h) is at only 27 km while the patrol is at 30 km coming back, closing at 10 km/h over a 3 km gap, meeting after 18 minutes at about 28.8 km. The headline answer of 14:00 and 36 km is therefore conditional on the turning point, and a candidate who quotes it without checking the turning point has not solved the problem.
Advice: the cyclist takes six hours over a message. If the message is time-critical, the patrol's own radio or a relayed voice message is the correct channel and the cyclist is a confirmation copy only. Send the cyclist for the written record, but never make him the primary means.