Lesson 8.6 — Replenishing Inventory and Changing Rates
补充库存与变化的速率
A stock can be used while more is arriving. When the number of users or the incoming rate changes, pause the story, update what remains, and begin a new period.
See inventory as a changing stock
Use a steady-flow model: supplies arrive and are used continuously at constant rates within each period. Each employee uses the same amount per day. Alternative trials begin with identical stock, and the stated run-out time is the first time the stock reaches zero.
A replenishing inventory has an amount already present, an inflow that adds more, and an outflow that removes it.
The stock changes because incoming items are added and outgoing use removes items.
Three quantities
Starting stock, S: inventory already present at day 0.
Incoming rate, g: inventory added each day.
Outgoing rate, n: inventory used or removed each day.
Stock after t days = S + gt − ntIdentify each quantity
Use two run-out times to reveal daily replenishment
A vending machine receives the same number of drinks every day. Its stock lasts 30 days for 5 employees, but only 20 days for 6 employees.
Measure in employee-days
One employee-day unit is the amount of drinks one employee uses in one day.
5 × 30 = 150 units used 6 × 20 = 120 units usedBoth totals include the same starting stock. The 30-day story also includes ten extra days of replenishment.
g = (150 − 120) ÷ (30 − 20) = 3The machine receives enough drinks each day for 3 employees for one day.
Subtracting the bars removes the same starting stock and isolates ten days of replenishment.
Build the comparison
Recover the stock that was there at the start
Once the replenishment rate is known, remove all newly added inventory from either complete scenario.
Use the 30-day scenario
S = 5×30 − 3×30 = 60Check with the 20-day scenario
S = 6×20 − 3×20 = 60Both routes reveal the same starting stock:
S = 60 employee-day unitsThe 30-day story contains 60 starting units and 90 units added during the 30 days.
Check the hidden stock
When demand changes, split the timeline
Start again with the 60-unit stock found in Mission 3. The company first has 4 employees. After 30 days, 2 more employees join. The replenishment remains 3 employee-day units per day.
Period 1 — four employees
Net depletion = 4 − 3 = 1 per day 60 − 1×30 = 30 units remainPeriod 2 — six employees
Net depletion = 6 − 3 = 3 per day 30 ÷ 3 = 10 more days 30 + 10 = 40 days altogetherSolve for four employees followed by six (regardless of the slider)
Apply the model to a replenished warehouse
In separate trials with identical starting stock and constant incoming deliveries, removing four truckloads per day empties a warehouse in 9 days, or five per day empties it in 6 days. Each truck can remove one truckload per day. Find the time one truck needs to carry only the original stock, ignoring later deliveries for that calculation.
Reveal the daily delivery into the warehouse
4×9 = 36 truckload-units 5×6 = 30 truckload-units g = (36−30) ÷ (9−6) = 2Recover the original stock
S = 36 − 2×9 = 18 truckloadsOne truck can carry the original stock alone in:
18 ÷ 1 = 18 daysThis question asks how long one truck needs for the original 18 truckloads, not for an endlessly replenished warehouse.
Check the warehouse model
Experiment with two periods
Change the starting stock, incoming rate, outgoing rate, and switch day. The graph will tell you whether the stock empties before the switch, after the switch, or never.
State the strategy
When the incoming rate changes instead
A company has typing work already waiting. Five equally productive typists finish in 24 days or nine in 12 days, with the same constant inflow and starting work. How many typists, working at a constant rate, would finish in 40 days if inflow stays unchanged for 8 days and then halves for the remaining 32 days?
Recover the original system
5×24 = S + 24g 9×12 = S + 12g g = (120−108) ÷ 12 = 1 S = 108−12 = 96Now split the 40-day timeline
For the first 8 days, new work arrives at 1 unit per day. For the next 32 days, it arrives at half that rate:
40n = 96 + 8×1 + 32×0.5 40n = 120, so n = 3 typistsThe outgoing rate stays the same; only the incoming work changes after day 8.
Complete the piecewise model
Recognize the same balance in pipes and escalators
Inventory is only one story. The same mathematics appears whenever a background flow adds to or works against a person’s action.
Pipe challenge
An initially empty pool has an inlet running alone for x minutes, without overflowing. It then stays open. With one identical outlet pipe open, the pool empties 12 minutes later. With two identical outlet pipes open, it empties 4 minutes later. An outlet’s rate may differ from the inlet’s rate.
The two outlet arrangements are separate trials beginning with the same stored water. Let inlet rate be a and one outlet rate be b.
ax = 12(b−a) ax = 4(2b−a)Equating the right sides gives:
12(b−a)=4(2b−a), so b=2a ax=12a, so x=12 minutesThe stored water is the starting stock; the inlet is continuing inflow; the outlets are continuing outflow.
Escalator moving downward
In separate trips on the same downward-moving escalator, a boy walks from bottom to top at 2 steps per second and takes 100 seconds. A girl walks from bottom to top at 3 steps every 2 seconds and takes 200 seconds. Walking rates are measured relative to the moving steps; all rates are constant, and each counted step advances one stair spacing.
Let the escalator move downward at e steps per second and let N be the number of visible steps when stopped.
N=(2−e)×100=(1.5−e)×200 e=1, and N=100Two people on an upward escalator
A brother starts at the top of an upward-moving escalator and walks to the bottom, taking 100 steps. His sister starts at the bottom at the same time and walks to the top, taking 50 steps. Each walking step advances one stair spacing relative to the moving escalator. All rates are constant, and the brother’s walking rate is twice the sister’s.
The brother takes 100 ÷ (2r) = 50/r seconds, and the sister takes 50 ÷ r = 50/r seconds. Their journey times are equal. If the escalator contributes E steps of upward movement during their shared time:
N=100−E=50+E 2E=50, so E=25 and N=75The escalator’s movement is a background rate: it can help or oppose the walker.
Check the transfer
A new replenishing supply — independent investigation
Try the eight steps yourself, using a small hint if needed. Check an attempt before opening worked review. Correct all eight to complete this investigation.
A supply receives the same amount each day. Six employees use it up in 12 days; eight use it up in 8 days. Each employee uses one unit per day. The separate trials begin with the same stock and end at the first run-out time. In a fresh trial, start again with this stock and unchanged replenishment: five employees use it for 6 days, followed by nine employees.
Write quantities in one consistent unit. Enter decimals or fractions such as 30/7; a decimal within 0.001 is accepted.
Hint
Multiply rate by time.
Worked solution
6 × 12 = 72.
Hint
Compare totals only after using the same unit.
Worked solution
8 × 8 = 64.
Hint
Use the two trials and cancel their identical starting stock.
Worked solution
(72 − 64) ÷ (12 − 8) = 2.
Hint
Track the extra time or remove all growth from total use.
Worked solution
72 − 12 × 2 = 48; also 64 − 8 × 2 = 48.
Hint
Subtract the daily replenishment from five employees’ daily use.
Worked solution
5 − 2 = 3 per day.
Hint
Subtract six days of net use from the starting stock.
Worked solution
48 − 6 × 3 = 30.
Hint
Subtract unchanged replenishment from nine employees’ daily use.
Worked solution
9 − 2 = 7 per day.
Hint
Divide remaining stock by its net depletion rate; check the time or capacity requested.
Worked solution
30 ÷ 7 = 30/7 ≈ 4.285714 additional days. Total time is 6 + 30/7 = 72/7 ≈ 10.285714 days.
Exit ticket
Answer all five questions. A perfect score completes the exit ticket. The certificate also requires the other core missions and the independent investigation.
Optional reflection — not automatically graded
Certificate of Mastery
Dynamic-Inventory Systems Engineer
This certifies that the learner can recover replenishment and starting stock, split a story when rates change, distinguish original stock from continuing inflow, and solve a fresh two-period inventory problem.
Freeze the timeline → update what remains → restart with the new ratesLesson 8.6 • Grade 5 Math Enrichment
Teaching notes
The typist, pipe, and escalator investigations are based on the chapter exercises that follow. The pipe exercise is phrased here with the mathematically consistent interpretation made explicit: the inlet stays open, the outlet pipes are identical to one another, and an outlet’s rate may differ from the inlet’s rate.