Chargeable weight: the same divisor gives two totals

One shipment can carry one divisor and two chargeable weights, depending on where the sum is taken. At a divisor of 6000 the cartons read 260 kg one way and 240 kg the other.

Two ways to add the same cartons

The per-carton calculation adds 20.00 kg when each carton takes its greater weight before summing at divisor 6,000.00, giving 260.00 kg against 240.00 kg for the shipment total.

Declared divisorcm³/kg

From the quote, 1000 to 20000 cm³/kg

One carton per row; enter its volume in cubic metres

Carton 1

Gross weightkg

Volumem³

Carton 2

Gross weightkg

Volumem³

Extra weight, kg

Divisor, cm³/kg

Per-carton total minus shipment total

Your divisor: 6,000.00

Largest gap: 5,000.00

Assumptions: Only the divisor moves; the marked values are listed below the figure

Values at the marked divisors

Values at the marked divisors
Divisor, cm³/kg	Per carton, kg	Shipment, kg	Gap, kg
1,000.00	1,200.00	1,200.00	0.00
4,000.00	300.00	300.00	0.00
5,000.00	276.00	240.00	36.00
6,000.00	260.00	240.00	20.00
8,000.00	240.00	240.00	0.00
10,000.00	240.00	240.00	0.00

Each carton, then sum

260.00 kg

Sum, then take the greater weight

240.00 kg

Extra weight from the first method

20.00 kg

Where the methods separate

The gap is positive only above 4,000.00 and below 8,000.00 cm³/kg; both endpoints give equal totals.

The largest gap is 36.00 kg at 5,000.00 cm³/kg, where the shipment's gross and volumetric weights are equal.

The calculation receipt

Carton 1: Gross, kg180.00; Volumetric, kg120.00; Gross weight180.00 kg

Carton 2: Gross, kg60.00; Volumetric, kg80.00; Volumetric weight80.00 kg

1.The typed volumes total 1.200 m³, or 1,200,000 cm³.
2.Taking the greater gross or volumetric weight for each carton gives 260.00 kg in total.
3.The gross total is 240.00 kg and the volumetric total is 200.00 kg, so the greater total is 240.00 kg.
4.The first method adds 20.00 kg, or 8.3% of the shipment-total reading.

The maths and its limits

Why taking the maximum first adds weight

For each carton, divide its cubic-centimetre volume by the declared divisor and compare that volumetric weight with gross kilograms. Selecting the larger value carton by carton preserves both the dense carton's gross-weight excess and the light carton's volume-weight excess; pooling first lets those excesses offset.

Here a is gross kilograms, v is volume in cubic metres, V is volume in cubic centimetres, d is the divisor, P is the per-carton total and S is the shipment total.

V_i = 10^6 v_i

P(d) = \sum_i \max(a_i,V_i/d)

S(d) = \max\!\left(\sum_i a_i,\frac{\sum_i V_i}{d}\right)

G(d) = P(d)-S(d) \ge 0

The boundary and the peak are exact

The sum of the individual maxima is at least the gross-weight sum and at least the volumetric-weight sum. A positive gap requires a carton governed by gross weight and another governed by volume. Below the shipment's own threshold the gap can only rise; above it the gap can only fall, so its maximum is at that threshold.

\begin{gathered}G(d)>0\\\Updownarrow\\\min_i(V_i/a_i)<d<\max_i(V_i/a_i)\end{gathered}

d_* = \frac{\sum_i V_i}{\sum_i a_i}

Cartons with identical volume to weight ratios may share one row using their summed weight and volume, because they change governing value at the same divisor. Combining different ratios would erase the mechanism being measured.

Opening record, to the digit

The cartons weigh 180.00 and 60.00 kg and occupy 0.720 and 0.480 m³; their individual thresholds are 4000.00 and 8000.00 cm³/kg.
At 6000.00, their volumetric weights are 120.00 and 80.00 kg, so their separate maxima total 260.00 kg; the shipment compares 240.00 kg gross with 200.00 kg volumetric and takes 240.00 kg.
The difference is 20.00 kg, or 8.3%; it peaks at 36.00 kg at divisor 5000.00, and is zero at both 4000.00 and 8000.00.

Source basis

The site's chargeable-weight glossary entry defines gross and volumetric weight; maximum and addition give the comparison proved here. The entered divisor supplies the convention, and the example is synthetic.

Basis: Deterministic maximum and addition, with no intermediate rounding
Model version: 1.0.0

The whole record prints. Your browser can save it as a PDF instead of printing it.

This compares two unrounded weight calculations; it does not decide the applicable tariff, rounding or invoice.

Chargeable weight record

Inputs entered

Declared divisor: 6,000.00 cm³/kg
Carton 1: 180.00 kg; 0.720 m³
Carton 2: 60.00 kg; 0.480 m³

Computed reading

Each carton, then sum: 260.00 kg
Sum, then take the greater weight: 240.00 kg
Extra weight from the first method: 20.00 kg; 8.3%

Numbered calculation trace

1. Convert volume: The typed volumes total 1.200 m³, or 1,200,000 cm³.
2. Compare before summing: Taking the greater gross or volumetric weight for each carton gives 260.00 kg in total.
3. Sum before comparing: The gross total is 240.00 kg and the volumetric total is 200.00 kg, so the greater total is 240.00 kg.
4. Subtract the two totals: The first method adds 20.00 kg, or 8.3% of the shipment-total reading.

Record

Model version: 1.0.0
Date