A 55 × 45 × 40 cm carton — 0.099 m³ — holds 24 units and weighs 14 kg on the scale. The air carrier's tariff quotes a DIM divisor of 139 in³/lb and a rate of $3.20 per kilogram. Decode the quote the way the carrier's software does: the divisor as an honest density, the carton's dimensional weight, the weight the carrier actually charges, and the freight cost carried by each unit in the box.
Every number in this problem is editable — change any value below and the whole chain recalculates.
Given
d = 139 in³/lb — Tariff DIM divisor
V = 99,000 cm³ — Carton volume (55 × 45 × 40 cm)
W_a = 14 kg — Scale weight
R = 3.2 $/kg — Freight rate
n = 24 units — Units in the carton
Determine
(a)the divisor as a density
(b)the carton's dimensional weight
(c)the chargeable weight
(d)the freight cost per unit
Step 1 of 4(a) · solve for Dimensional density
A divisor of 139 in³/lb is a density in a trench coat: one pound per 139 cubic inches is 199.1 kg/m³. Saying it as a density makes the tariff's threat legible — pack lighter than 199 kg/m³ and you will be billed for air.
Rearranged for ρ_dim
ρdim=dVbmb
Your values, in your units
ρdim=(139)⋅(1in3)(1lb)
Converted to base units
ρdim=(139)⋅(16.3871cm3)(0.453592kg)
Answer
ρdim=199.14kg/m3
Carried onward at full precision, not this rounded figure.
The carton's volume priced at the tariff's density: 0.099 m³ × 199.1 kg/m³ is 19.7 kg of dimensional weight. The box only weighs 14 kg — it is packed at 141 kg/m³, thirty percent fluffier than the carrier's threshold.
Rearranged for W_dim
Wdim=Vρdim
199.14 kg/m³carried from step 1
Your values, in your units
Wdim=(99,000cm3)⋅(199.136kg/m3)
Answer
Wdim=19.714kg
Carried onward at full precision, not this rounded figure.
The carrier charges the LARGER of scale and dimensional weight — max(14, 19.7) = 19.7 kg. Five and a half of those kilograms are empty space, billed at full rate: the aircraft sells volume and merely tolerates mass.
Rearranged for W_c
Wc=max(Wa,Wdim)
19.714 kgcarried from step 2
Your values, in your units
Wc=max((14kg),(19.7145kg))
Answer
Wc=19.714kg
Carried onward at full precision, not this rounded figure.
Landed on the contents: $3.20 × 19.7 kg across 24 units is $2.63 each — of which about 76 cents is the packaging's airiness, not the product's mass. That 76 cents is the number a redesigned carton competes against.
Rearranged for c_u
cu=nRWc
19.714 kgcarried from step 3
Your values, in your units
cu=(24units)(3.2$/kg)⋅(19.7145kg)
Answer
cu=2.6286$
Carried onward at full precision, not this rounded figure.
Therefore the tariff's 139 in³/lb is a 199.1 kg/m³ density, the carton dims out at 19.7 kg against 14 kg on the scale, the carrier charges the 19.7, and each unit inside carries $2.63 of freight — 76 cents of it paying to fly empty space.
Why this order
Dimensional weight exists because an aircraft cubes out long before it grosses out: the hold runs out of room while the wings could still lift more. The divisor is the carrier's break-even packing density stated in customary units, and converting it to kg/m³ — step (a) — is the single most useful move in freight arithmetic, because it turns a tariff rule into a design target. Every carton has a packing density; the tariff has a threshold; whichever is lower decides who pays for the air.
The trap is comparing quotes by rate alone. A carrier at $3.00/kg with a 110 divisor (251 kg/m³) beats this one on rate and loses on this carton — the fluffier the freight, the more the divisor matters and the less the rate does. Quote comparisons happen at chargeable weight, and chargeable weight is a function of your box, not just their tariff.
Carried values move at full precision, not the rounded figure shown — chaining rounded numbers compounds error.
