Unit Pricing

Put the packs side by side and let the division happen. Then move the offer slider and watch two headline percentages that look the same turn out to be worth very different amounts.

Worked example — Unit Pricing

Cheapest per unit	2.30 for 500 g
Dearest vs cheapest	13% more

These are the values the calculator shows when the page opens. Change any input above to recompute them.

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The division is one keystroke, which is why it is made awkward 🖖

Price divided by quantity. That is the whole calculation, and there is no version of shopping where it is not the number you want. Everything else on the packaging exists to make it inconvenient: sizes that do not halve, one product priced by weight and its neighbour by count, a multipack whose per-item price is nowhere on the label. The shelf edge in many countries is legally required to print the unit price precisely because the calculation is easy and doing it in front of a display is not. Type three packs in and the column appears — and then notice how rarely the ranking is the one the display suggests.

The big box is not reliably the cheap one 🖖

Press The big box is dearest. The 500 g works out at 0.46 per 100 g and the 1 kg at 0.52, so the bulk pack costs thirteen per cent more per gram than the middle size. This is common enough to have a name — a quantity surcharge — and it survives because the assumption that bigger is cheaper is strong enough that most people do not check. The readout above says outright whether the largest pack is the cheapest one, which is the single question worth asking before reaching for it.

Extra free and off are not the same offer 🖖

Slide the offer to 50 and read the line under it. 50% extra free saves you 33.3% per unit, because the same price is spread over one and a half times as much. 50% off saves you 50%. To match the discount with extra product you would need 100% extra free — twice the packet. The gap grows with the headline: at 10% the two are within a point of each other, at 50% they are seventeen points apart, and at 100% off there is no amount of extra free that matches, because free is free. This is the same reciprocal relationship that makes a 50% fall need a 100% rise to recover, which <a href="/math/cross-multiplication">Cross-Multiplication Grid</a> takes as its subject.

Problem solved in full

Two jars of the same coffee at 200 g or 300 g 7 steps
Two jars of the same coffee: 200 g at 4.50, or 300 g at 6.30 with 20% extra free on top. Which is cheaper per gram, and by how much?
1. Take the plain jar first. One division, and the answer is a price per gram.
  
  $\frac{4.50}{200} = 0.0225 \Rightarrow 2.25 / 100$
2. The second jar needs its size fixing before anything else. Twenty per cent extra free on 300 g is 360 g, and the price does not change.
  
  $300 \times 1.20 = 360$
3. Now the same division on the real quantity.
  
  $\frac{6.30}{360} = 0.0175 \Rightarrow 1.75 / 100$
4. The bigger jar wins, at 1.75 per 100 g against 2.25. That is a saving of 22.2% per gram, and it is worth noticing that the 20% extra free did not deliver a 20% saving on its own.
  
  $1 - \frac{1.75}{2.25} = 22.2%$
5. Separate the two effects. Without the offer, 300 g at 6.30 is 2.10 per 100 g, already 6.7% cheaper than the small jar. The extra free then takes that 2.10 down to 1.75.
  
  $\frac{6.30}{300} = 2.10 \Rightarrow 1 - \frac{2.10}{2.25} = 6.7%$
6. And that step is the one the label overstates. Twenty per cent extra free is a saving of 20/120, which is 16.7%, not 20%.
  
  $\frac{20}{120} = 16.7% \neq = 20%$
7. Which leaves the question the arithmetic cannot answer. The cheaper jar is only cheaper if you drink 360 g of coffee before it goes stale, and a saving on quantity you will not use is not a saving. Unit price ranks what is on the shelf; it says nothing about what is in your cupboard in three months.
  
  $\frac{x}{1 + x} < x (x > 0)$
Answer

The 360 g jar, at 1.75 per 100 g against 2.25 — a saving of 22.2% per gram. Of that, 6.7% comes from the larger size and the rest from the offer, which is worth 16.7% rather than the 20% on the label.

References (2)

The quantity surcharge measured in real shops, rather than asserted: S. Clerides and P. Courty, "Sales, Quantity Surcharge, and Consumer Inattention." The Review of Economics and Statistics 99(2), 357–370, 2017.
How common it is, and how large, once shopping moved online: H. Estelami, "Incidence and Magnitude of Quantity Surcharges and Quantity Discounts in Online Shopping." Journal of Consumer Affairs 52(3), 517–539, 2018.

By the 42Math team · Updated 2026-08-26

Reviewed against published sources. Corrections welcome. Editorial policy · Corrections

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Example problems

The big box is dearest - The 500 g is the best value at 0.46 per 100 g, and the 1 kg is the worst at 0.52 — thirteen per cent more per gram for the pack that looks like the sensible bulk buy.
Multipack vs single - A six-pack of 330 ml against a single bottle and a one-litre. The multipack wins here, but not by as much as the shelf implies, and the litre is the one to check.
Priced by the item - Eggs priced by the item. Buying twelve saves about fifteen per cent an egg over buying six, and buying one at a time costs more than either.