A compound unit describes a relationship between quantities. Litres per minute measures volume flow; kilograms per litre measures density; kilowatt-hours measure energy. The arithmetic becomes clearer when these units remain visible throughout the calculation.
Use the meaning and units of each quantity to decide whether to multiply or divide, then check that the resulting quantity answers the question. The basic methods remain in Rates and Unit Rates and Dimensional Analysis.
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Read “per” as a denominator with a meaning
A rate of 12 L/min means twelve litres for each minute under the stated flow model. Over four minutes at a constant rate, the delivered volume is
The minutes cancel. Multiplying by four hours without converting the time would use the right-looking number with the wrong physical meaning.
When a rate is only an interval average, multiplying by that interval’s duration recovers the interval total. It does not justify extrapolating the same rate into a different interval with different conditions.
Combine a flow rate with a density
A liquid flows at 12 L/min and has stipulated constant density 0.9 kg/L. Multiplying the rates gives mass flow:
Over four minutes the transferred mass is kg. The same result follows by finding volume first: .
Calculating mass from flow rate, time and density
Diagram explanation and exact data
Multiplying litres per minute by minutes gives litres. Multiplying that volume by kilograms per litre gives kilograms. The minute unit cancels first and the litre unit cancels second.
| Step | Calculation | Result |
|---|---|---|
| Volume | 12 L/min × 4 min | 48 L |
| Mass | 48 L × 0.9 kg/L | 43.2 kg |
| Equivalent mass flow | 12 L/min × 0.9 kg/L | 10.8 kg/min |
This is a chain of relationships. The litre cancels between the flow and density factors, leaving kilograms per minute. Dividing flow by density would instead give units of squared litres per kilogram-minute, which are not the requested mass rate.
Distinguish power from energy
Power is the rate at which energy is transferred. A kilowatt is a power unit. A kilowatt-hour is the energy transferred by one kilowatt sustained for one hour:
The NIST unit tables list the corresponding conversions. A kilowatt per hour, kW/h, is different: it describes how power changes with time, not accumulated energy.
For a hypothetical device drawing a constant 60 W for five hours per day over 30 days,
At an invented usage charge of $0.16 per kWh, the energy charge would be $1.44. That is not a complete utility bill: a real tariff could also contain fixed charges, time-varying prices and other terms. The example isolates the usage calculation rather than assuming those terms are zero in every contract.
Interpret “litres per 100 kilometres” literally
A stated consumption rate of 7.5 L per 100 km gives, over a modelled 240 km trip,
The factor 100 belongs to the denominator’s numerical scale. Do not treat the rate as 7.5 litres for every kilometre.
Consumption of 8 L per 100 km is reciprocal to efficiency of km/L. A lower L/100 km value indicates less volume used for the same distance; a higher km/L value indicates more distance for the same volume. Comparisons must use the same convention and comparable conditions.
Pool the quantities before calculating the combined rate
Suppose one trip uses 8 L over 100 km and another uses 12 L over 200 km. Their separate rates are 8 L/100 km and 6 L/100 km. The combined rate is
The unweighted mean of 8 and 6 is 7, but it gives the shorter trip the same weight as the longer trip. Pool total volume and total distance, or use distance weights explicitly.
The same principle appears in speed, density and cost calculations. Always identify the denominator that gives the correct weights. There is no universal rule that “averages should always be averaged”.
Separate area-based and length-based quantities
A stipulated material application of 30 g/m² over 12 m² requires 360 g. A fabric with mass per length of 0.4 kg/m over 15 m has mass 6 kg. Those are different physical descriptions, even when the material is sold on a roll.
A rectangular sheet priced at $18/m² and sized 1.2 m by 0.8 m costs $17.28 under an area-pricing model. A price per linear metre would also require a stated width. Units prevent the two pricing systems from being silently confused.
These examples are arithmetic models, not product-use instructions. Coverage, safety limits and suitable materials must come from the relevant application rather than being inferred from a unit rate alone.
Use dimensions to reject a calculation, not to certify it
The NIST quantity conventions distinguish quantity equations from equations for numbers in specified units. Preserve that distinction when copying formulas.
A result with the wrong dimensions cannot be the requested physical quantity. A result with the right dimensions can still use the wrong object, wrong average, missing factor or false assumption. For example, dividing total solution mass by volume and dividing solute mass by volume share dimensions but calculate density and concentration respectively.
For every compound rate, ask what its numerator counts, what its denominator measures, over which interval it applies, and whether the relevant relationship is being held constant.
Practice questions
- A stipulated constant 1.5 kW load runs for 40 minutes. Find the energy in kWh.
- A flow of 8 L/min has density 1.2 kg/L. Find the mass delivered in five minutes.
- Two trips use 5 L over 100 km and 21 L over 300 km. Find combined consumption in L/100 km.
Check your answers
- Forty minutes is h. Energy is kWh.
- The mass flow is kg/min. Five minutes delivers 48 kg.
- Total volume is 26 L and total distance is 400 km. The combined rate is L/100 km, not the unweighted mean of 5 and 7.
Related topics
Use Mathematical Modelling to test whether a constant-rate assumption is justified. Financial Mathematics extends unit and timing discipline to charges and cash flows.