Unit conversion pitfalls: gallons, tons, cups, GB vs. GiB

Many unit conversion mistakes aren’t arithmetic slips. The factor is right, but it belongs to a different unit that happens to share the name. “Gallon”, “ounce”, “ton”, “cup”, “gigabyte” and “degree” each cover two or more sizes, and some cover different quantities altogether. Before you multiply, settle three things: which system the unit comes from (US, imperial or metric), what it measures (volume, mass, a temperature reading or a temperature change), and, for data, whether the prefix is decimal or binary.

Why are there two gallons, pints and fluid ounces?

In 1824 Britain replaced its gallons with the new imperial gallon, while the US kept the older wine gallon of 231 cubic inches. The US gallon is exactly 3.785411784 L. The imperial gallon, which NIST lists as the Canadian and UK gallon, is exactly 4.54609 L under the UK’s Weights and Measures Act, about 20% larger. Every unit built from the gallon inherits the split.

The fluid ounce is the twist. The imperial gallon is split into 160 fluid ounces and the US gallon into 128, so the imperial fluid ounce is the smaller one even though every larger imperial unit is bigger. A UK pint holds 20 of the smaller ounces, a US pint 16 of the larger ones, and the UK pint still comes out 95.08 mL ahead.

For the British recipe in the table, 1 pint of milk is 568.26 mL, or 568.26 ÷ 236.59 = 2.40 US cups. A US pint (2 cups) gives 16.73% less milk than the recipe expects. The gap reads 20.09% one way and 16.73% the other because the base changes, as the guide to percentage mistakes explains.

Is a fluid ounce the same as an ounce?

No. A fluid ounce measures volume. An ounce (avoirdupois) measures mass: 1/16 of a pound, exactly 28.349523125 g. The two match closely only for water measured in imperial units: NIST Handbook 44 describes the imperial gallon as the volume of 10 pounds of water, so its 160 fluid ounces of water weigh about 160 ounces. For anything denser or lighter than water, 8 fl oz on a measuring cup and 8 oz on a scale are different amounts.

The weights are USDA FoodData Central portion figures. There’s also a third ounce: the troy ounce (oz t), the standard unit for trading precious metals, is 31.1034768 g, 9.71% heavier than the ordinary ounce. Handbook 44 says an ounce with no qualifier is the avoirdupois ounce and a troy unit should be labeled troy.

Which ton is meant: short, long or metric?

In the US, “ton” normally means the short ton of 2,000 lb, exactly 907.18474 kg; NIST notes that the long ton has only limited commercial use there. The UK’s ton is the long ton of 2,240 lb, exactly 1,016.0469088 kg. The metric ton, or tonne, is 1,000 kg, and NIST notes that the US calls the tonne a “metric ton”.

On an order for 30 tons, reading long tons where short tons were meant overstates the load by 3,265.87 kg, because a long ton is 12% heavier than a short ton. If a document says just “tons”, check where it came from and what the industry in question uses, or ask.

Why can’t cups of flour be converted to grams exactly?

Because a cup measures volume and a gram measures mass. Linking them takes the ingredient’s density, which differs from one ingredient to the next, and published figures for the same ingredient disagree. USDA FoodData Central lists a cup of all-purpose flour at 125 g, while King Arthur Baking’s ingredient chart uses 120 g. Both are averages, not properties of your flour or the way you filled your cup.

Take an example recipe with 3 cups of all-purpose flour and 1½ cups of granulated sugar:

The two references differ by 15 g of flour, about 4%, for the same 3 cups. If a recipe gives weights, weigh. If it gives only cups, scale it in cups, and if you want grams for next time, weigh your own cupful once and write it down. For the same reason, the recipe scaler converts cups to milliliters but won’t turn cups into grams.

How big is a cup or a tablespoon?

It depends on where the recipe comes from, and sometimes on what it’s for:

A metric cup is 5.67% larger than a US cup, about 40 mL over 3 cups. The tablespoon gap is bigger: an Australian tablespoon holds 35.3% more than a US one, which shows in strong ingredients such as salt, baking powder or yeast. Labels round too: for nutrition labeling, US rules treat a fluid ounce as 30 mL and an ounce as 28 g, so label figures won’t match an exact conversion.

Why does a 2 TB drive show only about 1.8 TB?

The drive maker counts in decimal units, and Windows counts in binary units but still labels them GB and TB. A decimal terabyte (TB) is 10¹² bytes. A binary tebibyte (TiB) is 2⁴⁰ = 1,099,511,627,776 bytes, 9.95% more.

For an example 2 TB drive:

1. Capacity in bytes: 2 × 10¹² = 2,000,000,000,000.
2. In gibibytes: 2,000,000,000,000 ÷ 1,073,741,824 = 1,862.65 GiB.
3. In tebibytes: 2,000,000,000,000 ÷ 1,099,511,627,776 = 1.82 TiB.

By the same arithmetic, a 1 TB drive shows about 931 GB: 10¹² ÷ 1,073,741,824 = 931.32 GiB.

No bytes are missing; the same capacity is counted in a larger unit. Formatting also uses some space, as Apple’s support note points out, and Apple’s own systems up to iOS 10 and OS X Leopard reported binary sizes too, so one drive can show different numbers on different devices. The gap grows with each prefix step: a KiB is 2.40% larger than a kB, a GiB 7.37% larger than a GB and a PiB 12.59% larger than a PB.

Why does a download take longer than the Mbps figure suggests?

Because connection speeds are quoted in bits per second and file sizes in bytes, and a byte is 8 bits. Network speeds also use decimal prefixes: 1 Mbps is 1,000,000 bits per second. A 300 Mbps connection moves at most 37.5 MB, or 35.76 MiB, per second.

$$\text{time in seconds} = \frac{\text{size in bytes} \times 8}{\text{speed in bits per second}}$$

For an example 4.7 GB file on a 300 Mbps connection:

1. Size in bytes: 4.7 × 10⁹ = 4,700,000,000.
2. Size in bits: 4,700,000,000 × 8 = 37,600,000,000.
3. Time: 37,600,000,000 ÷ 300,000,000 = 125.33 seconds, about 2 minutes 5 seconds.

Dividing 4,700 MB by 300 gives 15.67 seconds, which treats megabits as megabytes and is 8 times too fast. If the 4.7 “GB” came from Windows, the file is really 4.7 GiB and takes 134.58 seconds. And 125.33 seconds is the floor: protocol overhead and other traffic take a share of the line. Assuming, for example, 10% overhead, the download takes 125.33 ÷ 0.9 = 139.26 seconds, about 2 minutes 19 seconds.

How do you convert a temperature change instead of a temperature?

Multiply by 9/5 (or 5/9) and leave out the 32. The 32 only shifts the zero point between the two scales, and when you subtract one reading from another it cancels. NIST lists a Fahrenheit interval as 5/9 of a Celsius interval, and a Celsius interval as equal to a kelvin.

$$t_{\mathrm{F}} = \tfrac{9}{5}\,t_{\mathrm{C}} + 32 \qquad \Delta t_{\mathrm{F}} = \tfrac{9}{5}\,\Delta t_{\mathrm{C}}$$

Here $t$ is a reading on a thermometer and $\Delta t$ is a change or difference.

Readings still need the full formula: a thermostat set to 22 °C reads 71.6 °F. Rates such as degrees per hour are differences, so they use the difference formula.

Ratios are a related trap. 30 °C is not “50% warmer” than 20 °C, because 0 °C and 0 °F are not the bottom of the scale: 0 °C is 273.15 K above absolute zero. Measured in kelvin, the rise from 293.15 K to 303.15 K is 3.41%.

Why do UK and US mpg figures differ?

UK figures are miles per imperial gallon, and the imperial gallon holds 20.09% more fuel, so the same car shows a higher mpg in the UK. Convert the gallon before you compare. Liters per 100 km run the other way: they measure fuel per distance, so a lower number is better.

$$\text{US mpg} = \text{UK mpg} \times \frac{3.785411784}{4.54609} \approx \text{UK mpg} \times 0.8327$$ $$\text{L/100 km} = \frac{235.215}{\text{US mpg}} = \frac{282.481}{\text{UK mpg}}$$

As an example, car A is quoted at 52 mpg in a UK review and car B at 45 mpg on a US label. Assume 10,000 miles of driving a year:

1. Put car A in US gallons: 52 × 3.785411784 ÷ 4.54609 = 43.30 US mpg.
2. Compare: car B’s 45 US mpg is higher, so car B goes farther on a gallon. Car A’s 52 was bigger only because its gallon is bigger.
3. Check in L/100 km: car A uses 282.481 ÷ 52 = 5.43 L/100 km and car B 235.215 ÷ 45 = 5.23 L/100 km.
4. Fuel for 10,000 miles: car A needs 10,000 ÷ 43.2991 = 230.95 US gallons (874.25 L); car B needs 10,000 ÷ 45 = 222.22 US gallons (841.20 L).
5. Difference: car A uses 8.73 US gallons (33.05 L) more, 3.93% more fuel.

Reading 52 as US mpg would predict 192.31 US gallons for car A, 38.64 gallons short. Even after the gallon is fixed, the two figures aren’t measured the same way. US ratings come from EPA laboratory driving cycles, and UK figures from the WLTP laboratory test, so a comparison across countries is approximate, and your own driving will differ from both.

What conversion can’t fix

• An unnamed unit. If a label says “gallon”, “ton” or “cup” and nothing else, no factor tells you which one. The country, date and industry of the source usually do.
• False precision. 5 US gallons is 18.93 L, but if the 5 was a rough estimate, so is the answer. Round the result to match the input. The same goes for grams-per-cup averages and rounded label equivalents: converting them precisely doesn’t make them more accurate.

Try it

The results below are rounded, and the converters can show more digits.

• Fuel economy and trip cost calculator: choose One way, enter a distance of 10000 miles, then choose mpg (UK) as the fuel economy unit and enter 52. Any fuel price will do, because it only changes the cost. Fuel used is 230.95 US gallons (874.25 L), and the equivalents show 43.30 mpg (US) and 5.43 L/100 km. Choose mpg (US) and enter 45 to get 222.22 US gallons (841.20 L) and 5.23 L/100 km.
• Volume converter: convert 1 from UK imperial pints to see 568.26 mL, 2.40 US cups and 1.20 US pints.
• Weight converter: convert 30 from short tons to see 27,215.54 kg, 27.22 metric tons and 26.79 long tons.
• Data storage converter: with Sizes chosen, 2 TB shows 1,862.65 GiB and 1.82 TiB. With Transfer time chosen, a file size of 4.7 GB at 300 Mbps with 0% overhead takes 0:02:05, and with 10% overhead 0:02:19.
• Temperature converter: with A change chosen, 1.5 °C gives 2.7 °F and 25 °F gives 13.89 °C. With A temperature chosen, 22 °C gives 71.6 °F.
• Recipe scaler: replace the ingredients with 3 cups flour, give Original servings and New servings the same number and choose Metric. The flour comes out as 710 mL (709.76 mL exactly), a volume; the scaler won’t give grams without a density.