Units and Measures | Scoins.net | DJS

## Units and Measures

As of 2012, it is assumed you know, at Middle School level, that 5 miles is 8 kilometres, that an inch is 25.4 mm and that a kilogramme is 2.2 pounds.

This is a long page and it might be better written and more useful as four or five shorter worksheets.

An inch really is 25.4 mm exactly, so you can do exercise 1.

Exercise 1:  12 inches = 1 foot

an inch is the width of my thumb, your foot is a bit less than ‘a foot’ long (probably).

3 feet = I yard    - about the distance from your nose to the end of your arm.
22 yards = 1
chain  -  a typical garden-sized unit
10 chains = 1 furlongone of the rare occasions where a ten appears in imperial units;
8 furlongs = 1
mile  - a horsing unit, still seen in racing. (As is a hand, below)

1.    Convert all of these to millimetres, centimetres or metres as appropriate. Check: 1 mile = 1609344mm.

2.   Draw a triangular table converting all large units into the smaller ones. E.g. a mile = 63360 inches = 5280 feet = 1760 yards = 80 chains = 8 furlongs

3.   Construct a reverse table for American readers, explaining what a metre is in inches, a kilometre in yards and in miles. Use units and rounding that they would grasp.

Less well-known units of length:
1
hand = 4 inches  (horse heights are measured in hands),
1
fathom = 6 feet (fathoms are used for depths of water),
1 rod, pole or perch = 16.5 feet = 5.5 yards = ¼ chains.  Rods, etc are used in big timber work, farming and the history is from keeping birds. Rods are still used in canoeing, to describe portages – the distance you might carry a boat around an obstacle.  A canoe might well be a rod in length. Modern definition (from wikipedia) of the pole and perch says they have been used as labels for length area AND volume. My old maths exercise books had unit printed on the back and this was all assumed information:  Rod=pole=perch = 16½ feet.

Measures of area:
An
acre is a chain x a furlong, so 4840 sq yards. A hectare is (100m)² so 10,000 sq m. and 100 hectares make a square kilometre.  An acre is allegedly the area ploughed by an ox in a day; one are [pronounced as in ‘air’] is a metre by a hundred metres, from (I think) the Dutch.
A
rood is a rod x a furlong, or 40 square rods. Allotments (growing land in urban areas not gardens) are still often measured in roods or square poles. I have heard an Irish farmer refer to his “12 rood field”. I wasn't very old at the time and thought he was defining field by the amount of swearing it took to work it.

One rare unit of area is the KiloWale, used for measuring really large areas. Only used in the plural, one wale is the area of Wales, 20779km². The loss of forest, deforestation, per year (2015, wikipedia) is about 7.3 million hectares, 73,000 sq km, 3.5 times the area of Wales. Brazil is 409 times bigger than Wales (so about half a KiloWale).  Mars is close to 7 KiloWales. Earth is between 24 and 25 KiloWales. The sun’s surface is about 12000 times bigger, close to 300 MegaWales. Jupiter’s surface is about 3 MegaWales, but we have some trouble deciding where the surface is.

Exercise 2:

4.   How many square metres was the 12 rood field? What would be the appropriate metric measure?

5.   How many square rods in a square mile? How many roods in a square mile? How many square yards in a rood?

6.   France has a land area of 547,030 km². Convert this to square miles. The UK has an area of 94,526 sq miles and Great Britain (UK less Northern Ireland) is 80,823 sq miles; convert these to square kilometres. The results are surprisingly similar numbers. How many times bigger is France over the UK?

7.   How many roods in a square kilometre? Why might this be a nice measure to use?

8.   Which is bigger, a hectare or an acre? Divide the bigger one by the smaller one and give this multiplier to 3 significant figures.

9.    A nautical mile, 10 cables, was 6080 feet. How many metres is this? A newer definition (1970) makes the nautical mile 1852 metres. How big was the correction, in feet?. If the equator is 40 million metres long, how many nautical miles is this? The equator is 360 degrees and there are 60 minutes to a degree, so how many minutes around the equator? This tells you how the nautical mile was originally defined.

Measures of volume: the imperial measures of pint and gallon remain in use, mainly for counting fluid volume (such as car fuel). There are eight pints to a gallon; “a pint of water weighs a pound and a quarter” so a gallon weighs ten pounds [this is my second ’special ten’ in Imperial units]. A cubic foot weighs 62.5 pounds and is 6.25 gallons (imperial units use fractions, so 6¼ gallons usually).

Now a litre of water has the same mass as a kilogramme, as near as makes no difference, so you should be able to do these, given that a pound is 454 grammes, 0.454 kg (that’s not exact, it's 3sf)

Exercise 3A:

10.   How many litres in a gallon?

11.   The US gallon is different from the Imperial gallon; both have four quarts and eight pints but the Americans commonly use quarts (e.g. of ice cream) where the Brits use pints (e.g. of beer). A US gallon is 231 cubic inches. Show this is about 3¾ litres and give an answer to more decimal places.

12. A British / Imperial pint is 20 fluid ounces where a US pint is 16. Is this consistent with your figures from Questions 11 & 12?

13.  The current definition of an Imperial gallon is 4.546092 litres; using this figure, how many cubic inches is this? You can manage at least 7 sig.fig.; does that conflict with any of the earlier information? If so, what do you think happened?

14.  There’s a huge aquifer in the US with the name Ogallala. It may be (have been; it is shrinking) one million billion gallons of underground water. Convert that to cubic kilometres and compare it with the one under Egypt and Libya at 500 million km³. The Americans tell me that the same water is 3.3 billion acre-feet; 3.3 billion acres to a depth of one foot of water. Show the working that checks or disagrees with that calculation.

There are many odd units of volumes measure. Many of the terms already used have several different definitions, such as the ‘dry’ gallon; from the history of measuring things like corn, wood & ales we also have the peck and the bushel (and the cord, barrel and hogshead); for small volumes we have the gill (pronounced Jill, five fluid ounces), and such things as the tablespoon, teaspoon and dram. For really big things we might use cubic miles.

Inside the 20 fluid ounce (British) pint there are 4 gills, 8 fluid drachms (also drams), 24 fluid scruples and 480 minims. These disappeared officially in 1971.

I was asked about kitchen units, spoons, cups and so on. These are almost necessarily less than exact units and I have picked the round(er) numbers.

5ml to a teaspoon; two teaspoons to a dessert spoon, four teaspoons to a tablespoon and fifty teaspoons to a ‘cup’, 250 ml in Australia and Canada but 284.1 ml in Britain, 240 ml according to the FDA (US) but 236.6 elsewhere in the USA. Think of ‘cup’ as 250ml and you’ll be okay, though many cups are a bit smaller.

The US fluid measures are a shade different from the UK ones; 96 drops make a teaspoon, [this one is 4.93ml not the previous 5ml]; 3 teaspoons to the tablespoon; 3 tablespoons to the jigger, 1.5 fl oz, where a cup is 8 fl oz, 236.59ml, as above. So these are pretty equivalent labels, apart from the pint at 20% (or 25%, depending on which one you take as standard) different. The fluid ounces are different by 4%; I didn’t say that they were the same, just implied it: a US fluid ounce is 29.574g while a UK fluid ounce is 28.413g.

Exercise 3B:

15.    If I tell you that a gallon is 4.54609 litres, show that a fluid ounce is about 28 millilitres and a minim about 60 microlitres.

16.    Show that a pint is about 0.57 litres and that a litre is about 61 cubic inches. One way would be to imagine a 10cm cube of water...

17.   How many British cups to a pint? How many American cups to an American pint? Now (harder), how many American cups to a UK gallon and British cups to a US one?  I didn’t say that a US fluid ounce, 29.574g was the same as a UK fluid ounce, 28.413g; they are just close.

Measures of weight: Imperial weight and mass are confused; physicists who want the newton as weight and the kilogram or kilogram as mass, need to not say “pound weight”, or get used to the idea of multiplying by 32 from pounds to poundals. Recent updates to wikipedia tell us to use the terms pound mass, pound weight and pound force. I say go metric, but understand the problem Americans must have.

16 ounces [oz.] = 1 pound [lb.]
14 pounds [lbs] = 1 stone [st.] = 224 ounces
8 stones [st]      = 1 hundredweight [cwt] = 112 lbs
20 hundredweight = 1 ton = 160 cwt = 2240 lbs

Looking smaller than an ounce we have a drachm [dr], at 1/16 ounce, and a grain [gr] at 1/7000 to the pound. A grain is exactly 64.79891mg. An ounce is therefore ever so slightly less than 28.35 grams and a pound is exactly 453.59237 grammes [that’s the exact version of the 454 I gave earlier]. Now I understand why the exact definition of a pound in grammes divides by seven.

Exercise 4:

18.     Write a ton in tonnes (1.0...)  and a tonne in tons (0.9...).

19.    An American (US & Canada) ton is based on 100 not 112 pounds in a hundredweight, so there are 2000 lbs in a ‘short’ ton; how many kg is this, to 6 d.p.?

20.  Find the difference between a ton and a tonne in both pounds and kilograms.

21.   Use the definition of a pound to find how many pounds to the kilogramme, and therefore how many pounds to the tonne. We learn 2204 lbs/tonne but notice that 2205 is closer.

Don’t get upset about these differing units; it is a piece of history reflecting that the US was independent of Great Britain when Britain defined weights and measures in 1824; the US stayed with the units as understood at the time of occupation (or conquest, or settlement, depending on your point of view).

Between 1971 and 2007 the only countries left using the imperial units are American; Canada is switching steadily to metric units and mixes them much as Britain does. Members of the Commonwealth are changing steadily and a mixture of units will be found across Malaysia and India. Gallons are still used in the Caribbean, while the Arab nations that were using gallons seem to be switching over to litres.

Despite those comments, there are some weird situations in Britain. Here are some examples:

22.  I think the price of diesel just passed £6 per gallon. Identify the pump price per litre to the nearest penny.

23. I just pumped up my tyres and discover I’ve been stupid enough to let the tyres run a whole 10psi below the recommended figures. This makes an improvement of a whole 4% in fuel consumption.

(i) Explain ‘psi’, and convert 38psi into the correct SI unit

(ii) My fuel consumption, on a recent long drive, was 57.1mpg.  With the tyres now correctly inflated, what do I expect consumption to be if I repeat this trip tomorrow?

(iii) How would a Frenchman describe this new consumption figure? [Technically, that is a ‘hard’ question, because it demands that your previous answer was right, so concentrate on showing how you process that answer, so as not to be penalised twice for not being right, or not right enough]. ‘Show your working’ is mandatory in this situation.

DJS 20130422

3.  A metre is 39.37 inches; a kilometre is 1100 yards (1093.61 yards), just under 5 furlongs (4.97 furlongs), or 0.621 miles. A yard is exactly 0.9144 metres.

4.  12 x 40 x 5.02922 = 12140 m² = 1.2 ha.

5.  320²=102400 rod² = 2560 roods = 1 sq mile. 1210 sq yards = 1 rood.

6.  UK = 244717 km²; France = 211209 miles². Great Britain = 209331 km². France/UK = 2.61323. To 2sf the area of France is 210,000 sq miles and the area of GB is 210,000 sq km.

7.  I rod = 5.0292 metres. 39537 sq rods / km² suggests 988.42 roods per sq km, not quite 1012 sq m to the rood. Which in turn makes a rood, at nearly 1000 to the sq km, quite a nice unit to use.1.

8.  A hectare is bigger, by 2.5 (2.47, 2.471 054) times. An acre = 0.405 hectares.

9.   About 4 feet shorter, 1.184 metres.  21598.272 n.mi. 21600 = 360x60.

14. Ogallala: 1x1015 gallons x 4.54609 litres/gallon x 10-3 m3/litre x 10-9 km3/m3 = 4500km³.

17.  1 Two of each. 4546.09/236.59 = 19.215 US cups/UK gallon;  3785.41/284.1 = 13.3 Uk cups per US gall. Each would have 16 of their own cup unit per gallon.

18.  1 imperial ton = 1.016 tonnes, 1.01605 for people that googled it. Reverse is 0.984207.

19. One US short ton is 907.185 kg.  2000x0.454=9080; 200x0.453592=907.184 Told to use 6 s.f., so must find a 6s.f. precision conversion figure. 2000 x 0.45359237 = 907.18474, so 907.185 is even better so 6s.f. Teacher to decide marking scheme!!

20. about 16kg, which is around 35 lbs.         21. exercise in using figures already produced.

22. £6.00 / 4.54 (or 4.54609) => 1.3215 (1.3198) so £1.32 per litre

23 psi= pounds per square inch, pounds force per square inch. Correct unit is pascal, but here kliopascals are appropriate. 1 psi = 6.89476 kPa, so 38psi= 262kpa, though 2s.f. is correct, so 260 is a ‘better’ answer. 1 atmosphere = 101325 Pa = 760 torr = 14.6959 psi.
57.1*1.04 = 59.4 to 3 s.f.           Conversion is to litres per 100km, answer is 4.8 L/100km using internet. Calculation requires sight of an equivalent to 4.54 / 1600, preferably with more precision, such as 4,546/1609  These need correcting by 10⁵ to put the decimal point in the right place, giving the conversion figure 282.5 to 4 s.f. There will be disagreement around the edges, so to speak, as students choose different values. the point here is to see the conversions put in the right places, not a magically found 282.48 (for example), which i would penalise for laziness. Using US units is a fail (235.21).

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