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Rounding and Significant Figures

Item 1: Buses

A school wants to take 236 students on a school excursion and each bus will hold 28 students. They divide 236 by 28 and get 8.42857. Should they order 8.42857 buses?

This is an example where the numbers have been counted rather than being measured but rounding is required to make sense in the problem. Rounding to the nearest whole number (8) is not appropriate as this leaves 12 students at the bus stop. So 9 buses is a better answer (unless the school predicts a lot of absentees). In this case, rounding up to the nearest whole number is sensible.

Item 2: Robbers

A witness told police that the bank robber is about 6 foot tall. The police put out a description for a man who is 182.88 cm tall.

Using a conversion of 2.54 cm to 1 inch and then 12 inches to 1 foot gives 6 feet as 182.88 cm. However, the witness only estimated the height as about 6 feet and the degree of accuracy conveyed by writing 182.88 cm is inappropriate. The following tape measure marked in both feet and cm suggests that the robber's height may be between 175 and 190 cm and so saying 180 cm would be more suitable. The result from the calculator has thus been rounded to the nearest 10 cm. (Note that 182.88 cm is a valid height, but it could only be obtained using a very accurate measuring device. The smallest mark on a normal measuring tape is in millimetres, which are tenths of centimetres, and so the most accurate measurement using a tape would be something like 182.9 cm.)

Item 3: Parking the car

How close to a fire hydrant can you park a car? The Victorian traffic rules (1995) state that it is not permitted to park within 1 metre of a fire hydrant, within 3 metres of a letter box, within 9 metres of the departure side of a bus stop or within 18 metres of the approach side of a bus stop. (Victorian Traffic Handbook, VicRoads, 1995.) Does it assume that people can estimate distances to the nearest 1 metre? When ought a police officer book you?

When the conversion to metric occurred in Australia, the traffic rule books were reprinted with measurements given in the new system. So instead of being unable to park within 4 feet of a fire hydrant, it became not within 1.2192 metres of a fire hydrant. Converting inches and feet to centimetres as above would give 4 ft as 1.2192 m, however, it is impossible for a driver to make this estimate as it involves tenths of millimetres! Using the tape measure below suggests 1.2 m would be a suitable answer. As in Item 2 above, 1.2192 m is a valid distance but it could only be obtained using a very accurate measurement device.

Item 4: Cutting String

a) A 12 m length of string was cut into 3 equal pieces, so each piece was 4 m long.
b) A 14 m length of string was cut into 3 equal pieces, so each piece was 4.666666667 m long.

The original measurements of 12 m and 14 m were made with some kind of measuring device and the level of accuracy or confidence in the measurement is indicated in how the length is written. For example, if the lengths were accurate to the nearest millimetre, they should be written as 12.000 m or 12 000 mm. The accuracy of the first measurement affects the accuracy of the answer, or how much confidence we have in our answer after calculations are performed.

a) How was the 12 m string measured in the first place? Just because it given as a whole number 12, don't be tricked into treating it as a counting number, rather than as a distance! The string is originally between 11 1/2m and 12 1/2 m and so when the string is cut into 3 equal pieces the string would then be between 3 5/6m and 4 1/6m. While the answer of 4 m is acceptable, 4.00 m (400 cm) would be unsuitable, as this suggests the original length was accurate to the nearest centimetre too.

b) The calculator may display results of calculations to 9 decimal places, but this is not always appropriate. As in (a) we cannot pretend that accuracy to the nearest millimetre has been achieved as the original length of 14 m did not indicate this. The following number lines represent the original length and the length after cutting. The shaded region around the 14 on the top line indicates some inaccuracy, and the corresponding region on the lower line likewise. It would be sensible to give an answer like 4.7 m rather than 4.666666667 m.

Item 5: Strange but True

When Mount Everest's height was carefully calculated to the nearest foot they found that it was 29 000 feet.

Usually a number written like this (with zeros) indicates that the measurement was made to the nearest thousand feet, and that the actual height is somewhere between 28 1/2 thousand and 29 1/2 thousand feet. Rather than convey this false message, it was decided to record the height as 29 002 feet, indicating the level of accuracy of the measurement. It is ironic that in an attempt to convey the degree of accuracy, they actually introduced a deliberate error!

Sainsbury's recipe for lentil and tomato soup includes 1 large onion and 397-gram of chopped tomatoes.

The vagueness of one of the masses (the large onion) contrasts bizarrely with the extreme precision of the other, especially since Sainsbury's sells 400 g cans of tomatoes! The measurement 397 grams implies it is accurate to the nearest 1 g. Giving a measurement of 400 grams instead would imply less accuracy is required - very sensibly. (Source: New Scientist, 7 Feb 1998, p 63).

Item 6: When is 1 cm = 0.3937 inch?

In the following letter published in New Scientist, the inappropriate conversion of 1 cm to 0.3937 inches is discussed:

"Talking of excessive precision..., when I was a child, my father had an English translation of a manual on violin playing by the great Hungarian-German teacher Carl Flesch. It told budding violinists to lift their fingers 0.3937 inches from the finger board. I still have occasional visions of music students trying to measure this with micrometers." (Source: Kenneth Goodare, Letters, New Scientist 2 May, 1998, p53)

Item 7: When is 32/64 = 1/2?

In this letter published in New Scientist, the topic is the precision implied by the way the number is written:

"If you look at Charles Darwin's notebooks, you find meticulously recorded measurements such as 3 32/64 inch. As a schoolboy I would have been taught to simplify this to 3 1/2 inch, one half inch being preferred to the cumbersome thirty-two sixty-fourths of an inch. However, Darwin recorded the length in sixty-fourths to indicate the level of precision of his measurements." (Source: John Aitken, Letters, New Scientist 2 May, 1998, p53).