The art of approximating fractions (pdf)
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Transcript of The art of approximating fractions (pdf)
I call this the ART of decimalisation because I want you to understand, right up front, that this is not a precise business with a particular, RIGHT way of doing it. This is an art that requires judgement. Deciding how to get the answer is as important as getting the answer, and that’s what makes it fun.
Take a nasty fraction and associate it with a friendlier fraction; that means a fraction with a denominator that is easy to work with.
300
?
309
47 What number should I put on top to
preserve accuracy?
The denominator is being made smaller by 9 parts in 309, which is about 9 parts in 300, which is about 3 parts in 100, or 3 percent. Therefore the numerator has to be made smaller by the same amount.
300
?
309
47
3 percent of 47 is about 3 percent of 50 ...which is 1.5
(multiply the 3 and the 5 and adjust the decimal point)
...so 47 reduces to 45.5
300
?
309
47
This might not look much better, but you can make it better by turning 300 into 100. That means dividing top and bottom by 3.
300
5.45
309
47
100
15
.150
300
5.45
309
47
Don’t be precise! Round it off as you go.
This is already pretty accurate, but if I want to I can make it better simply by adding back the little bit that I took off to simplify the fraction.
100
15
.150
300
5.45
309
47
100
15
.150 ..00166.0 600
1
300
0.5 add
.151670
( 0.3 % error )
300
5.45
309
47
The point is to shift the numerator and denominator to friendlier numbers in such a way that reasonable accuracy is preserved. Fortunately the process is very robust and not affected by small errors.
309
47 is pretty close to 1/6, so the shift to the numerator should be 1/6 of the shift to the denominator. (so if you subtract 9 from the bottom, you should subtract 1.5 fro the top)
Therefore what you need to do is start with a simple observation of how big the denominator is compared to the numerator.
555.0100
5.55
595
5.253
95
53 is pretty close to 1/2, so the shift to the numerator should be half of the shift to the denominator.
Therefore what you need to do is start with a simple observation of how big the denominator is compared to the numerator.
87
65 is pretty close to 3/4, so the shift to the numerator should be 3/4 of the shift to the denominator.
75.0 100
75
300
224
150
112
75
56
1287
965
Therefore what you need to do is start with a simple observation of how big the denominator is compared to the numerator.
Alternatively, I can try this:
You see that there’s no single ‘right’ way to do it.
74.0 100
74
1387
965
87
65
If mathematical thinking is easy then people will think mathematically more often, even when they don’t need to. If calculations can be achieved in an eyeblink, with as little time and effort as an eyeblink, people won’t even consider the fact that they are thinking mathematically but will simply call it thinking. It will be like thinking of other places without fetching an atlas, or thinking of words without fetching a dictionary. THAT is when maths switches from being a science to a living art; something that I feel should be high on the list of objectives for mathematical teachers. -DC
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