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March 19, 2013

Recovering ratios from two decimal place approximations

Filed under: mathematics — ckrao @ 11:52 am

This post is inspired by a question I asked myself during a recent one day international cricket game I watched on TV. It was pointed out that Pakistani batsman Nasir Jamshed had an impressive average of 50.26. I knew that he hadn’t played that many games so I was surprised that his average ended in .26 which suggests that he was dismissed more than just a few times. For example if his average were 50.22 I would have imagined that this was 50 and 2/9, so I could guess he had scored 452 runs and been dismissed 9 times.

So how many dismissals could he have had if his average were 50.26, assuming it is small? This was the way I thought about it mentally: the number 0.26 is just over 1/4, so it is probably of the form k/(4k-1), then it’s a question checking numbers of this form. To two decimal digits we have

  • 2/7 = 0.29
  • 3/11 = 0.27
  • 4/15 = 0.27
  • 5/19 = 0.26 (aha!)

Hence I guessed that he had been dismissed 19 times and had scored 50 \times 19 + 5 = 955 runs. Later on I checked that this was in fact the case. I thought it was cool that from a single average to two decimal places one could make a good guess of both the number of dismissals and the runs scored by the player (i.e. both the dividend and divisor), just based on the assumption that the ratio is simple. Of course if the decimal had been something like 50.25, it would have been more difficult to guess whether the he had scored 201, 402, 603, … runs for 4, 8, 12, … dismissals.

Below is a table showing for each value from 0.01 to 0.50 (in 0.01 increments) the simplest fraction that rounds to that value correct to two decimal places. Interestingly apart from fractions close to 0 and 0.5, the only case where the denominator exceeds 20 is for 0.34.

Decimal Simplest ratio Decimal Simplest ratio Decimal Simplest ratio Decimal Simplest ratio Decimal Simplest ratio
0.01 1/67 0.11 1/9 0.21 3/14 0.31 4/13 0.41 7/17
0.02 1/41 0.12 2/17 0.22 2/9 0.32 6/19 0.42 5/12
0.03 1/29 0.13 1/8 0.23 3/13 0.33 1/3 0.43 3/7
0.04 1/23 0.14 1/7 0.24 4/17 0.34 10/29 0.44 4/9
0.05 1/19 0.15 2/13 0.25 1/4 0.35 6/17 0.45 5/11
0.06 1/16 0.16 3/19 0.26 5/19 0.36 4/11 0.46 6/13
0.07 1/14 0.17 1/6 0.27 3/11 0.37 7/19 0.47 7/15
0.08 1/12 0.18 2/11 0.28 5/18 0.38 3/8 0.48 10/21
0.09 1/11 0.19 3/16 0.29 2/7 0.39 7/18 0.49 17/35
0.10 1/10 0.20 1/5 0.30 3/10 0.40 2/5 0.50 1/2

To show how one of the above ratios is calculated (apart from trial and error), we will show as an example how to find fractions m/n whose decimal equivalent to two places is 0.26. In other words, 0.255 \leq m/n < 0.265 or equivalently 51n \leq 200m < 53n. We first find a fraction that is close to m/n, in this example 1/4 is appropriate. Using this, we set n = 4m - d (more generally for the approximation p/q choose pn = qm \pm d) and find that 204m - 51d \leq 200m < 212m - 53d or equivalently,

\displaystyle \frac{53d}{12} < m \leq \frac{51d}{4}.

Hence for d = 1, m ranges from 5 to 12, corresponding to the fractions 5/19, 6/23, \ldots, 12/47 (note that 12/47 \approx 0.2553 which rounds to 0.26 while 13/51 \approx 0.2549 which rounds down to 0.25).

For d = 2, m ranges from 9 to 25, corresponding to the fractions 9/34, 10/38, \ldots, 25/98. We can continue to find all fractions satisfying the above inequality but it is apparent that the simplest will correspond to d=1 and m = \lceil 53/12 \rceil = 5.

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