Fractions Revisited (Including Simpson's Paradox)
Department of Mathematics
York College (CUNY)
Jamaica, New York 11451
Students perennially have difficulty with the topic of adding fractions. Many students find it tempting when adding 3/10 and 4/15 to add the numerators and denominators of the fractions and get the answer 7/25. They often will do this for fractions such as 6/7 and 2/3 where the "answer" they get from adding numerators and denominators 8/10 is clearly smaller than the sum of 6/7 and 2/3 which they should easily see is bigger, not smaller, than 1.
Perhaps students would be less likely to use this approach, if when they were shown how to add fractions, they were also shown this remarkable, but simple fact
if (positive) a/b is less then c/d then (a+c)/(b+d) is a fraction which always lies between a/b and c/d in size.
Consider the sum of 6/7 and 2/3. Notice that 8/10 = 4/5. Also notice that 2/3 is less than 6/7. Let us check how far 4/5 is from 2/3.
4/5 is bigger than 2/3 as can be seen by computing (4/5) - (2/3) which equals 2/15,
4/5 is larger than 6/7 as can be seen by computing (6/7) - (4/5) which equals 2/35.
Hence in this example, 4/5 lies strictly between 2/3 and 6/7.
Note that while 1/4 and 2/8 are different fractions, they are the same rational number. Consider what happens in the two calculations below analagous to what we did above.
We know 1/4 is less than 1/2 and (1+1)/(4+2) = 2/6 = 1/3 which lies between 1/4 and 1/2
2/8 is less than 1/2 and (2+1)/(8+2) = 3/10 which lies between 2/8 and 1/2
but 3/10 is not equal to 1/3.
If one shows the observation of the theorem to students, they may be less likely to add fractions by adding numerators and denominators.
The theorem can easily be proved from the basic rules of inequalities but it can be shown to students who don't yet know how to work with inequalities. The "process" of creating a "mediant" fraction that lies between two fractions also has lots of additional interesting aspects. One of the most intriguing directions here is to learn about the fractions known as Farey Fractions.
Another fascinating aspect of this discussion is the way these ideas can be used to help illustrate Simpson's Paradox, named for Edward Simpson but also known to other scholars such as the statistician George Udny Yule.
Consider the table below:
Here is the "story" behind this data. A college is doing hiring by its English and Mathematics departments. The fractions shown indicate (row 1) that English had 5 male applicants of whom 1 was hired, while it had 8 women applicants of whom 2 were hired. Notice that 1/5 is less than 2/8.
The Mathematics department (row 2) had 8 applicants of whom 6 were hired, while it had 13 woman applicants of whom 6 were hired.
In English 25 percent of those hired were women while only 20 percent of the men were hired, relatively more women than men.
In Mathematics 80 percent of the women were hired while only 75 percent of the men were hired.
Hence, in each department individually relatively more women than men were hired.
Now let us lump together the two departments data to get university-wide data. This shows up in the table above as the last row. Note that here 7/13 is bigger than 6/13, indicating that 54 percent of the hires were men while only 46 percent of the hires were women. For the "whole" we see the "reverse" of what was true of the parts.
When lumped the data seems to have one message while its separate parts indicate something else; this is an indication that "Simpson's Paradox" is at work.