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Write-up


Title
Splitting Fractions in “two”

Problem Statement
2/5 can be written as the sum of two unique unit fractions. For example, 2/5 = 1/3 + 1/15. Try to find two unique unit fractions whose sum is 2/7. What about 2/11? 2/13? Is there a pattern?

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Problem setup

Remember, unit fraction refers to the notion that there is one part of the fraction being considered (there must be a one in the numerator). 

 First I checked the example given: 2/5 = 1/3 + 1/15.  Finding the LCD of 15 enabled me to verify that the equation is true.

 

Plans to Solve/Investigate the Problem

First I thought of the denominator of the second addend, which is 15.  The denominators of 5 * 3 gave you 15.  Will this work for the denominator of 7?  7 * 3 = 21.  This however didn’t work as 1/7 does not equal 1/3 + 1/21.

 

Investigation/Exploration of the Problem

Next try the denominators of 7 * 4, which gives you 28.  It works!  2/7 =1/4 + 1/28.  I found it by multiplying the denominator 7 by 4 (in the first problem you multiplied the denominator 5 by 3) to the other addend’s denominator of 28.

I suspect a pattern involving consecutive numbers multiplied by prime numbers.  The denominators in the problems given suggest  this pattern of prime numbers.  You’re given 5,7,11, and 13.  5 * 3 gives you the other addend’s denominator of 15.  7 * 4 gives you the other addend’s denominator of 28.  I suspect 11 * 5 will give me the other addend’s denominator of 55 for the next fraction 2/11.

Doesn’t work.  2/11 does not equal 1/5 + 1/55.

The instructor Sarah Ledford lends me some help.  With the use of Excel to generate some patterns, we learn that 2/11 = 1/6 +1/66.  Here’s the pattern:

 Pattern:

2/5 = 1/3 + 1/15

2/7 =1/4 + 1/28

(I skipped this fraction of 2/9 in the pattern as it is not given in the original problem.) 

2/11 = 1/6 +1/66

2/13 = 1/7 + 1/91

2/15 = 1/8 + 1/120

2/17 = 1/9 + 1/153

The pattern is found by multiplying the successive odd numbers in the denominator by consecutive numbers (the second denominator) to get the final denominator.  Giving you denominators of 5,7,11,and 13 (and not 9) suggested the pattern would have to do with prime numbers, but this is not the case.  Now we can go back and fill in the missing equation:

 2/9 = 1/5 + 1/45.