Notice that each positive fraction has an even numerator and each negative fraction has an odd numerator. Let's list the numerators from 19 to 1!
19, 18, 17, 16, 15, 14, 13, 12, 11
10, 9, 8, 7, 6, 5, 4, 3, 2, 1
We can see that there are 10 positive fractions with even numerators and 9 negative fractions with odd numerators. Next, let's recall the Associative Property of Addition.
Property
Description
Using Algebra
Associative Property of Addition
Changing the grouping of addends does not change the sum.
(a+b)+c=a+(b+c)
This means that we can use the Associative Property of Addition to group each negative fraction with a corresponding positive fraction. For example, we can rewrite the first two fractions in the following way.
19/20+ (- 18/20)&=[ 19/20+ (- 18/20)] [0.5em]
&=19+(- 18)/20
&=19-18/20
&=1/20
Let's group the first 18 fractions in a similar way.
Notice that each pair of grouped addends simplifies to the fraction 120. Looking at the diagram, we can also see that the last addend in the sum is also equal to 120. Finally, we can calculate the sum. To do so, we will add 10 fractions of 120. Let's do it!
