It is considered that a certain fraction is expressed by the formula of addition of some fractions whose numerator is 1 and whose denominator is different such as the following [Example].
[Example]
2/3 = 1/2 + 1/6, 2/3 = 1/3 + 1/4 + 1/12, etc.
13/20 = 1/2 + 3/20 = 1/2 + 1/7 + 1/140,
13/20=(10 + 2 + 1)/20 = 1/2 + 1/10 + 1/20, etc.
Find one formula of addition of some fractions regarding the fraction of the following (1) and (2).
[Example]
2/3 = 1/2 + 1/6, 2/3 = 1/3 + 1/4 + 1/12, etc.
13/20 = 1/2 + 3/20 = 1/2 + 1/7 + 1/140,
13/20=(10 + 2 + 1)/20 = 1/2 + 1/10 + 1/20, etc.
Find one formula of addition of some fractions regarding the fraction of the following (1) and (2).
(1) 13/18
(2) 5/13
(2) 5/13
Answer
Solution
(1) 1/2 is contained in 13/18.
13/18 - 1/2 = 4/18 = 2/9 remains.
According to 9/2 = 4.5, since 1/5 is smaller than 2/9, it is found that 1/5 is contained in 2/9.
Since 2/9 - 1/5 = 1/45, 13/18 = 1/2 +1/5 + 1/45.
（2） It is considered as same as (1), according to 13/5=2.6, 1/3 is contained in 5/13.
5/13 - 1/3 = 2/39 remain.
According to 39/2 = 19.5, 1/20 is contained in 2/39.
2/39 - 1/20 = 1/780.
Therefore, 5/13 = 1/3 + 1/20 + 1/780.
(1) 13/18 = 1/2 +1/5 + 1/45
(2) 5/13 = 1/3 + 1/20 + 1/780
(2) 5/13 = 1/3 + 1/20 + 1/780
(1) 1/2 is contained in 13/18.
13/18 - 1/2 = 4/18 = 2/9 remains.
According to 9/2 = 4.5, since 1/5 is smaller than 2/9, it is found that 1/5 is contained in 2/9.
Since 2/9 - 1/5 = 1/45, 13/18 = 1/2 +1/5 + 1/45.
（2） It is considered as same as (1), according to 13/5=2.6, 1/3 is contained in 5/13.
5/13 - 1/3 = 2/39 remain.
According to 39/2 = 19.5, 1/20 is contained in 2/39.
2/39 - 1/20 = 1/780.
Therefore, 5/13 = 1/3 + 1/20 + 1/780.
