Considering 1 can be written as 1/1, it can also be written as any fraction where the numerator is equal to the denominator. This means that the value closest to 1 would have the numerator closest to the common denominator.
To solve this, I first found the lowest common multiple between the denominators. The first step of this is to list all of the prime factors.
5 : 5
6 : 2x3
8 : 2x2x2
15 : 3x5
20 : 2x2x5
25 : 5x5
The second step is to calculate the lowest common multiple by multiplying the factors that occur the most in the separate prime factor lists.
2 occurs three times in 8 : 2x2x2
3 only ever occurs once : 3
5 occurs twice in 25 : 5x5
2x2x2x3x5x5 = 600 = LCM = lowest common multiple.
This tells me that all fractions have a common denominator of 600. I then calculated the fractions with a denominator of 600, by multiplying the numerator by 600/denominator.
1/6 : 100/600
1/25 : 24/600
3/5 : 360/600
3/20 : 90/600
4/15 : 160/600
5/8 : 375/600
To find the the pairs of fractions, I now had to find the two numerators that when added together, equalled the value closet to 600. This list shows examples of each numerator, and how close they can be to 600 when another numerator is added.
24 + 375 = 399
90 + 375 = 465
100 + 375 = 475
160 + 375 = 535
360 + 375 = 735
The value closest to 600 is 535, which is made from the fractions 160/600 + 375/600. In the question, these fractions are simplified to 4/15 and 5/8. This means that the answer is the pair 4/15 and 5/8.