I took a break during some work to show a colleague (who is looking to become a teacher) this website. We couldn't resist from having a go at this ourselves! An enjoyable problem.
We felt we could probably solve this by programming some software to try every iteration (Brute force), but that might not be in the spirit of the problem!
So after trying to find some clever algebraic way to solve the problem, we abandoned it in favour of a good old "Try combinations" method, hopefully with some clever thinking along the way.
We began by converting all the fractions to have the same denominators. (I used a prime factorisation of each denominator to find the Lowest Common Multiple). This yielded :
1/6 = 100/600.
1/25 = 24/600
3/5 = 360/600
3/20 = 90/600
4/15 = 160/600
5/8 = 375/600
Now we've simplified the problem in to a counting issue. Which of the numerators can we add together to get closest to 600 in order to make the fraction 1.
In fact, we started by adding all the numbers together and calculating the difference of the total from our target number. (The Sum is 1109, and the target is 600, so the difference is 509.) 509 is a smaller number and so instinctively it felt easier to find the numbers we wouldn't use.
First we added together all the lowest numbers (100+24+90+160=374). This is further away from our answer than 375 by itself, and if we add 350 or 375 to this sum we will go way over our target. In one step this shows us that 360 or 375 will be part of the numbers we're looking for (remember we're using this step to calculate the numbers we're not going to use in our final solution). Thankfully with 360 and 375 as an accepted part of the answer this gave us much fewer iterations to try!
we tried several combinations using 360 as a base, only a few were needed to find
360 + 160 = 520, which is 11 above our target value.
Using 375 as our base we then found 375+100+24 = 499 which is below our target by 10! This was the best we found (Proved by exhaustion of the remaining posibilities).
And so by removing 375,100 and 24. From our solution. We are left with :
90/600+160/600+360/600 = 610/600
Or in the original Fractions : 3/20+4/15 +3/5 = 1.017 (3dp).