Here it is typed up (and slightly edited)
KEEP IT SIMPLE
Unit fractions can be written as the sum of two different unit fractions.
E.g. 1/2=1/3+1/6
Charlie's Rule
"The denominator of the last fraction is the product of the denominators of the first two fractions"
Are all the examples correct?
"1/2=1/10+1/20"
1. To add fractions of different denominators, we have to convert them first. 1/10=2/20
2. Now, we can add them together: 2/20+1/20=3/20
3. We can rewrite the sum to "1/2=3/20". This is obviously incorrect, but let's be sure (anyway). 3.5 Let's make them have the same denominator , so it is easier to compare. 1/2=10/20. Now, it states that 10/20=3/20. However, 10 ≠ 3, so 10/20 ≠ 3/20, so he is incorrect.
We already answered the initial problem (no, they are not all correct), but I like maths, so let's keep going, but with shorter explanations.
"1/3=1/4+1/12"
1. Convert 1/4 into12ths- 1/4=3/12
2. Now we can add them together: 3/12+1/12=4/12
3. Now, if you have a good understanding of fractions, you can tell they're equal. But not everyone does, so here we go... 3.5. 4/12=1/3 Now the sum is 1/3=1/3, so this is correct.
"1/3=1/7+1/21"
1. Convert 1/7 into 21sts- 1/7=3/21
2. Add them together- 3/21+1/21=4/21
3. This is the sum now: 4/21=1/3. Let's make it clearer. 3.5. 1/3=7/21 However, 7 ≠ 4, so 7/21 ≠ 4/21, so he is incorrect.
"1/4=1/5+1/20"
1. Convert 1/5 into 20ths- 1/5=4/20
2. Now add them together- 4/20+1/20=5/20
3.The sum can be rewritten as 1/4=5/20. That is true, but let's make sure :) 3.5. 5/20= 1/4, 1/4=1/4, so this one is true.
What do I notice?
In the correct sums, the second fraction always has a denominator 1 larger than the first fraction, treating the denominators as integers. Now, we have a possible formula: 1/n=1/n+1 + 1/n(n+1) . This could be wrong, so lets test it.
Find some other correct examples...
Lets do 7, because it should be enough, and 7 is a nice number
1/2=1/3+1/6
1. 1/3=2/6
2. 1/6+2/6=3/6
3. 3/6=1/2. This is correct. Let's double-check. 3.5. 3/6=1/2
1/5=1/6+1/30
1. 1/6=5/30
2. 5/30+1/20=6/30
3. 6/30=1/5. Let's make sure. 3.5. 6/30=1/5. This is correct.
1/6=1/7+1/42
1. 1/7=6/42
2. 6/42+1/42=7/42
3. 7/42=1/6. That's correct. Double-check: 3.5. 7/42=1/6
1/7=1/8+1/56
1. 1/8=7/56
2. 7/56+1/56=8/56
3. 8/56=1/7 Correct. Double-check 3.5. 8/56=1/7
1/8=1/7+1/72
1. 1/9=8/72
2. 8/72+1/72=9/72
3. 9/72=1/8 Correct. Double-check- 3.5. 9/72=1/8
1/9=1/10+1/90
1. 1/10=9/90
2. 9/90+1/90=10/90
3. 10/90=1/9 Correct. Double-check~ 3.5. 10/90=1/9
1/10=1/11+1/110
1. 1/11=10/110
2. 10/110+1/110=11/110
3. 11/110=1/10 Correct. Making sure- 3.5. 11/110=1/10
How would I explain this to Charlie?
Generally (I think) you can plug in any number as n in this formula: 1/n=1/n+1 + 1/n(n+1)
As far as I've experimented, the formula works. To explain it more simply, goes like this: The first fraction's denominator can be any integer, and the second fraction's denominator is that the first fraction's denominator +1. The third fraction's denominator is the product of the first two (fraction's) denominators.
Here's a diagram with 1/2 = 1/3 + 1/6
1/2 = 1/3 + 1/6 can be written as 3/6 = 2/6 + 1/6
Because 3/6 and 2/6 have a difference of one, that means the difference of the fractions will always be the product of the denominators.
Alison's Discovery
For some fractions, there were multiple ways of doing the sum.
Charlie's turn
1/8=1/9+?
Reconstruct the problem: ?=1/8-1/9. The lowest common denominator between the two is 72 (8*9=72). Conversions- 1/8=9/72 1/9=8/72. So, ?=9/72-8/72. 9-8=1, so 9/72-8/72=1/72. ?=1/72. 1/8=1/9+1/72...This one works~
1/8=1/10+?
Reconstruction ?=1/8-1/10. A common denominator between the two is 80 (8*10=80). Conversions- 1/8=10/80 1/10=8/80. So, ?=10/80-8/80. 10-8=2, so 10/80-8/80=2/80. We can simplify- 2/80=1/40. ?=1/40. 1/8=1/10+1/40...This one works too :)
1/8=1/11+?
Reconstruct ?=1/8-1/11. A common denominator between the two is 88 (8*11=88). Conversions- 1/8=11/88 1/11=8/88. So, ?=11/88-8/88. 11-8=3, so 11/88-8/88=3/88. This means that ?=3/88. We cannot simplify further, and this is not a unit fraction, so this one is wrong.
Can all unit fractions be mad in mare than 1 way, like this?
My first thought is no, so let's try to prove that wrong.
1/2=1/3+1/6
I really can't think of another way to do this, so let's assume there is only one way to make 1/2 out of 2 other different unit fractions. Now we (hopefully) solved the initial problem, let's try to find a rule to double-check.
1/3=1/4+1/12
Again, I really can't think of another way. My brain is broken, so I might be wrong. Let's say I'm right... There are a few number sequences with 2s and 3s (e.g. Fibonacci), but the one I think is most relevant is the Prime numbers, as they have only 2 factors- thus, there should be fewer solutions (if they were the denominator). Let's test it out.
1/5=1/6+1/30
Well, the pattern is working (I think). Let's try a fraction with a composite (not prime) denominator.
1/12=1/13+1/156. Here's one. But I can think of more... 1/12=1/14+1/84 1/12=1/15+1/60 1/12=1/16+1/48 12=1/18+1/36 1/12=1/20+1/30 1/12=1/21+1/28
Here are some more that I think work, so there is more than one way of making all unit fractions with composite denominators (I think). I noticed something that further proves my statement- all of the fractions' denominators in each sum (except the 1st one) all have a factor in common with another fraction in the sum. Another interesting thing I found is that the last sum I wrote (1/12=1/21+1/28) doesn't have a common factor (above 1) between all three fractions, which is odd. Anyway, because all of the sums -excluding the one achieved with the formula- have common factors in between them, a sum with a prime denominator only has factors linking to the last fraction- there is no common factor (except 1) between the first two fractions (because a prime his only two factors~ 1 and itself).
Lets triple-check with an absurdly large prime (for a fraction).
1/97=1/98+1/9506
Ok, I don't think there's any other solution for this one. Now (I hope) we can make our final statement.
No, not all unit fractions can be made in more than one way like this- unit fractions that have a prime denominator can only be made in one way(using the formula), while composite denominators can be made in several ways.