#### irene12

##### New member
Hello, I have an extra credit problem I would like to do, but I don't know where to start. Here it is :Find solution to 3/7=1/a+1/b+1/c where a, b, and c are positive intergers and a<b<c . Thank you. Any help would be great.

#### Denis

##### Senior Member
irene12 said:
Find solution to 3/7=1/a+1/b+1/c where a, b, and c are positive intergers and a<b<c .
1/a + 1/b + 1/c = 3/7 ; using LCD:
(bc + ac + ab) / abc = 3/7
Simplifying, in terms of c:
c = 7ab / (3ab - 7a - 7b)
At this point, you can tell that 3ab > 7a + 7b
Hope that "gets you going"...

EDIT: in case you're worried there might be no solutions, don't: there are 6 different solutions :idea:

#### soroban

##### Elite Member
Hello, irene12!

$$\displaystyle \text{Find solutions to: }\;\frac{1}{a} + \frac{1}{b} + \frac{1}{c}\:=\:\frac{3}{7}\; \text{ where }a, b, c\text{ are positive intergers and }a<b<c.$$

I found a number of solutions:

. . $$\displaystyle \frac{1}{3} + \frac{1}{12} + \frac{1}{84} \;=\;\frac{3}{7}$$

. . $$\displaystyle \frac{1}{3} + \frac{1}{11} + \frac{1}{231} \;=\;\frac{3}{7}$$

. . $$\displaystyle \frac{1}{3} + \frac{1}{14} + \frac{1}{42} \;=\;\frac{3}{7}$$

. . $$\displaystyle \frac{1}{3} + \frac{1}{15} + \frac{1}{35} \;=\;\frac{3}{7}$$

. . $$\displaystyle \frac{1}{4} + \frac{1}{6} + \frac{1}{84} \;=\;\frac{3}{7}$$

. . $$\displaystyle \frac{1}{4} + \frac{1}{7} + \frac{1}{28} \;=\;\frac{3}{7}$$

Do a Google search for "Egyptian fractions".

#### Denis

##### Senior Member
Soroban, I think you missed Irene's "an extra credit problem ";
I don't think we're allowed to give solution on these, only help.

Any other opinions?

#### irene12

##### New member
Thank you for the help with the problem. Both answers were very helpful because they both helped me solve the problem and hopefully, I will be able to solve this kind of problem on my own next time. With both answers, I can clearly see how and what to do. Thank you, Irene

#### mmm4444bot

##### Super Moderator
Staff member
Denis said:
Any other opinions?

My opinion is that soroban has already provided the regulars here with sufficient evidence to show that he doesn't care what we think.

#### Subhotosh Khan

##### Super Moderator
Staff member
Denis said:
That was clever - one "get out of corner now" pass for you ......