Between them, three brothers -Michael, Daniel and Nicholas - eat 2/3 of a pie. If Michael eats 1/4 the amount eaten and Daniel eats 1/5 of what Nicholas eats, what fraction of the pie does Nicholas eat?
Can someone start me off in the right direction. What I have to start is (1/4)(2/3)-(1/5)(1/4)(2/3)-x=0. This is clearly not the correct equation. What am I doing wrong?
m + d + n = 2/3
m = 1/4 d
d = 1/5 n
I know this doesn’t help much, but I get five twelfths doing it in my head. Is that the right answer? 
Between them, Daniel and Nicholas ate three quarters of two thirds of the pie. If Nicholas’ share of this amount was x, then:
x + (x/5) = (3/4) x (2/3)
Nitpick: Your second line should be “m = 1/4 * 2/3”; it says Michael eats “1/4 the amount eaten”, not “1/4 the amount Daniel eats”.
Did it in your head, I am quite jealous.
We have 6x=30/12.
x = 5/12.
You’re right, so it does.
For what it’s worth, ignoring the 2/3 for the moment…
Michael ate 1/4 of the total
That leaves 3/4, which Daniel and Nicholas consumed together.
Daniel ate 1/5 of what Nicholas eats.
n + n/5=3/4
5n/5 + n/5=3/4
6n/5=3/4
24n=15
n=15/24 or 5/8
So Nichlolas at 5/8 and Daniel ate 1/8
5/8 * 2/3 =10/24, or 5/12=Nicholas’ portion of the total pie
Checking…
Daniel ate one-fifth of what Nicholas ate, 1/12
Michael ate 1/4 of 2/3, = 2/12.
5/12 + 1/12 + 2/12=8/12, reduces to 2/3 of the overall pie.
Or, to simplify, Daniel and Nicholas ate half the pie. So if Daniel ate 1/5 of what Nicholas ate, Daniel ate 1/6 of the pie and Nicholas ate 5/6. As others have already said.
ETA: Not 1/6 and 5/6, I meant 1/12 and 5/12.
I would solve it this way. Divide the pie into 36 equal pieces. 24 (2/3) were eaten by the three brothers
Michael eats 1/4 of 24 or 6 pieces.
18 pieces are left.
Nichalas eats 5 times as many pieces as Daniel.
Daniel eats 15 pieces and Nicholas eats 3 pcs.
Michael eats 6 out 36 pcs or 1/6 of the pie.
Nicholas eats 15 out 36 pcs or 5/12 of the pie.
Daniel eats 3 out 36 pcs or 1/12 of the pie.
FWIW, Nicholas sounds like my brother. He was a pig too.