[Combinatorics]
Problem: Eleanor has 100 marbles, each of which is black or gold. The ratio of the number of black marbles to the number of gold marbles is $1: 4$. How many gold marbles should she add to change this ratio to $1: 6$ ?

Solution: Since Eleanor has 100 marbles which are black and gold in the ratio $1: 4$, then $\frac{1}{5}$ of her marbles are black, which means that she has $\frac{1}{5} \cdot 100=20$ black marbles.

When more gold marbles are added, the ratio of black to gold is $1: 6$, which means that she has $6 \cdot 20=120$ gold marbles.

Eleanor now has $20+120=140$ marbles, which means that she added $140-100=40$ gold marbles.

Final answer: 40