The decimal $0.76$ is equal to the fraction $\frac{4b+19}{6b+11}$, where $b$ is a positive integer. What is the value of $b$?
Let's begin by multiplying both sides of the equation by $6b+11$. That would give us $0.76(6b+11) = 4b+19$. Expanding the left side gives us $4.56b + 8.36 = 4b+19$. Now we can subtract $4.56b$ from both sides and get $8.36 = 0.44b + 19$. And now we divide both sides by $0.56$. Yes, and we get $b=19$.

