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 $b$?
Setting $\frac{4b+19}{6b+11}$ equal to $0.76=\frac{76}{100}=\frac{19}{25}$, we have \begin{align*} \frac{4b+19}{6b+11}&=\frac{19}{25}
\\\Rightarrow\qquad 25(4b+19)&=19(6b+11)
\\\Rightarrow\qquad 100b+475&=114b+209
\\\Rightarrow\qquad -14b&=-266
\\\Rightarrow\qquad b&=\boxed{19}
\end{align*}.
19
Level 3
Algebra
Problem: 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 $b$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To solve for the value of \( b \), we start by equating the decimal \( 0.76 \) to the fraction \( \frac{4b+19}{6b+11} \). This requires converting the decimal into a fraction. The decimal \( 0.76 \) can be expressed as \( \frac{76}{100} \). 

Next, we set up the equation:

\[
\frac{4b + 19}{6b + 11} = \frac{76}{100}
\]

To eliminate the fractions, we cross-multiply:

\[
100(4b + 19) = 76(6b + 11)
\]

Expanding both sides gives:

\[
400b + 1900 = 456b + 836
\]

Next, we rearrange the equation to isolate terms involving \( b \):

\[
400b - 456b = 836 - 1900
\]

This simplifies to:

\[
-56b = -1064
\]

Dividing both sides by \(-56\) yields:

\[
b = \frac{1064}{56}
\]

Calculating the right-hand side, we simplify \( \frac{1064}{56} \):

\[
b = 19
\]

Since \( b \) must be a positive integer, we confirm that \( b = 19 \) is valid. Thus, the final answer is \( 19 \).<|END_OF_LATENT|> The final answer is $19$.