To solve the equation
\[ \frac{b}{9} + \frac{1}{4} = 19, \]
we can start by isolating the term \(\frac{b}{9}\).
- Subtract \(\frac{1}{4}\) from both sides:
\[ \frac{b}{9} = 19 - \frac{1}{4}. \]
- To combine the right side, we should have a common denominator. The common denominator between \(1\) and \(4\) is \(4\). Rewrite \(19\) as a fraction:
\[ 19 = \frac{76}{4}. \]
Now we can combine the fractions:
\[ 19 - \frac{1}{4} = \frac{76}{4} - \frac{1}{4} = \frac{76 - 1}{4} = \frac{75}{4}. \]
So we have:
\[ \frac{b}{9} = \frac{75}{4}. \]
- To solve for \(b\), multiply both sides by \(9\):
\[ b = 9 \cdot \frac{75}{4} = \frac{675}{4}. \]
- This can also be expressed as:
\[ b = 168.75. \]
- Now, let's check our solution by substituting \(b\) back into the original equation:
\[ \frac{675/4}{9} + \frac{1}{4}. \]
Calculating \(\frac{675/4}{9}\):
\[ \frac{675}{4} \cdot \frac{1}{9} = \frac{675}{36} = \frac{75}{4}. \]
Now substituting back into the equation:
\[ \frac{75}{4} + \frac{1}{4} = \frac{75 + 1}{4} = \frac{76}{4} = 19. \]
Since both sides are equal, our solution is verified. Thus, the final answer is:
\[ b = \frac{675}{4} \text{ or } 168.75. \]