Solve: p(3/4a + 5/8b/p) + 9 ÷ (2/a + 4/3b) Complex Fraction Equation

To solve this algebraic expression, we'll follow these steps:

• Step 1: Distribute $p$ over the terms in the parentheses:
  $p(\frac{3}{4}a + \frac{5}{8}\frac{b}{p}) = \frac{3}{4}ap + \frac{5}{8}b$. Here, $p$ cancels with the $p$ in the denominator in the second term.
• Step 2: Simplify $9:\frac{2}{a}$:
  Dividing by a fraction is equivalent to multiplying by its reciprocal:
  $9 \times \frac{a}{2} = \frac{9a}{2}$.
• Step 3: Add $\frac{4}{3}b$ to the expression:
  We now combine all terms: $\frac{3}{4}ap + \frac{5}{8}b + \frac{9a}{2} + \frac{4}{3}b$.
• Step 4: Combine like terms:
  - Since $\frac{5}{8}b$ and $\frac{4}{3}b$ are like terms, find a common denominator to combine. - Convert and sum the fractions: $\frac{5}{8} = \frac{15}{24}$ and $\frac{4}{3} = \frac{32}{24}$, so: $\frac{5}{8}b + \frac{4}{3}b = \frac{47}{24}b$. - Combine with $\frac{9a}{2}$ and rewrite: $\frac{9a}{2} = 4.5a$.
• Step 5: Expression now is:
  $\frac{3}{4}ap + \frac{47}{24}b + 4.5a$

Therefore, the fully simplified algebraic expression is:

$\frac{3}{4}ap + 1\frac{23}{24}b + 4\frac{1}{2}a$

The correct answer corresponds to choice 2.