Solve the Equation: Finding the Unknown Factor in ?(b+8) = a+8a/b

To solve this problem, we'll fill in the missing value in the equation:

$?(b+8)=a+8\frac{a}{b}$

Let's follow these steps:

• Step 1: Let's represent the missing value as $x$. Thus, the equation becomes $x(b + 8) = a + 8\frac{a}{b}$.
• Step 2: Distribute $x$ on the left-hand side: $xb + 8x = a + 8\frac{a}{b}$.
• Step 3: To equate both sides of the equation, observe that the expression on the right can be decomposed as $a + 8\frac{a}{b}$.
• Step 4: By factoring $a$ in the right-hand side expression: $a + \frac{8a}{b} = a\left(1 + \frac{8}{b}\right)$.
• Step 5: For the expressions $xb + 8x$ and $a\left(1 + \frac{8}{b}\right)$ to be equal, the factor $x$ must be equal to the common factor applied to $b + 8$ and the factored right-hand side.
• Step 6: Set each coefficient or common factor equal: $x = \frac{a}{b}$.

Therefore, the missing value in the equation is $\frac{a}{b}$.