Calculate the 5-Floor Buildings: Solving for Unknowns

Let's solve the problem step-by-step:

• First, summarize the data:

  • 2 buildings with 11 floors

  • 4 buildings with 2 floors

  • 5 buildings with 3 floors

  • Let $x$ be the number of buildings with 5 floors.

• Calculate the total number of buildings:

• $\text{Total buildings} = 2 + 4 + 5 + x = 11 + x$

• Compute the weighted sum of floors:

• $\begin{aligned} \text{Sum of floors} &= (2 \times 11) + (4 \times 2) + (5 \times 3) + (5 \times x) \\ &= 22 + 8 + 15 + 5x \\ &= 45 + 5x \end{aligned}$

• Set up the weighted average equation:

• $4.29 = \frac{45 + 5x}{11 + x}$

• Multiply both sides by $11 + x$ to eliminate the fraction:

• $4.29(11 + x) = 45 + 5x$

• Expand and solve the equation:

• $\begin{aligned} 4.29 \times 11 + 4.29x &= 45 + 5x \\ 47.19 + 4.29x &= 45 + 5x \\ 47.19 - 45 &= 5x - 4.29x \\ 2.19 &= 0.71x \end{aligned}$

• Solve for $x$:

• $x = \frac{2.19}{0.71} = 3$

Therefore, the number of buildings with 5 floors is $3$.