The question consists of two parts based on the given figure:

(a) If the area of rectangle $P$ is $6 \, \text{cm}^2$, what is the area of the whole figure?

Step-by-step solution:

1. Assumptions:

   • The whole figure consists of 3 identical squares.
   • $P$ and $Q$ are equal in area.
   • $\frac{1}{8}$ of the whole figure is shaded.

2. Key observations:

   • The area of $P$ is $6 \, \text{cm}^2$. This implies $\text{Area of } Q = 6 \, \text{cm}^2$ as they are equal.
   • The rectangles $P$ and $Q$ are formed by overlapping sections of the squares.

3. Fractional division of the area: Let the total area of one square be $S$. Since there are 3 identical squares: $\text{Total Area of the whole figure} = 3S$ Given that $\frac{1}{8}$ of the figure is shaded: $\text{Shaded Area} = \frac{1}{8} \times 3S$ This shaded area is part of $P$ and $Q$.

4. Relationship between $P, Q,$ and squares:

   • Since $P$ and $Q$ overlap parts of the squares and their combined area equals the shaded region: $P + Q = \frac{1}{8} \times 3S$

5. Solving for $S$: Since $P = Q = 6 \, \text{cm}^2$, the total area of $P + Q$ is: $P + Q = 6 + 6 = 12 \, \text{cm}^2$ Substituting into the equation: $12 = \frac{1}{8} \times 3S$ Solve for $S$: $S = \frac{12 \times 8}{3} = 32 \, \text{cm}^2$

6. Total area of the whole figure: Since the whole figure consists of 3 squares: $\text{Total Area} = 3S = 3 \times 32 = 96 \, \text{cm}^2$

(b) Express the total areas of $P$ and $Q$ as a fraction of one square.

1. From the above calculation, the area of one square is: $S = 32 \, \text{cm}^2$
2. The combined area of $P$ and $Q$ is: $P + Q = 12 \, \text{cm}^2$
3. Express as a fraction of one square: $\frac{P + Q}{S} = \frac{12}{32} = \frac{3}{8}$

Final answers:

(a) The total area of the whole figure is $96 \, \text{cm}^2$.
(b) The total areas of $P$ and $Q$ as a fraction of one square is $\frac{3}{8}$.

Let me know if you want detailed steps or additional explanations!

5 Related Questions:

1. How is the overlapping area between $P$ and $Q$ calculated if it is explicitly provided?
2. Can you find the side length of each square given its area?
3. What is the area of the unshaded region of the entire figure?
4. How would the solution change if the shaded fraction were $\frac{1}{6}$ instead of $\frac{1}{8}$?
5. What is the relationship between the number of squares and the fraction of the shaded area?

1 Tip:

Always write down all known areas and relationships in geometric problems before solving equations—this reduces the chance of confusion!