1. **State the problem:** Maria has a rectangular board measuring 150 cm by 30 cm. She cuts it into four smaller rectangular pieces, each with the same length-to-width ratio as the original board. 2. **Identify the original ratio:** The original board's length-to-width ratio is $$\frac{150}{30} = 5:1$$. 3. **Scale factor for length and width:** Since the smaller pieces have the same ratio, the scale factor applies equally to length and width. Let the scale factor be $$k$$, so the smaller board's length is $$150k$$ and width is $$30k$$. 4. **Number of pieces and area relation:** The original board is cut into 4 equal pieces, so the total area is divided by 4. 5. **Calculate scale factor using area:** The original area is $$150 \times 30 = 4500$$ cm². Each smaller piece has area $$\frac{4500}{4} = 1125$$ cm². 6. **Express smaller piece area in terms of scale factor:** $$\text{Area of smaller piece} = (150k)(30k) = 4500k^2$$ Set equal to 1125: $$4500k^2 = 1125$$ 7. **Solve for $$k$$:** $$k^2 = \frac{1125}{4500} = \frac{1}{4}$$ $$k = \frac{1}{2}$$ (taking positive root since length is positive) 8. **Scale factor ratio:** The scale factor relating original length to smaller length is $$1 : \frac{1}{2}$$, which can be written as $$2 : 1$$ when expressed in whole numbers. 9. **Ratio of areas:** The ratio of the original area to the smaller piece area is $$4500 : 1125$$. Simplify by dividing both sides by 1125: $$\frac{4500}{1125} : \frac{1125}{1125} = 4 : 1$$ **Final answers:** - Scale factor length ratio: **2 : 1** - Area ratio: **4 : 1**