1. **State the problem:** A projector creates a 30 cm wide image when it is 50 cm from the screen. We want to find the width of the image when the projector is moved back 20 cm, so it is now 70 cm from the screen. 2. **Formula and concept:** The width of the image is proportional to the distance from the projector to the screen. This means: $$\frac{\text{width}_1}{\text{distance}_1} = \frac{\text{width}_2}{\text{distance}_2}$$ 3. **Assign known values:** $$\text{width}_1 = 30 \text{ cm}, \quad \text{distance}_1 = 50 \text{ cm}$$ $$\text{distance}_2 = 50 + 20 = 70 \text{ cm}$$ 4. **Set up the proportion:** $$\frac{30}{50} = \frac{\text{width}_2}{70}$$ 5. **Solve for $\text{width}_2$:** Multiply both sides by 70: $$\text{width}_2 = 70 \times \frac{30}{50}$$ 6. **Simplify the fraction:** $$\text{width}_2 = 70 \times \frac{\cancel{30}}{\cancel{50}} = 70 \times \frac{3}{5}$$ 7. **Calculate the width:** $$\text{width}_2 = 70 \times 0.6 = 42 \text{ cm}$$ **Final answer:** The picture will be 42 cm wide when the projector is moved back 20 cm.