1. **State the problem:** A projector creates a 30cm wide image when it is 50cm from the screen. We want to find the width of the image when the projector is moved back 20cm, making the distance 70cm. 2. **Formula and concept:** The width of the image is proportional to the distance from the screen because the projector creates similar triangles. This means: $$\frac{\text{width}_1}{\text{distance}_1} = \frac{\text{width}_2}{\text{distance}_2}$$ 3. **Assign known values:** $$\text{width}_1 = 30 \text{ cm}, \quad \text{distance}_1 = 50 \text{ cm}, \quad \text{distance}_2 = 50 + 20 = 70 \text{ cm}$$ 4. **Set up the proportion:** $$\frac{30}{50} = \frac{\text{width}_2}{70}$$ 5. **Solve for 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.