Let me explain these examples in a simplified and intuitive manner.

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Example 1: Enlarging a photograph

The photograph is originally 12 cm wide and 8 cm tall, and it is enlarged in the ratio 3:2.

Step 1: Understanding the ratio

• The ratio 3:2 means the new image is 1.5 times larger than the original (since $\frac{3}{2} = 1.5$).

Step 2: Enlarging the width

• Start with the width of the photograph: 12 cm.
• Multiply it by $1.5$: $12 \times 1.5 = 18$.
• So, the enlarged width is 18 cm.

Step 3: Enlarging the height

• Start with the height of the photograph: 8 cm.
• Multiply it by $1.5$: $8 \times 1.5 = 12$.
• So, the enlarged height is 12 cm.

Conclusion

The enlarged photograph has dimensions 18 cm wide and 12 cm tall.

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Example 2: Projecting an image

A photographic transparency is 5 cm wide and 3 cm tall. It is projected onto a screen, where the width of the image becomes 1.5 m (or 150 cm). Let’s figure out the enlargement ratio and the new height.

Step 1: Calculate the enlargement ratio

• The original width is 5 cm, and the new width is 150 cm.
• Divide the new width by the original width to get the ratio: $\text{Enlargement ratio} = \frac{\text{new width}}{\text{original width}} = \frac{150}{5} = 30$
• The enlargement ratio is 30:1, meaning the image is 30 times larger.

Step 2: Calculate the height of the image

• The original height of the transparency is 3 cm.
• Multiply the original height by the enlargement ratio: $\text{New height} = 3 \times 30 = 90 \, \text{cm}$
• So, the height of the projected image is 90 cm.

Conclusion

The projected image is enlarged 30 times, and its dimensions are 150 cm wide and 90 cm tall.

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Key Takeaways:

1. Ratios represent a proportional scaling factor—here, the new dimensions are calculated by multiplying the originals by the scaling factor.
2. Always use the same units for all dimensions (e.g., convert meters to centimeters) to ensure consistency in calculations.

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Let me know if you’d like more examples or explanations! Here are 5 related questions to test your understanding:

1. If a 7 cm wide image is enlarged by a ratio of 4:3, what is the new width?
2. What happens to an image's height if it is reduced in the ratio 5:6?
3. If the enlargement ratio is 20:1 and the original image height is 4 cm, what is the new height?
4. A poster 9 cm wide is scaled to 135 cm. What is the enlargement ratio?
5. If a 10 cm x 6 cm photograph is scaled down by a ratio of 1:2, what are the new dimensions?

Tip: Ratios often represent scaling factors directly. Just multiply the original measurements by the factor to find the new dimensions!