Megan creates a scale drawing of a car. The ratio of her scale drawing length to actual car
length is 3 cm: 2 m. The length of the car in her drawing is 6 cm.
Ivan draws a scale drawing of the same car. His ratio of scale drawing length to actual car
length is 2 cm :1 m.
What is the length of the car in Ivan’s scale drawing?
The Correct Answer and Explanation is :
To find the length of the car in Ivan’s scale drawing, we need to establish the relationship between the lengths in the scale drawings and the actual length of the car based on their respective ratios.
Step 1: Understand Megan’s Scale Drawing
Megan’s ratio of scale drawing length to actual car length is given as (3 \text{ cm} : 2 \text{ m}). This means for every 3 cm in her drawing, the actual length of the car is 2 m.
We can convert the actual length from meters to centimeters for easier comparison. Since 1 m equals 100 cm, 2 m becomes:
[
2 \text{ m} = 2 \times 100 \text{ cm} = 200 \text{ cm}
]
Now, we can express the relationship in the following way:
[
\frac{\text{Drawing length}}{\text{Actual length}} = \frac{3 \text{ cm}}{200 \text{ cm}}
]
Step 2: Find the Actual Length of the Car
According to the problem, the length of the car in Megan’s drawing is 6 cm. Using the ratio we established:
[
\frac{6 \text{ cm}}{\text{Actual length}} = \frac{3 \text{ cm}}{200 \text{ cm}}
]
To find the actual length, we can set up a proportion:
[
6 \text{ cm} \cdot 200 \text{ cm} = 3 \text{ cm} \cdot \text{Actual length}
]
Calculating this gives:
[
1200 \text{ cm} = 3 \cdot \text{Actual length}
]
Dividing both sides by 3:
[
\text{Actual length} = \frac{1200 \text{ cm}}{3} = 400 \text{ cm}
]
Step 3: Understand Ivan’s Scale Drawing
Ivan’s ratio of scale drawing length to actual car length is (2 \text{ cm} : 1 \text{ m}). Converting 1 m to cm gives us:
[
1 \text{ m} = 100 \text{ cm}
]
So Ivan’s ratio is (2 \text{ cm} : 100 \text{ cm}), which simplifies to:
[
\frac{2 \text{ cm}}{100 \text{ cm}} = \frac{1}{50}
]
Step 4: Calculate the Length of Ivan’s Drawing
Using the same proportion method as before, we can set up the relationship for Ivan:
[
\frac{\text{Drawing length}}{400 \text{ cm}} = \frac{2 \text{ cm}}{100 \text{ cm}}
]
Setting this up as a proportion:
[
\text{Drawing length} \cdot 100 = 2 \cdot 400
]
Calculating gives:
[
\text{Drawing length} \cdot 100 = 800
]
Thus,
[
\text{Drawing length} = \frac{800}{100} = 8 \text{ cm}
]
Conclusion
The length of the car in Ivan’s scale drawing is 8 cm. This solution illustrates how to apply ratios to find corresponding lengths in scale drawings, emphasizing the importance of unit conversion and proportional reasoning in practical applications.