To convert from one unit to another, we need to determine the ratio of the two units, which is known as the representative fraction (RF). The RF is often expressed as a fraction where the numerator and denominator are in the same units.
a. 2 cm to 4 km
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Convert both measurements to the same unit (let's convert everything to centimeters).
- 4 km = 4,000 meters
- 4,000 meters = 4,000 × 100 cm = 400,000 cm
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Now we have:
- 2 cm = 2 cm
- 4 km = 400,000 cm
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The RF is the ratio of the distance on a map (or model) to the actual distance: \[ \text{RF} = \frac{\text{Distance on map}}{\text{Actual distance}} = \frac{2 \text{ cm}}{400,000 \text{ cm}} = \frac{2}{400,000} = \frac{1}{200,000} \]
So, the representative fraction for 2 cm to 4 km is 1:200,000.
b. 3 inches to 6 miles
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Convert both measurements to the same unit (let's convert everything to inches).
- 6 miles = 6 × 5,280 feet = 31,680 feet
- 31,680 feet = 31,680 × 12 inches = 380,160 inches
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Now we have:
- 3 inches = 3 inches
- 6 miles = 380,160 inches
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The RF is the ratio of the distance on a map (or model) to the actual distance: \[ \text{RF} = \frac{3 \text{ inches}}{380,160 \text{ inches}} = \frac{3}{380,160} \]
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Simplifying that fraction:
- Dividing both the numerator and the denominator by 3 gives: \[ \frac{1}{126,720} \]
So the representative fraction for 3 inches to 6 miles is 1:126,720.
Summary:
- a. 2 cm to 4 km → 1:200,000
- b. 3 inches to 6 miles → 1:126,720