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Difficulty: MediumRatios, Rates, and Proportions

An architectural firm is building a scale model of a new office building. The ratio of the height of the model to the height of the actual building is 1:501:50. The actual building will have a rectangular base with a perimeter of 270270 meters. If the ratio of the length of the base to the width of the base is 5:45:4, what is the length, in centimeters, of the base of the scale model?

  1. A
    1.5
  2. B
    120
  3. 150Answer
  4. D
    300
  5. E
    1,500

Answer

The length of the base of the scale model is 150150 centimeters.
The correct answer represents the scaled-down length of the base converted to centimeters. First, the sum of the actual length and width is half the perimeter: 270÷2=135270 \div 2 = 135 meters. Since the ratio of length to width is 5:45:4, we divide 135135 meters into 99 equal parts of 1515 meters each. The actual length is 55 parts, which equals 7575 meters. Applying the scale of 1:501:50 gives a model length of 75÷50=1.575 \div 50 = 1.5 meters. Finally, converting this to centimeters yields 1.5×100=1501.5 \times 100 = 150 centimeters.

Step-by-Step Solution

1
Calculate the sum of the actual length and width from the given perimeter.
L+W=135L + W = 135 meters.
The perimeter of a rectangle is 2(L+W)2(L + W), so the sum of one length and one width is half of the perimeter: 270÷2=135270 \div 2 = 135 meters.
2
Use the length-to-width ratio to find the actual length of the building's base.
L=75L = 75 meters.
The ratio L:W=5:4L:W = 5:4 means L=5xL = 5x and W=4xW = 4x. Substituting these into the sum gives 5x+4x=135⇒9x=135⇒x=155x + 4x = 135 \Rightarrow 9x = 135 \Rightarrow x = 15. Therefore, the actual length is 5×15=755 \times 15 = 75 meters.
3
Scale the actual length down to the model's scale.
Model length =1.5= 1.5 meters.
The scale of the model is 1:501:50, so the dimensions of the model are found by dividing the actual dimensions by 5050: 75 meters÷50=1.575 \text{ meters} \div 50 = 1.5 meters.
4
Convert the model length from meters to centimeters.
Model length =150= 150 centimeters.
Since 11 meter is equal to 100100 centimeters, multiplying the length in meters by 100100 gives the length in centimeters: 1.5×100=1501.5 \times 100 = 150 centimeters.

Key Concept

Solving multi-step ratio and proportion problems involving scale drawings, geometric perimeter, and unit conversions.

Alternative Method

Instead of finding the actual dimensions first, you can scale the total perimeter first: the perimeter of the model is 270 meters÷50=5.4 meters=540 centimeters270 \text{ meters} \div 50 = 5.4 \text{ meters} = 540 \text{ centimeters}. Since the ratio of length to width is 5:45:4, the perimeter consists of 2(5+4)=182(5 + 4) = 18 equal parts. The length represents 55 of these parts, so the model length is 518×540=150\frac{5}{18} \times 540 = 150 centimeters.
Estimated Time:1m 30s
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Topics

Ratios, Rates, and Proportions

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