Question

An architect makes a model of a new house. The model shows a tile patio in the backyard. In the​ model, each tile has length one third in.
and width one fourth in.
The actual tiles have length one fourth ft
and width three sixteenths ft
. What is the ratio of the length of a tile in the model to the length of an actual​ tile? What is the ratio of the area of a tile in the model to the area of an actual​ tile? Use pencil and paper. Describe two ways to find each ratio.
Question content area bottom
Part 1
The ratio of the length of a tile in the model to the length of an actual tile is
  
one ninth
.
​(Type the ratio as a simplified​ fraction.)
Part 2
The ratio of the area of a tile in the model to the area of an actual tile is
  
one ninth
.
​(Type the ratio as a simplified​ fraction.)

Answer 1

Part 1 — length ratio
Answer: 1/9.

Two ways:
1. Convert actual length to inches: 1/4 ft = 12*(1/4) = 3 in. Then ratio = (1/3 in) ÷ (3 in) = (1/3)/3 = 1/9.
2. Convert model length to feet: 1/3 in = (1/3)*(1/12) ft = 1/36 ft. Then ratio = (1/36) ÷ (1/4) = (1/36)*(4/1) = 4/36 = 1/9.

Part 2 — area ratio
Answer: 1/81.

Two ways:
1. Compute areas in the same unit (inches): model area = (1/3)*(1/4) = 1/12 in^2. Actual: 1/4 ft = 3 in and 3/16 ft = (3/16)*12 = 9/4 in, so actual area = 3*(9/4) = 27/4 in^2. Ratio = (1/12) ÷ (27/4) = (1/12)*(4/27) = 1/81.
2. Use the length scale factor: length ratio = 1/9, so area ratio = (1/9)^2 = 1/81.