An architect

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 of 1/2 inch and width of 1/6 inch. The actual tiles have length of 2/3 feet and width 2/9 feet. What is the ratio of the length of a tile in the model to the length of the actual tile? What is the ratio of the area of a tile in the model to the area of an actual tile?

Correct result:

k₁ =  1:16
k₂ =  1:288

Solution:

$$\begin{gathered} a=1\mathrm{/}2={\displaystyle \frac{1}{2}}=0.5\text{ inch } \\ b=1\mathrm{/}6={\displaystyle \frac{1}{6}}\doteq 0.1667\text{ inch } \\ {a}_2=2\mathrm{/}3={\displaystyle \frac{2}{3}}\doteq 0.6667\text{ ft } \\ {b}_2=2\mathrm{/}8={\displaystyle \frac{1}{4}}=0.25\text{ ft } \\ x=a_2\to \text{ inch}=a_2\cdot 12\text{ inch}=0.6667\cdot 12\text{ inch}=8\text{ inch } \\ y=b_2\to \text{ inch}=b_2\cdot 12\text{ inch}=0.25\cdot 12\text{ inch}=3\text{ inch } \\ {k}_1=a\mathrm{/}x=0.5\mathrm{/}8={\displaystyle \frac{1}{16}}=0.0625=1:16 \\ \text{ Verifying Solution: } \\ {K}_1=b\mathrm{/}y=0.1667\mathrm{/}3={\displaystyle \frac{1}{18}}\doteq 0.0556 \\ {K}_1\mathrm{\ne }{k}_1 \end{gathered}$$

$$\begin{gathered} S_1=a\cdot b=0.5\cdot 0.1667={\displaystyle \frac{1}{12}}\doteq 0.0833{\text{inch}}^2 \\ {S}_2=x\cdot y=8\cdot 3=24{\text{inch}}^2 \\ {k}_2=S_1\mathrm{/}{S}_2=0.0833\mathrm{/}24={\displaystyle \frac{1}{288}}\approx 0.0035=1:288 \end{gathered}$$

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