Negative and postcard

The image on the negative has dimensions of 36x24 mm and is enlarged to a postcard format of 13.5x9 cm. In what proportion have the lengths increased? In what proportion has the image area increased?

Final Answer:

k =  15:4
q =  225:16

Step-by-step explanation:

$$\begin{gathered} x_1=36\text{mm} \\ {x}_2=13.5\text{cm}\to \text{mm}=13.5\cdot 10\text{mm}=135\text{mm} \\ {y}_1=24\text{mm} \\ {y}_2=9\text{cm}\to \text{mm}=9\cdot 10\text{mm}=90\text{mm} \\ k=x_2/x_1=135/36=\frac{15}{4}=3\frac{3}{4}=3.75=15:4 \\ \text{ Verifying Solution: } \\ {k}_1=y_2/y_1=90/24=\frac{15}{4}=3\frac{3}{4}=3.75 \end{gathered}$$

$$\begin{gathered} q=k\cdot k=3.75\cdot 3.75=\frac{225}{16}=14\frac{1}{16}=14.0625=225:16 \\ \text{ Verifying Solution: } \\ {S}_1=x_1\cdot y_1=36\cdot 24=864{\text{mm}}^2 \\ {S}_2=x_2\cdot y_2=135\cdot 90=12150{\text{mm}}^2 \\ {q}_2=S_2/S_1=12150/864=\frac{225}{16}=14\frac{1}{16}=14.0625 \end{gathered}$$

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