To find the surface area of the new box after applying a scale factor, we can use the relationship between the scale factor and surface area.
If the scale factor for the linear dimensions of an object is \( k \), then the surface area of the object changes by a factor of \( k^2 \).
In this case, the scale factor is \( \frac{3}{4} \). Therefore, the change in surface area can be calculated as follows:
\[ \text{New Surface Area} = \text{Original Surface Area} \times k^2 \]
Given that the original surface area is \( 120 , \text{cm}^2 \):
\[ \text{New Surface Area} = 120 \times \left( \frac{3}{4} \right)^2 \]
Calculating \( \left( \frac{3}{4} \right)^2 \):
\[ \left( \frac{3}{4} \right)^2 = \frac{9}{16} \]
Now, we substitute this back into our equation:
\[ \text{New Surface Area} = 120 \times \frac{9}{16} \]
Calculating this:
\[ \text{New Surface Area} = \frac{120 \times 9}{16} = \frac{1080}{16} = 67.5 \]
Therefore, the surface area of the new box is
\[ \boxed{67.5} , \text{cm}^2 \]
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