To find the dimensions of the postcard after Pam's reductions, we start with the original dimensions of the painting, which are 16 inches by 40 inches.
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First reduction by a scale factor of \( \frac{1}{4} \):
- For the width: \[ 16 \times \frac{1}{4} = 4 \text{ inches} \]
- For the height: \[ 40 \times \frac{1}{4} = 10 \text{ inches} \]
After the first reduction, the dimensions of the image are 4 inches by 10 inches.
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Second reduction by a scale factor of \( \frac{1}{2} \):
- For the width: \[ 4 \times \frac{1}{2} = 2 \text{ inches} \]
- For the height: \[ 10 \times \frac{1}{2} = 5 \text{ inches} \]
After the second reduction, the dimensions of the postcard are 2 inches by 5 inches.
Thus, the smaller dimension first, followed by the larger dimension, gives us:
2 inches by 5 inches.