To find the radius of the original circle when a dilated circle has a radius of 10 inches and a scale factor of \( \frac{2}{3} \), you can use the formula for dilation:
\[ \text{Radius of the original circle} = \frac{\text{Radius of the dilated circle}}{\text{Scale factor}} \]
Substituting the known values:
\[ \text{Radius of the original circle} = \frac{10 , \text{inches}}{\frac{2}{3}} \]
Dividing by a fraction is the same as multiplying by its reciprocal:
\[ \text{Radius of the original circle} = 10 , \text{inches} \times \frac{3}{2} = 15 , \text{inches} \]
Thus, the radius of the original circle is \( 15 , \text{inches} \).
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