The recipe calls for 1 1/4 dozen cupcakes, which is equal to:
\[ 1 \frac{1}{4} \text{ dozen} = 1.25 \text{ dozen} = 1.25 \times 12 = 15 \text{ cupcakes} \]
Next, the recipe requires 1 2/3 cups of flour, which can be expressed as:
\[ 1 \frac{2}{3} \text{ cups} = \frac{5}{3} \text{ cups} \]
Now, to find out how many cupcakes Rashida can make with just 1 cup of flour, we need to determine the amount of flour used for each cupcake.
To find the amount of flour needed for one cupcake, we divide the total amount of flour required by the total number of cupcakes:
\[ \text{Flour per cupcake} = \frac{\frac{5}{3} \text{ cups}}{15 \text{ cupcakes}} = \frac{5}{3} \div 15 = \frac{5}{3} \times \frac{1}{15} = \frac{5}{45} = \frac{1}{9} \text{ cups per cupcake} \]
Now, we will determine how many cupcakes Rashida can make with her 1 cup of flour:
\[ \text{Number of cupcakes} = \frac{1 \text{ cup}}{\frac{1}{9} \text{ cups per cupcake}} = 1 \times \frac{9}{1} = 9 \text{ cupcakes} \]
Therefore, Rashida can make 9 cupcakes. The answer is:
\[ \boxed{9} \]
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