To evaluate the multiplication of a fraction and an integer variable term, such as $$\frac{11}{3}(-9a)$$, first determine the sign of the product. Since the signs are different, the product will be negative. Next, write the integer term as a fraction, $$-\frac{9a}{1}$$, and multiply the numerators and denominators to get $$-\frac{11 \cdot 9a}{3 \cdot 1}$$. Rewrite the numerator to reveal the common factor of $$3$$: $$-\frac{11 \cdot 3 \cdot 3a}{3}$$. Dividing out the common factor of $$3$$ simplifies the expression to the final product, $$-33a$$.

Claude

To evaluate the product of a fraction and an integer, such as $$-\frac{12}{5}(-20x)$$, first determine the sign. Since both factors are negative, their product is positive. Next, write the integer term $$20x$$ as a fraction by placing it over $$1$$, resulting in $$\frac{12}{5} \cdot \frac{20x}{1}$$. Multiply the numerators and denominators to get $$\frac{12 \cdot 20x}{5 \cdot 1}$$. To simplify, rewrite $$20$$ as $$5 \cdot 4$$ to expose the common factor of $$5$$: $$\frac{12 \cdot 5 \cdot 4x}{5}$$. Dividing out the common factor of $$5$$ leaves $$12 \cdot 4x$$, which simplifies to $$48x$$.

Example of Multiplying $$-\frac{12}{5}(-20x)$$

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Example of Multiplying $$\frac{11}{3}(-9a)$$

To evaluate the multiplication of a positive fraction and a negative integer variable term, such as $$\frac{13}{7}(-14b)$$, first determine the sign of the product. Since the terms have different signs, the product is negative. Write the integer term as a fraction, $$-\frac{14b}{1}$$, and multiply the numerators and denominators to yield $$-\frac{13 \cdot 14b}{7 \cdot 1}$$. Rewrite the numerator to expose the common factor of $$7$$, giving $$-\frac{13 \cdot 2 \cdot 7b}{7}$$. Finally, divide out the common factor of $$7$$ to simplify the expression, resulting in $$-26b$$.

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