Select the correct answer.
What is the product of $-3 \frac{1}{3}$ and $-8 \frac{7}{10}$?
The problem asks for the product of two negative mixed numbers, \(-3 \frac{1}{3}\) and \(-8 \frac{7}{10}\). First, recall that the product of two negative numbers is always positive. To multiply these numbers, it is most efficient to convert each mixed number into an improper fraction. For \(-3 \frac{1}{3}\), we write it as \(-\frac{3 \times 3 + 1}{3} = -\frac{10}{3}\). For \(-8 \frac{7}{10}\), we write it as \(-\frac{8 \times 10 + 7}{10} = -\frac{87}{10}\).
Now, set up the multiplication of the two improper fractions:
\[ \left( -\frac{10}{3} \right) \times \left( -\frac{87}{10} \right) \]
Since both numbers are negative, the result will be positive:
\[ \frac{10}{3} \times \frac{87}{10} \]
Before performing the full multiplication, observe that we can simplify the expression by canceling common factors in the numerator and the denominator. The 10 in the numerator and the 10 in the denominator cancel out:
\[ \frac{1 \times 87}{3 \times 1} = \frac{87}{3} \]
Finally, divide 87 by 3 to find the product:
\[ \frac{87}{3} = 29 \]
Thus, the product of \(-3 \frac{1}{3}\) and \(-8 \frac{7}{10}\) is 29.
Understanding how to convert and multiply fractions is essential in fields like cooking or construction, where you might need to adjust recipe portions or measure materials using mixed numbers. For instance, if you have a blueprint where a length is scaled by a factor that involves a mixed number, knowing how to multiply these values precisely ensures that your measurements are accurate and your project is successful. Mastering this algebraic skill provides a foundation for more complex calculations in geometry and engineering.