- Recognize that \(-3 \frac{5}{8}\) is equal to \((-3) + (-\frac{5}{8})\).
- Apply the distributive property of multiplication: \(15 \times ((-3) + (-\frac{5}{8})) = 15(-3) + 15(-\frac{5}{8})\).
- Compare the result with the given choices.
- The equivalent expression is \(\boxed{A}\).
Explanation
- Analyzing the Mixed Number Expression
The given expression is \(15 \left(-3 \frac{5}{8}\right)\). A mixed number such as \(3 \frac{5}{8}\) is equivalent to the sum \(3 + \frac{5}{8}\). Therefore, the negative mixed number \(-3 \frac{5}{8}\) represents the value \(-\left(3 + \frac{5}{8}\right)\), which is equal to \(-3 - \frac{5}{8}\) or \((-3) + \left(-\frac{5}{8}\right)\).
- Applying the Distributive Property
To rewrite the entire expression \(15 \left(-3 \frac{5}{8}\right)\), we substitute the expression for the mixed number:
\[15 \left( (-3) + \left( -\frac{5}{8} \right) \right)\]
By the distributive property of multiplication over addition, which states that \(a(b + c) = ab + ac\), we can distribute the multiplier \(15\) to each part of the sum inside the parentheses:
\[15(-3) + 15\left(-\frac{5}{8}\right)\]
This expression is equivalent to the original one.
- Conclusion and Selecting the Correct Option
Now, we compare our derived expression, \(15(-3) + 15\left(-\frac{5}{8}\right)\), with the given options:
A\(15(-3) + 15\left(-\frac{5}{8}\right)\)
B\(15(-3) + 15\left(\frac{5}{8}\right)\)
C\(15(3) - 15\left(\frac{5}{8}\right)\)
D\(-15(-3) + -15\left(-\frac{5}{8}\right)\)
Comparing these, we see that option A matches our derived expression exactly.
Examples
Understanding how to distribute multiplication over mixed numbers is useful in everyday scenarios, such as calculating total costs. For instance, if you need to buy 15 items that each cost \(\$3.625\) (which is \(\$3 \frac{5}{8}\)), you might calculate the cost by thinking of it as \(15 \times (3 + 0.625)\) or \(15 \times 3 + 15 \times 0.625\). Mastering this distributive property allows you to simplify complex arithmetic and calculate totals more efficiently without necessarily needing a calculator.