- First, calculate the target sum: \(54 + 30 = 84\).
- Apply the distributive property \(a(b + c) = ab + ac\) to each given option.
- Evaluate \(6(9 + 5) = 6 \times 9 + 6 \times 5 = 54 + 30 = 84\).
- Conclude that the correct choice is \(\boxed{a}\).
Explanation
- Understanding the Problem and Evaluating the Sum
The goal is to factor the numerical expression \(54 + 30\) using the distributive property, which states that \(a(b + c) = ab + ac\). To factor the sum, we need to find a common factor for both 54 and 30 and rewrite the expression as the product of that common factor and the sum of the remaining parts. First, we calculate the total sum: \(54 + 30 = 84\). Then, we evaluate each of the given options to see which one results in the sum of 84 using the distributive property.
- Evaluating the Options
We evaluate each option by distributing the multiplier:
Using the distributive property: \(6(9) + 6(5) = 54 + 30 = 84\).
This matches our target sum.
Using the distributive property: \(3(17) + 3(10) = 51 + 30 = 81\).
This does not match 84.
Using the distributive property: \(6(8) + 6(5) = 48 + 30 = 78\).
This does not match 84.
Using the distributive property: \(3(16) + 3(10) = 48 + 30 = 78\).
This does not match 84.
- Conclusion
After evaluating all given options, we find that only the first option, \(6(9 + 5)\), correctly factors the expression \(54 + 30\) to result in 84. Thus, the correct expression is \(6(9 + 5)\). The final answer is \(\boxed{a}\).
Examples
In daily life, the distributive property is useful when calculating total costs or quantities. For instance, if you are buying 6 packs of pencils where each pack contains 9 red pencils and 5 blue pencils, you can calculate the total number of pencils by either adding the total per pack first, \(6 \times (9 + 5) = 6 \times 14 = 84\), or by adding the total of each color, \((6 \times 9) + (6 \times 5) = 54 + 30 = 84\). Both methods provide the same total, demonstrating how the property helps in simplifying complex arithmetic.