Verified Answer :

  • Expand the given options using the distributive property: \(k(m + n) = km + kn\).
  • Calculate each expression to find which one equals the original sum \(20 + 24 = 44\).
  • Identify that \(4(5 + 6) = 20 + 24 = 44\) is the correct factorization.
  • Final answer: \(\boxed{4(5 + 6)}\).

Explanation

  1. Problem Analysis

The objective is to identify the correct factored form of the expression \(20 + 24\). Factoring a sum of two numbers involves finding a common factor \(k\) such that the expression can be rewritten as \(k(m + n) = km + kn\). We are given four options: \(4(5 + 7)\), \(4(5 + 6)\), \(2(10 + 11)\), and \(2(10 + 14)\). We will verify each option by expanding it to see which one results in the original sum \(20 + 24 = 44\).

  1. Evaluating the Options

We will test each of the provided options by applying the distributive property, \(k(m + n) = km + kn\):

  • For \(4(5 + 7)\): \(4 \times 5 + 4 \times 7 = 20 + 28 = 48\).
  • For \(4(5 + 6)\): \(4 \times 5 + 4 \times 6 = 20 + 24 = 44\).
  • For \(2(10 + 11)\): \(2 \times 10 + 2 \times 11 = 20 + 22 = 42\).
  • For \(2(10 + 14)\): \(2 \times 10 + 2 \times 14 = 20 + 28 = 48\).

Comparing these results to the target sum \(44\), we see that only the second option yields the correct value.

  1. Final Conclusion

Since expanding \(4(5 + 6)\) gives \(20 + 24\), we conclude that \(4(5 + 6)\) is the correct factored form of \(20 + 24\). Thus, the correct choice is \(\boxed{4(5 + 6)}\).

Examples

Factoring is useful in everyday shopping scenarios. For instance, if you are buying 4 packs of pens that cost \(\$5\) each and 4 packs of notebooks that cost \(\$6\) each, you can calculate the total cost in two ways: either by adding the individual subtotals, \(\$20 + \$24 = \$44\), or by factoring out the number of packs, \(4 \times (\$5 + \$6) = 4 \times \$11 = \$44\). Understanding this allows you to simplify calculations and manage your budget more efficiently.

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