Answered

3 and 10 are a factor pair of 30 because $30=3 \times 10$. Break apart the 10 to find another factor pair of 30.



$\begin{aligned}

30 & =3 \times 10 \\

& =3 \times(5 \times 2) \\

& =(3 \times 5) \times 2

\end{aligned}$



Which expression correctly shows multiplying inside the parentheses?


A15 × 2
B6 × 3
C6 × 2
D15 × 3

Verified Answer :

  • Start with the decomposition of 30: \(30 = 3 \times 10 = 3 \times (5 \times 2)\).
  • Use the associative property to group the factors: \((3 \times 5) \times 2\).
  • Calculate the product inside the parentheses: \(3 \times 5 = 15\).
  • Conclude that the resulting expression is \(15 \times 2\), which corresponds to option \(\boxed{a}\).

Explanation

  1. Analyze the Factorization and Grouping

We are given that \(30 = 3 \times 10\). We are asked to break apart the factor \(10\) into its prime factors, \(5 \times 2\), to find another way to group the factors of \(30\). We start with the expression:
\[30 = 3 \times 10\]
Substituting \(10 = 5 \times 2\) into the equation, we get:
\[30 = 3 \times (5 \times 2)\]
Using the associative property of multiplication, we can regroup the factors to obtain:
\[30 = (3 \times 5) \times 2\]

  1. Evaluate and Select the Correct Expression

Now we evaluate the expression inside the parentheses, \((3 \times 5)\):
\[3 \times 5 = 15\]
Substituting this value back into the expression \((3 \times 5) \times 2\), we get:
\[15 \times 2\]
This matches option a. Thus, the expression that correctly shows the result of multiplying the numbers inside the parentheses is \(15 \times 2\).

  1. Final Answer Formulation

Comparing our result \(15 \times 2\) with the given options:

A\(15 \times 2\)
B\(6 \times 3\)
C\(6 \times 2\)
D\(15 \times 3\)

The result corresponds to option a. Therefore, the correct expression is \(15 \times 2\). The final answer is \(\boxed{a}\).

Examples

When you need to multiply several numbers, you can group them differently using the associative property of multiplication. For instance, if you want to calculate \(3 \times 5 \times 2\), you can first multiply \(3 \times 5 = 15\), then multiply that result by \(2\) to get \(30\). This technique makes complex mental arithmetic much easier by breaking problems into simpler, manageable steps.

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