- Identify that the goal is to substitute the factor \(10\) with its own factor pair, which is \(2 \times 5\).
- Rewrite the original expression \(30 = 3 \times 10\) by replacing \(10\) with \((5 \times 2)\).
- Observe that the resulting expression is \(3 \times (5 \times 2)\), which matches option c.
- Conclude that the correct choice is \(\boxed{c}\).
Explanation
- Problem Analysis and Objective
To solve the problem, we need to express the factor \(10\) in the equation \(30 = 3 \times 10\) as a product of its own factors. This process is called "breaking apart" a factor, and it is a useful technique to find different ways to represent a product. Our goal is to identify which of the provided options correctly shows this breakdown of the number \(10\).
- Breaking Apart the Factor
The number \(10\) is a composite number, meaning it can be written as the product of smaller whole numbers. The most common pair of factors for \(10\) is \(2\) and \(5\), because \(2 \times 5 = 10\). To "break apart" the factor \(10\) in our original equation, we substitute the product \((2 \times 5)\) or \((5 \times 2)\) in place of the number \(10\). Substituting this into the original expression gives us \(30 = 3 \times (5 \times 2)\).
- Evaluating the Options
Now let's compare our result, \(3 \times (5 \times 2)\), with the given choices:
- a. \((2 \times 3) \times 10\): This is incorrect because it changes the first factor and keeps the \(10\).
- b. \((2 \times 5) \times 10\): This is incorrect because it adds an extra product and keeps the original \(10\).
- c. \(3 \times (5 \times 2)\): This is correct because it replaces the factor \(10\) with the product \((5 \times 2)\).
- d. \(3 \times (4 \times 2)\): This is incorrect because \(4 \times 2 = 8\), not \(10\).
Therefore, the expression that correctly shows breaking apart the factor \(10\) is \(3 \times (5 \times 2)\).
- Final Answer
The expression that correctly shows breaking apart the factor \(10\) is \(3 \times (5 \times 2)\), which corresponds to option c. Thus, the correct choice is \(\boxed{c}\).
Examples
In real life, breaking apart factors is like grouping items. If you have 3 boxes, and each box contains 10 apples, you have a total of \(3 \times 10 = 30\) apples. If you decide to organize those apples further by putting them into smaller bags inside each box, such that each box now contains 2 smaller bags with 5 apples each, you are essentially "breaking apart" the factor 10 into \(2 \times 5\). You still have the same total of \(3 \times 2 \times 5 = 30\) apples, but they are now organized in a different way!