- The expression \(30 = 3 \times (5 \times 2)\) represents the number 30 broken down into its prime factors or a set of factors.
- By applying the associative property of multiplication, we reorder the parentheses to group the factors as \((3 \times 5) \times 2\).
- Comparing this to the given choices, option c is the only one that matches this regrouping.
- The correct option is \(\boxed{c}\).
Explanation
- Analyze the given expression
The number 30 is expressed as the product of two factors, 3 and 10, meaning \(30 = 3 \times 10\). We are then asked to break apart the factor 10 into its own factors, specifically 5 and 2, which gives us \(10 = 5 \times 2\). Substituting this into the original expression, we get \(30 = 3 \times (5 \times 2)\).
- Apply the Associative Property of Multiplication
To find another factor pair or a different grouping, we use the associative property of multiplication. This property states that for any three numbers \(a\), \(b\), and \(c\), the order in which we group the factors does not change the product: \(a \times (b \times c) = (a \times b) \times c\). In our case, where \(a = 3\), \(b = 5\), and \(c = 2\), the expression \(3 \times (5 \times 2)\) can be rewritten as \((3 \times 5) \times 2\).
- Compare and Select the Correct Expression
Now, let's examine the provided options:
A\((3 \times 2) \times 10\)
B\((5 \times 10) \times 2\)
C\((3 \times 5) \times 2\)
D\((5 \times 3) \times 10\)
By comparing our result \((3 \times 5) \times 2\) to these options, we see that it matches option c. Options a, b, and d do not represent the same regrouping of the factors derived from breaking down 10 into 5 and 2.
- Final Answer
The correct expression that demonstrates the regrouping of the factors of 30 after breaking apart 10 is \((3 \times 5) \times 2\). Therefore, the correct option is \(\boxed{c}\).
Examples
Imagine you are arranging 30 chairs in a room. You could put them in 3 rows of 10. If you decide to break those 10 chairs into 2 smaller groups of 5, you have effectively created 3 groups of 2 sets of 5. By re-grouping, you can see that you still have the same total of 30 chairs, just arranged differently: \((3 \times 5) \times 2 = 15 \times 2 = 30\). This helps in organizing space efficiently by understanding how factors can be shifted while keeping the total amount constant.