Fill in the blanks to find the products and explain why they are equivalent.
Understanding the first blank
In the first equation, we need to fill in the blank in:
\[10 \times \frac{3}{4} = 10 \times \square \div \square = \]
When we multiply by a fraction like $\frac{3}{4}$, we can think of it as multiplying by the numerator and dividing by the denominator:
\[10 \times \frac{3}{4} = 10 \times 3 \div 4\]
So the first blank is 3 and the second blank is 4.
Calculating the result of the first equation
\[10 \times \frac{3}{4} = 10 \times 3 \div 4 = 30 \div 4 = 7.5\]
Understanding the second blank
In the second equation, we need to fill in the blanks in:
\[10 \times \frac{3}{4} = 10 \div \square \times \square\]
Another way to think about multiplying by a fraction is to divide by the denominator and then multiply by the numerator:
\[10 \times \frac{3}{4} = 10 \div 4 \times 3\]
So the first blank is 4 and the second blank is 3.
Calculating the result of the second equation
\[10 \times \frac{3}{4} = 10 \div 4 \times 3 = 2.5 \times 3 = 7.5\]
Explaining why these are equivalent
These approaches are equivalent because multiplication and division are both associative operations when properly ordered. In the first approach, we multiply first and then divide (following the order of operations). In the second approach, we divide first and then multiply.
Mathematically, this is because:
\[a \times \frac{b}{c} = \frac{a \times b}{c} = \frac{a}{c} \times b\]
Both methods yield the same result of 7.5, demonstrating that the order of these operations can be rearranged as long as we maintain the relationship between the numbers.
\(\boxed{
\begin{array}{l}
10 \times \frac{3}{4} = 10 \times 3 \div 4 = 7.5 \\
10 \times \frac{3}{4} = 10 \div 4 \times 3 = 7.5
\end{array}
}\)
These are equivalent because multiplication and division are associative operations when properly ordered, and both approaches represent different but valid ways to calculate the product of a number and a fraction.
\(\boxed{
\begin{array}{l}
10 \times \frac{3}{4} = 10 \times 3 \div 4 = 7.5 \\
10 \times \frac{3}{4} = 10 \div 4 \times 3 = 7.5
\end{array}
}\)
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