Multiplication of fractions is the process of finding the product of two or more fractions by multiplying the numerators together and multiplying the denominators together. This operation is used to find the result when multiplying fractions in various mathematical contexts.
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To multiply fractions, you multiply the numerators together and then multiply the denominators together.
The resulting fraction has a numerator that is the product of the original numerators and a denominator that is the product of the original denominators.
Multiplying fractions is commutative, meaning the order of the factors does not affect the final result.
Multiplying a fraction by 1 does not change the value of the fraction, as 1 is the multiplicative identity.
Multiplying a fraction by 0 results in 0, as 0 is the multiplicative absorbing element.
Review Questions
Explain the step-by-step process for multiplying two fractions.
To multiply two fractions, you first multiply the numerators together to get the new numerator, and then multiply the denominators together to get the new denominator. For example, to multiply the fractions $\frac{1}{2}$ and $\frac{3}{4}$, you would multiply the numerators (1 x 3 = 3) and then multiply the denominators (2 x 4 = 8), resulting in the product $\frac{3}{8}$.
Describe how the commutative property applies to the multiplication of fractions.
The commutative property of multiplication states that the order of the factors does not affect the final product. This principle also applies to the multiplication of fractions. For instance, multiplying $\frac{1}{2}$ by $\frac{3}{4}$ will yield the same result as multiplying $\frac{3}{4}$ by $\frac{1}{2}$, which is $\frac{3}{8}$. The order of the fractions being multiplied does not change the final product.
Analyze the effects of multiplying a fraction by 1 or 0, and explain the reasoning behind these results.
Multiplying a fraction by 1 does not change the value of the fraction, as 1 is the multiplicative identity. This means that $\frac{1}{2} \times 1 = \frac{1}{2}$. Conversely, multiplying a fraction by 0 results in 0, as 0 is the multiplicative absorbing element. This can be expressed as $\frac{1}{2} \times 0 = 0$. The reasoning behind these results is that multiplying by 1 leaves the original value unchanged, while multiplying by 0 always produces a result of 0, regardless of the other factor.