Operaciones fraccionarias

Combining Fractional Operations

Fractions can seem like complex entities when we first encounter them, but they become easier to understand once you know how to perform basic mathematical operations with them. This guide will walk you through combining four key fractional operations: division, multiplication, subtraction, and addition. Our goal is to demystify these processes by illustrating each operation step-by-step.

Division with Fractions

To divide fractions, swap the dividend (the number above the line) and divisor (the number below the line), then flip both numbers upside down. Multiply this new expression just created, and your result will be the original dividend divided by the original divisor.

For example, let's calculate [\frac{6}{7} \div \frac{2}{9}]. First, swap the pieces and turn them upside down: [-\frac{7}{6} \cdot \frac{9}{2}]. Now multiply: [ -\frac{63}{12}= -\frac{31}{6}.] Flipping back a negative sign gives us our final answer: [\frac{31}{6}].

Multiplication with Fractions

Multiplying fractions involves finding the product of their numerators while keeping track of the denominators separately. Then, simplify the resulting fraction if possible.

Continuing from the previous section's example, multiplying two fractions instead:[\left( \frac{6}{7} \right)\times\left(\frac{2}{9}\right)=\frac{12}{63}=\frac{4}{21}].

Subtraction with Fractions

Subtracting fractions requires converting one fraction into an equivalent fraction with opposite signs, adding, then flipping the sign again. Let's try it out using [\frac{8}{15}-\frac{3}{10}]:

First, make the second fraction have the same denominator as the first (in this case (15)): [\frac{8}{15} - \frac{3\times\frac{3}{5}}{10} = \frac{8}{15}-\frac{9}{50}]. Next, combine the expressions and keep track of the denominator: [\frac{8\times\frac{5}{5} - 9}{50} = \frac{\cancelto{40}{8\times\frac{5}{5}} - 9}{\cancelto{50}{50}}]= [\frac{40-9}{50}=\frac{31}{50}].

Addition with Fractions

Adding fractions follows essentially the same steps as subtraction; however, since unlike terms are added without any changes, the process is typically more straightforward.

Let's compute [\frac{5}{12}+\frac{1}{6}]:

[ \frac{5}{12}+\frac{1\times\frac{2}{3}}{6} =\frac{5}{12}+\frac{2}{18}=\frac{5+2}{12\times\frac{3}{3}}=\frac{7}{12}].

In summary, performing operations on fractions may initially appear daunting, especially given their symbolic nature. However, by understanding the principles behind each operation, working through examples, and applying these concepts consistently, you'll find handling fractions becomes much simpler.