Fraction

A fraction is the ratio of two numbers. Most commonly, we consider rational numbers, those fractions which are the ratio of two integers or decimals.

An example of a fraction is $\frac34$. In the example, the numerator is $3$ and represents the number of individual parts of a given fraction, and the denominator is $4$ and represents the individual parts needed for the fraction to be one whole. A fraction where the numerator is greater than the denominator is called an improper fraction. A fraction where the numerator is the same as the denominator such as $\frac{3}{3}$ is always equal to $1$.

Converting Fractions

We convert fractions to fractions of a different denominator by multiplying or dividing the numerator and denominator by the same quantity (since we're essentially multiplying by 1).

When, after converting fractions, the numerator and denominator have a GCD equal to $1$, the fraction is in simplest form, or in lowest terms. Converting a fraction to a fraction of simplest form is known as simplifying a fraction.

Comparing Fractions

There are various methods to compare fractions.

One method is to compare fractions by common denominator. When two fractions have the same denominator, the one with the higher numerator is greater (because there are more of said quantity).

Another method is to compare fractions by common numerator. When two fractions have the same numerator, the one with the lower denominator is greater (because the "pieces" are bigger).

Cross-multiplication is also used to compare fractions. During cross-multiplication, we are essentially multiplying both sides by both the denominators.

Operations with Fractions

Addition and Subtraction

Before adding and subtracting fractions, if the denominators are different, then the fractions should be converted have the same denominator. This is because only quantities with the same units can be combined (we can't add 1 banana and 1 orange, but we can add 1 fruit and 1 fruit).

When the denominators are the same, we can add or subtract the numerators (since the numerators represent the number of "pieces").

Multiplication

When multiplying fractions, the resulting numerator is the product of the numerators multiplied, and the resulting denominator is the product of the denominators multiplied. Sometimes, during multiplication, we can cross-cancel to simplify fractions midway through.

Division

By definition, dividing a number means multiplying by its reciprocal. So after finding the reciprocal of the divisor, we can proceed with multiplication.

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