Fractions

Fractions are a simple way to represent parts of a whole.

Imagine cutting a cake into 4 equal pieces: you can’t always express the amount of cake with whole numbers, and that’s where fractions come in. Each slice represents $ \frac{1}{4} $ (one-quarter) of the whole.

A fraction consists of two key numbers, each playing a specific role:

• Numerator: The top number in the fraction, showing how many parts of the whole are being considered. For example, in \( \frac{1}{4} \), the numerator "1" indicates that you are taking one part out of four.
• Denominator: The bottom number, which tells you how many equal parts the whole is divided into. In the case of \( \frac{1}{4} \), the denominator "4" tells you the whole has been divided into four equal parts.

This structure allows you to easily represent portions of a whole, clarifying the relationship between the parts and the whole.

What are fractions used for? Fractions are helpful in everyday life when dividing objects or quantities into equal parts. Whether you’re sharing a cake, dividing a group of people, or allocating time, fractions help you describe exactly how much you’re dealing with compared to the whole. Understanding fractions is essential for tackling more advanced mathematical concepts, such as percentages, decimals, and algebra.

Types of Fractions

There are several types of fractions:

• Proper Fractions: The numerator is smaller than the denominator, representing a value less than one. For example, \( \frac{2}{3} \) means you have taken 2 out of the 3 equal parts that make up a whole.
• Improper Fractions: The numerator is greater than the denominator, representing more than one whole. These can be converted into a whole number plus a fraction. For instance, \( \frac{23}{5} \) can be rewritten as \( 4 \frac{3}{5} \), meaning 4 wholes and 3 parts out of 5.
• Apparent Fractions: When the numerator equals or is a multiple of the denominator, the fraction equals a whole number. For example, \( \frac{12}{3} \) equals 4. In such cases, there's no need to divide the whole further, as the division is already perfectly even.

Keep in mind that a fraction with zero in the denominator is undefined, as division by zero is not a valid mathematical operation: $$ \frac{a}{0} \quad \text{undefined} \ \ \text{when} \ a > 0 $$ For instance, you cannot write: $$ \frac{0}{0}, \ \frac{1}{0}, \ \frac{2}{0}, \ \dots $$ Conversely, you can write \(\frac{0}{n}\) because \(0\) divided by any non-zero number is always \(0\):
$$ \frac{0}{1} = \frac{0}{2} = \dots = \frac{0}{n} = 0 $$

Operations with Fractions

Let’s break down the basic operations with fractions using some practical examples.

• Adding and Subtracting Fractions
  To add or subtract fractions, you first need to have the same denominator (the bottom number of the fraction). If the denominators are different, you can't just add the numerators; you need to find a common denominator. For example, with \( \frac{1}{4} \) and \( \frac{2}{3} \), you can bring both fractions to a denominator of 12, the least common multiple of 4 and 3: $$ \frac{1}{4} + \frac{2}{3} = \frac{3}{12} + \frac{8}{12} = \frac{11}{12} $$ This simple method is essential for many math operations.
• Multiplying Fractions
  Multiplying fractions is more straightforward compared to adding or subtracting, as there’s no need to find a common denominator. You just multiply the numerators and denominators directly. For example: $$ \frac{2}{3} \times \frac{3}{5} = \frac{2 \times 3}{3 \times 5} = \frac{6}{15} = \frac{2}{5} $$ This operation is useful in various contexts.
• Dividing Fractions
  Dividing fractions involves one extra step: instead of dividing directly, you multiply the first fraction by the reciprocal of the second. The reciprocal is obtained by swapping the numerator and denominator. Here’s an example: $$ \frac{3}{4} \div \frac{2}{5} = \frac{3}{4} \times \frac{5}{2} = \frac{15}{8} $$

Fractions and Decimal Numbers

Fractions and decimal numbers are two sides of the same coin. Any fraction can be converted into a decimal number by simply dividing its numerator by its denominator.

Some fractions, like \( \frac{1}{2} \), result in exact decimals (\( 0.5 \)), while others, such as \( \frac{1}{3} \), produce repeating decimals (\( 0.333\ldots \)).

Let’s look at a few practical examples.

Take the fraction \( \frac{25}{4} \) as an example. It represents a division that ends with no remainder (\( r=0 \)), giving us a finite decimal as the result (\( 6.25 \)).

Now consider the fraction \( \frac{455}{63} \). This represents a division that never ends with a remainder of zero. As a result, the quotient (\( 7.\overline{2} = 7.222222... \)) is a repeating decimal, with a repeating block of digits (2) that continues indefinitely.

The relationship between fractions and decimals is particularly valuable in fields like finance, where percentages, interest rates, and discounts are often expressed in decimal form.

How can you determine whether a fraction produces a terminating or repeating decimal?

It’s all about the prime factorization of the denominator \( b \) when the fraction \( \frac{a}{b} \) is reduced to its simplest form.

• If \( b = 1 \), the fraction is an integer (e.g., \( \frac{5}{1} = 5 \)).
• If \( b \) is made up only of powers of \( 2 \) and/or \( 5 \), the decimal is terminating (e.g., \( \frac{2}{5} = 2.5 \)).
• If \( b \) contains prime factors other than \( 2 \) or \( 5 \), the decimal is repeating (e.g., \( \frac{1}{3} = 0.333333... \)).
• If \( b \) includes both powers of \( 2 \) and/or \( 5 \) along with other prime factors, the decimal is repeating but begins with a non-repeating part (e.g., \( \frac{1}{6} = \frac{1}{2 \cdot 3} = 0.1666666... \)).

Comparing Fractions: Which is Bigger?

To compare fractions with different denominators, first bring them to the same denominator, then compare the numerators.

For example, consider \( \frac{3}{8} \) and \( \frac{5}{12} \). The least common denominator is 24, so:

$$ \frac{3}{8} = \frac{9}{24}, \quad \frac{5}{12} = \frac{10}{24} $$

In this case, \( \frac{10}{24} \) is greater than \( \frac{9}{24} \).

$$ \frac{9}{24} < \frac{10}{24} $$

So, you can conclude that \( \frac{5}{12} \) is larger than \( \frac{3}{8} \).

$$ \frac{3}{8} < \frac{5}{12} $$

It’s actually a lot simpler than it seems, don’t you think?

Instead, you can compare two fractions using the principle of proportions without finding a common denominator.

This principle states that in a proportion, the product of the extremes is equal to the product of the means.

If you have two fractions, \(\frac{a}{b}\) and \(\frac{c}{d}\), you can compare them by calculating the product of the extremes, \( a \cdot d \), and the product of the means, \( b \cdot c \):

• If \(a \cdot d > b \cdot c\), then \(\frac{a}{b} > \frac{c}{d}\).
• If \(a \cdot d < b \cdot c\), then \(\frac{a}{b} < \frac{c}{d}\).

For example, to compare \(\frac{3}{8}\) and \(\frac{5}{12}\) using this rule, start by writing the proportion:

$$ 3:8 = 5:12 $$

Next, calculate the product of the extremes:

$$ 3 \cdot 12 = 36 $$

Then, calculate the product of the means:

$$ 5 \cdot 8 = 40 $$

Since \( 36 < 40 \), you can conclude:

$$ \frac{3}{8} < \frac{5}{12} $$

This method is much quicker and avoids the need to calculate the least common denominator.