Fractions

Learn the *ingenious details about fractions*. Fractions are measure of "parts of a whole". Numerator is the count of the parts and Denominator represents the size of one part as *place-value* for the numerator. In this, learn the following.

• Fractions as numbers with specified place-values

• Fractions as directed-numbers (positive and negative)

• Fractions arithmetics : first principles

• fractions arithmetics : simplified procedures

Addition/Subtraction with LCM of denominators

Multiplication by multiplying numerators and denominators

Division as multiplication with multiplicative inverse

Fractions: Part of a whole

Fractions are introduced as part of a whole. Numerator and Denominator are introduced and explained in detail. This introduces fractions as count or number of pieces as numerator and the place-value of the count as denominator.

Fractions: Dividing a Group

Fractions are explained for a group of objects as a whole. It was earlier studied that fractions represent part of a whole. In that, the whole was considered to be one object and that was cut into many pieces.

This page introduces the concept of dividing a group of objects into smaller subgroups, and fractions are used to represent the smaller subgroups with respect to the entire group of objects.

Fractions as Directed Numbers

Fractions are also directed numbers. That is, Fractions with positive and negative values are explained.

Like and Unlike fractions

Based on the place values (denominators), the fractions are classified as like fractions and unlike fractions.

Proper, Improper, and Mixed fractions

Based on the value of the fraction in comparison with the whole, the fractions are classified as proper fractions, improper fractions and mixed fractions.

Equivalent Fractions and Simplest form of a fraction

Fractions that specify the same amount with different place values are equivalent fractions. In the set of equivalent fractions, the fractions without a common factor between numerator and denominator is the simplest form.

Converting UnLike fractions

In this page, converting two or more unlike fractions to like fractions is explained.

Simplest form of a fraction

The simplest form of equivalent fractions is explained and given a fraction, the procedure to find the simplest form is detailed.

Comparing Fractions

Comparing two fractions of different place values or denominators is explained.

Comparison in first principles -- matching two quantities to find one as smaller, equal, or larger than the other -- is extended for fractions (numbers of different place-values).

Based on this a simplified procedure (find LCM of denominators and convert to like fractions) is explained.

Addition and Subtraction of Fractions

Addition or Subtraction of fractions having different place values or denominator is explained. The following are explained for fractions.

• addition in first principles -- combining two quantities and measuring the combined

• subtraction in first principles -- taking away a part of a quantity and measuring the remaining quantity.

• Simplified procedure : Sign property of Addition and converting to like fractions

• Simplified procedure : Sign property of Subtraction and Converting to like fractions

Multiplication of fractions

Fractions are part of whole. Multiplication is repeated addition. How does it work for fractions?

Reciprocal of a fraction

In this page, reciprocal of a fraction is introduced.

Division of Fractions

In this page, division of fractions is explained.

Fraction: Simplification of Expressions

In this page, handling of fractions, both positive and negative fractions, in numerical expressions is explained.

The precedence order PEMDAS / BODMAS is explained.

For operations of same precedence order, the sequence of operation "simplification from left to right" is explained.

Fractions : Precedence Order PEMA / BOMA

Redefining the precedence order in arithmetics with PEMA or BOMA. As part of fractions, the revised version is in handling division -- handle division as inverse of multiplication.

eg: $x\xf7y\xf7x$ This cannot be simplified using PEMDAS / BODMAS, instead, it can be simplified as $x\times \left(\frac{1}{y}\right)\times \left(\frac{1}{x}\right)$ and applying properties of arithmetics, it is $\left(\frac{1}{y}\right)\times x\times \left(\frac{1}{x}\right)=\frac{1}{y}$