overview

Fractions are part of whole. Division is dividing a number into equal parts.

• division in first principles -- splitting a quantity into a number of parts and measuring one part

• simplified procedure : Division as multiplication by reciprocal

just know this

Division is the inverse of multiplication. And division of fractions can be easily explained with that.

$\frac{3}{4}\xf7\frac{2}{3}$

$=\frac{3}{4}\times \frac{3}{2}$ (multiply by inverse)

$=\frac{9}{8}$.

A learner may stop at this, but I would suggest to read the rest (too long, too hard) to understand how division works for fractions.

or understand this

Division $6\xf73=2$ is illustrated in the figure.

• $6$ is the dividend

• $3$ is the divisor

• $2$ is the quotient

Dividend $6$ is split into divisor $3$ parts and one part of that $2$ is the quotient.

One way to understand the integer division:

• Dividend $6$ is considered as $3$ parts

• in that $3$ is the divisor

• In the $3$ parts one part is taken.

The key in this explanation is " dividend is considered as divisor parts and one part is taken". The same can be extended for fractions.

Division $2\xf7\frac{1}{3}$ is illustrated in the figure.

Dividend $2$ is split as divisor $\frac{1}{3}$ parts. That is $2$ is the fraction $\frac{1}{3}$ and the full part for the fraction is found. The $\frac{1}{3}$ part is repeated to get the full part. This is shown in the figure.

In this, one part $6$ is the quotient. $2\xf7\frac{1}{3}=6$

Division $4\xf7\frac{2}{3}$ is illustrated in the figure.

Dividend $4$ is split as divisor $\frac{2}{3}$ parts.

In this, one part $6$ is the quotient.

$4\xf7\frac{2}{3}=6$

Division $\frac{3}{4}\xf7\frac{2}{3}$ is considered. The figure shows the dividend $\frac{3}{4}$. The division is illustrated in the next page.

Division $\frac{3}{4}\xf7\frac{2}{3}$ is illustrated in the figure.

Dividend $\frac{3}{4}$ is split as divisor $\frac{2}{3}$ parts. This is shown in the figure.

In this, the place value is $\frac{1}{8}$ and the count is 9.

The same can be simplified as follows.

$\frac{3}{4}\xf7\frac{2}{3}=\frac{3}{4}\times \frac{3}{2}=\frac{9}{8}$

yeah! enough of that

Division is inverse of multiplication.

**Division of Fractions: **Given two fractions $\frac{p}{q}$ and $\frac{l}{m}$, the division is

$\frac{p}{q}\xf7\frac{l}{m}$

$=\frac{p}{q}\times \frac{m}{l}$

$=\frac{p\times m}{q\times l}$

example

Divide $\frac{14}{18}\xf7\frac{4}{3}$.

The answer is '$\frac{7}{12}$'

summary

» $\frac{a}{b}\xf7\frac{c}{d}$ = $\frac{a}{b}\times \frac{d}{c}$

» $6\xf73$

→ $6$ is $3$ parts and find the group representing $1$ part

→ $=2$

» $2\xf7\frac{1}{3}$

→ $2$ is $\frac{1}{3}$ part and find the group representing $1$ part

→ $=6$

» $\frac{3}{4}\xf7\frac{2}{3}$

→ $\frac{3}{4}$ is $\frac{2}{3}$ part and find the group representing $1$ part

→ $=\frac{9}{8}$

*Making sense of the first principles of division by fraction is one of the exhilarating experiences of learning.*

» **Procedural Simplification**: Division is multiplication by inverse

→ $\frac{3}{4}\xf7\frac{2}{3}$

→ $=\frac{3}{4}\times \frac{3}{2}$

→ $=\frac{9}{8}$

Outline

The outline of material to learn "fractions" is as follows.

• * click here for detailed outline of Fractions *

→ __Part of whole__

→ __Dividing a group__

→ __Fractions as Directed numbers__

→ __Like and Unlike Fractions__

→ __Proper and Improper Fractions__

→ __Equivalent & Simplest form__

→ __Converting unlike and like Fractions__

→ __Simplest form of a Fraction__

→ __Comparing Fractions__

→ __Addition & Subtraction__

→ __Multiplication__

→ __Reciprocal__

→ __Division__

→ __Numerical Expressions with Fractions__

→ __PEMA / BOMA__