Fractions : Division of Fractions

maths > fractions

Division of Fractions

what you'll learn...

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.

$\frac{3}{4} \div \frac{2}{3}$ $3/4 -: 2/3$
$= \frac{3}{4} \times \frac{3}{2}$ $quad = 3/4 xx 3/2$ (multiply by inverse)
$= \frac{9}{8}$ $quad = 9/8$ .

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

$fraction division 6/3$

Division $6 \div 3 = 2$ $6 -: 3 = 2$ is illustrated in the figure.

• $6$ $6$ is the dividend

• $3$ $3$ is the divisor

• $2$ $2$ is the quotient

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

One way to understand the integer division:

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

• in that $3$ $3$ is the divisor

• In the $3$ $3$ 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.

$fraction division 2 by 1/3$

Division $2 \div \frac{1}{3}$ $2 -: 1/3$ is illustrated in the figure.

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

In this, one part $6$ $6$ is the quotient. $2 \div \frac{1}{3} = 6$ $2-:1/3 = 6$

$fraction division 4 by 2/3$

Division $4 \div \frac{2}{3}$ $4 -: 2/3$ is illustrated in the figure.

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

In this, one part $6$ $6$ is the quotient.
$4 \div \frac{2}{3} = 6$ $4-:2/3 = 6$

$fraction division 3/4$

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

$fraction division 3/4 by 2/3$

Division $\frac{3}{4} \div \frac{2}{3}$ $3/4 -: 2/3$ is illustrated in the figure.

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

In this, the place value is $\frac{1}{8}$ $1/8$ and the count is 9.
The same can be simplified as follows.

$\frac{3}{4} \div \frac{2}{3} = \frac{3}{4} \times \frac{3}{2} = \frac{9}{8}$ $3/4-:2/3 =3/4 xx 3/2 = 9/8$

yeah! enough of that

Division is inverse of multiplication.

Division of Fractions: Given two fractions $\frac{p}{q}$ $p/q$ and $\frac{l}{m}$ $l/m$ , the division is
$\frac{p}{q} \div \frac{l}{m}$ $p/q -: l/m$
$= \frac{p}{q} \times \frac{m}{l}$ $quad = p/q xx m/l$
$= \frac{p \times m}{q \times l}$ $quad = (p xx m)/(q xx l)$

example

Divide $\frac{14}{18} \div \frac{4}{3}$ $14/18 -: 4/3$ .
The answer is ' $\frac{7}{12}$ $7/12$ '

summary

» $\frac{a}{b} \div \frac{c}{d}$ $a/b -: c/d$ = $\frac{a}{b} \times \frac{d}{c}$ $a/b xx d/c$

» $6 \div 3$ $6 -: 3$
    → $6$ $6$ is $3$ $3$ parts and find the group representing $1$ $1$ part
    → $= 2$ $= 2$

» $2 \div \frac{1}{3}$ $2 -: 1/3$
    → $2$ $2$ is $\frac{1}{3}$ $1/3$ part and find the group representing $1$ $1$ part
    → $= 6$ $= 6$

» $\frac{3}{4} \div \frac{2}{3}$ $3/4 -: 2/3$
    → $\frac{3}{4}$ $3/4$ is $\frac{2}{3}$ $2/3$ part and find the group representing $1$ $1$ part
    → $= \frac{9}{8}$ $= 9/8$

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
    → $\frac{3}{4} \div \frac{2}{3}$ $3/4 -: 2/3$
    → $= \frac{3}{4} \times \frac{3}{2}$ $= 3/4 xx 3/2$
    → $= \frac{9}{8}$ $= 9/8$

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