overview

In this page, multiplcation of fractions is explained.

• multiplication in first principles -- repeatedly combining a quantity and measuring the combined and

• simplified procedure : Multiplying numerators and denominators

repeated times

Multiplication of integers or whole numbers: $4\times 3=12$

• $4$ is the multiplicand

• $3$ is the multiplier

• $12$ is the product

Multiplication is repeating the multiplicand the multiplier times -- to get the product.

$4$ repeated $3$ times =

$4+4+4=12$

Multiplication by a fraction : Consider $1\times \frac{1}{4}$

Multiplicand $1$ is multiplied by a multiplier $\frac{1}{4}$.

The multiplication $1\times \frac{1}{4}$ is illustrated.

The multiplicand is divided into $4$ parts (which is the place value of the multiplier) and product is calculated by selecting $1$ part (which is the numerator of the multiplier). The product is $\frac{1}{4}$.

Multiplication by a fraction : Consider $2\times \frac{1}{4}$

Multiplicand $2$ is multiplied by a multiplier $\frac{1}{4}$.

The multiplication process is illustrated in the next page.

The multiplication $2\times \frac{1}{4}$ is illustrated.

The multiplicand is divided into $4$ parts (which is the place value of the multiplier) and product is calculated by selecting $1$ part from each (which is the numerator of the multiplier).

The value of one piece with respect to a whole is $\frac{1}{4}$ and $2$ pieces are taken. That is, $2\phantom{\rule{1ex}{0ex}}\text{pieces of}\phantom{\rule{1ex}{0ex}}\frac{1}{4}=\frac{2}{4}$.

Multiplication by a fraction : Consider $1\times \frac{3}{4}$

Multiplicand $1$ is multiplied by a multiplier $\frac{3}{4}$.

The multiplication process is illustrated in the next page.

The multiplication $1\times \frac{3}{4}$ is illustrated.

The multiplicand is divided into $4$ parts (which is the place value of the multiplier) and product is found by selecting $3$ parts (which is the numerator of the multiplier).

The product is $3$ pieces in the place value $\frac{1}{4}$ which is $\frac{3}{4}$.

fraction by fraction

Multiplication by a fraction : consider $\frac{2}{3}\times \frac{1}{4}$

Multiplicand $\frac{2}{3}$ is multiplied by a multiplier $\frac{1}{4}$.

The multiplication process is illustrated in the next page.

The multiplication $\frac{2}{3}\times \frac{1}{4}$ is illustrated.

Each part of multiplicand is divided into $4$ parts (which is the place value of the multiplier) and product is found by selecting $1$ part (which is the numerator of the multiplier) from each.

The product is $2$ pieces in the place value $\frac{1}{12}$ which is $\frac{2}{12}$.

$\frac{2}{12}$ can be simplified to $\frac{1}{6}$.

Multiplication by a fraction : Consider $\frac{2}{3}\times \frac{3}{4}$

Multiplicand $\frac{2}{3}$ is multiplied by a multiplier $\frac{3}{4}$.

The multiplication process is illustrated in the next page.

The multiplication $\frac{2}{3}\times \frac{3}{4}$ is illustrated.

The each part of multiplicand is divided into $4$ parts (which is the place value of the multiplier) and product is found by selecting $3$ parts (which is the numerator of the multiplier) from each.

The product is $6$ pieces in the place value $\frac{1}{12}$ which is $\frac{6}{12}$.

$\frac{6}{12}$ can be simplified to $\frac{1}{2}$.

simplify

Having understood the first principles of multiplication of fractions, the procedural simplification for the same is :

• Numerators of multiplicand and multiplier are multiplied to numerator of product.

• Denominators of multiplicand and multiplier are multiplied to denominator of product.

Multiplication is understood in two steps

• The place value of multiplicand is modified by the place value of multiplier.

• The given number of multiplicand is multiplied by the given number of multiplier.

**Multiplication of Fractions: ** For two fractions multiplicand $\frac{p}{q}$ and the multiplier $\frac{l}{m}$

Every $\frac{1}{q}$ part of multiplicand is split into $m$ pieces making it $\frac{1}{q\times m}$ as the modified place value

)
From each of $p$ parts of multiplicand $l$ parts make the product making $p\times l$ parts

$p\times l$ parts in $\frac{1}{q\times m}$ place value gives the result $\frac{p\times l}{q\times m}$

example

Multiply $\frac{14}{18}\times \frac{3}{4}$.

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

summary

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

» $\textcolor[rgb]{}{2}\times \textcolor[rgb]{}{\frac{1}{4}}$

→ $2$ pieces

→ $\frac{1}{4}$ of each of those pieces

→ result is count $2$ in place value$\frac{1}{4}$

→ $=\frac{2}{4}$

» $\textcolor[rgb]{}{\frac{2}{3}}\times \textcolor[rgb]{}{\frac{3}{4}}$

→ $\textcolor[rgb]{}{2}$ pieces in place value $\textcolor[rgb]{}{\frac{1}{3}}$

→ $\textcolor[rgb]{}{\frac{3}{4}}$ of each of those pieces

→ result is count $6$ in place value $12$

→ $=\frac{6}{12}=\frac{1}{2}$

» **Procedural Simplification**: Multiply numerators and denominators. Cancel any common factors .

→ $\frac{2}{3}\times \frac{3}{4}$

→ $\frac{2\times 3}{3\times 4}$

→ common factors $3$ and $2$ canceled

→ $=\frac{1}{2}$

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__