# Cross multiplication

## Formula

$\dfrac{a}{b}$ $\,=\,$ $\dfrac{c}{d}$ $\,\,\,\Longleftrightarrow\,\,\,$ $a \times d$ $\,=\,$ $b \times c$

An equality rule in multiplying the quantities crossly to solve an equation that consists of fractions on both sides is called the cross multiplication.

### Introduction

The expressions on both sides of an equation are fraction in some cases.

$\dfrac{a}{b}$ $\,=\,$ $\dfrac{c}{d}$

The expressions in rational form on both sides of an equation create a problem while solving it. A special rule is required for overcoming it. However, there is an arithmetic property for solving the equations in which the rational expressions are there on both sides.

The products of the numerator of one side and the denominator of other side are equal when two fractions (or rational expressions) form an equation.

$\implies$ $a \times d$ $\,=\,$ $b \times c$

The equality property of quantities in product form is called the cross-multiplying method.

### Example

$\dfrac{2}{5}$ $\,=\,$ $\dfrac{6}{15}$

$2$ divided by $5$ and $6$ divided by $15$ are the rational numbers, which are the expressions on two sides of the equation. They both are equivalent fractions.

The numerator of left-hand side of the equation is $2$ and denominator of the right-hand side of the equation is $15$. Now, find the product of them.

$\implies$ $2 \times 15$ $\,=\,$ $30$

The numerator of right-hand side of the equation is $6$ and the denominator of the left-hand side of the equation is $5$. Now, calculate the product of them.

$\implies$ $6 \times 5$ $\,=\,$ $30$

It clears that

$\,\,\,\therefore\,\,\,\,\,\,$ $2 \times 15$ $\,=\,$ $6 \times 5$ $\,=\,$ $30$

#### Problems

Solve $\dfrac{2x-3}{4x+5}$ $\,=\,$ $\dfrac{1}{3}$

Learn how to use the cross multiplication to solve the fraction or rational expression basis equations.

#### Proof

Learn how to derive the cross multiplying system to solve the equation in which two fractions are involved.

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Jun 26, 2023

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