Math Doubts

Proof of Cross multiplication

Let $a, b, c$ and $d$ represent four quantities in algebraic form and they form two rational expressions $a$ divided by $b$ and $c$ divided by $d$. Let us take the quotients of them are equal and it is expressed in the following mathematical form.

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

It can be solved by the cross-multiplying method of the fractions or rational numbers. Let’s learn how to prove the cross multiply method in mathematics.

Release the expression from fraction form

Look at the fraction at the left-hand side of the equation. The rational expression can be released from the fraction form by multiplying it with the quantity in the denominator. It is not appropriate to include the quantity $b$ on one side of the equation for maintaining the equilibrium of the equation but it is acceptable to include a quantity on both sides of the equation appropriately.

Therefore, multiply the rational expression on the left-hand side of the equation by the literal $b$ and also multiply the fraction on the right-hand side of the equation by the same literal.

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

Now, split the fraction in the first factor of the left-hand side of the equation for releasing the rational expression from the fraction form.

$\implies$ $\dfrac{a \times 1}{b} \times b$ $\,=\,$ $\dfrac{c}{d} \times b$

$\implies$ $a \times \dfrac{1}{b} \times b$ $\,=\,$ $\dfrac{c}{d} \times b$

Now, multiply the second and third factors on the left-hand side of the equation.

$\implies$ $a \times \dfrac{1 \times b}{b}$ $\,=\,$ $\dfrac{c}{d} \times b$

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

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

$\implies$ $a \times 1$ $\,=\,$ $\dfrac{c}{d} \times b$

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

Release other side expression from fraction form

In the above step, the rational expression in the left-hand side of the equation is successfully released from the fraction form. Similarly, it is time to release the rational expression on the right-hand side of the equation. It can be done by multiplying the expressions on both sides with the quantity $d$.

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

The product of the second and third factors on the right-hand side of the equation can be written as follows as per the commutative property of multiplication.

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

Now, repeat same procedure to simply the mathematical expression on the right-hand side of the equation.

$\implies$ $a \times d$ $\,=\,$ $\dfrac{c \times 1}{d} \times d \times b$

$\implies$ $a \times d$ $\,=\,$ $c \times \dfrac{1}{d} \times d \times b$

$\implies$ $a \times d$ $\,=\,$ $c \times \dfrac{1 \times d}{d} \times b$

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

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

$\implies$ $a \times d$ $\,=\,$ $c \times 1 \times b$

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

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

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