Question

How could you use cross products to check the solution to this proportion $\dfrac{12}{30}=\dfrac{5}{x}$ ?

Answer 1

Hint: To find the solution of $\dfrac{12}{30}=\dfrac{5}{x}$ , we will use cross multiplication, also known as Cross product. An equation of the form $\dfrac{a}{b}=\dfrac{x}{d}$ can be solved by multiplying the numerators on both sides by the denominators of the opposite sides. Hence, we can write this equation as $ad=bx$ . Similarly, we can solve the given equation.

Complete step by step solution:
We have to use cross products to check the solution to this proportion $\dfrac{12}{30}=\dfrac{5}{x}$ . Let us see what cross product is. Cross product is also known as cross multiplication which is usually done when we want to solve equations which are not in linear form.
Let us consider an equation of the form $\dfrac{a}{b}=\dfrac{x}{d}$ . We have to find the value of x. Let us do cross multiplication. For this, we will multiply the numerators on both sides by the denominators of the opposite sides. Hence, we can write this equation as
$ad=bx$
Similarly, we can solve the given expression. Let us apply cross multiplication on the given expression.
$\Rightarrow 12x=5\times 30$
Let us solve the RHS by multiplying 5 and 30.
$\Rightarrow 12x=150$
Now, we can find the value of x by taking 12 to RHS.
$\Rightarrow x=\dfrac{150}{12}$
Let us cancel the common factor (2) from numerator and denominator.
$\Rightarrow x=\dfrac{75}{6}$
We can further solve by cancelling the common factor 3 from the numerator and denominator.
$\Rightarrow x=\dfrac{25}{2}$

Hence, the answer is $\dfrac{25}{2}$

Note: Students must know what a linear equation is to solve these kinds of problems A linear equation is represented as $ax+b=c$ , where a, b and c are constants. We can solve a linear equation faster. This is why we used the cross product in the given question so that it becomes in the form of a linear equation. We can check whether the answer obtained is correct or not. We are given that $\dfrac{12}{30}=\dfrac{5}{x}$ . So, let us substitute the value of x in this equation and see whether LHS is equal to RHS. Let us consider the LHS. LHS can be further simplified by cancelling the common terms first by 3 and then by 2.
$\Rightarrow \dfrac{12}{30}=\dfrac{4}{10}=\dfrac{2}{5}$
Let us consider the RHS.
\[\dfrac{5}{x}=\dfrac{5}{\dfrac{25}{2}}=5\times \dfrac{2}{25}=\dfrac{2}{5}\]
Hence, LHS=RHS.