Question

If $\sqrt {0.04 \times 0.4 \times a} = 0.4 \times 0.04 \times \sqrt b $, then find the ratio $\dfrac{a}{b}$.

Answer 1

Hint: Here, we will first square both sides of the given equations. Then we will simplify the equation in such a way that the equation comes in the required ratio form. We will simplify the equation further to get the required value. Ratio is defined as a comparison of two or more numbers that indicates their sizes relation to each other.

Complete step-by-step answer:
We are given that $\sqrt {0.04 \times 0.4 \times a} = 0.4 \times 0.04 \times \sqrt b $
By squaring on both the sides, we get
$ \Rightarrow {\left( {\sqrt {0.04 \times 0.4 \times a} } \right)^2} = {\left( {0.4 \times 0.04 \times \sqrt b } \right)^2}$
We know that the square of a square root of a number is the same number without any orders.
By simplifying the equation, we get
$ \Rightarrow 0.04 \times 0.4 \times a = {\left( {0.4} \right)^2} \times {\left( {0.04} \right)^2} \times {\left( {\sqrt b } \right)^2}$
We know that the square of a square root of a number is the same number without any orders.
By simplifying the equation, we get
$ \Rightarrow 0.04 \times 0.4 \times a = {\left( {0.4} \right)^2} \times {\left( {0.04} \right)^2} \times b$
By rewriting the equation, we get
$ \Rightarrow \dfrac{a}{b} = \dfrac{{{{\left( {0.4} \right)}^2} \times {{\left( {0.04} \right)}^2}}}{{0.4 \times 0.04}}$
By cancelling the same terms, we get
$ \Rightarrow \dfrac{a}{b} = \left( {0.04} \right) \times \left( {0.4} \right)$
Multiplying the terms, we get
$ \Rightarrow \dfrac{a}{b} = 0.16$
Therefore, the ratio of $\dfrac{a}{b}$is $0.16$ or $\dfrac{4}{{25}}$ .

Note: Ratio can be expressed in the form of fractions or decimal numbers. The ratio is the number which can be expressing the one quantity as the fraction of the other ones. Ratios should be represented in its simplest form. The ratio should always exist between the units of the same kind. While comparing two things, the units should be similar to both the quantities. The comparison of ratios can be performed only if the ratios are equivalent like the fractions. When two ratios are equivalent to one another, then the ratio is said to be Proportion. The first term is called an antecedent and the second term is a consequent.