Question

If $\left( {\dfrac{1}{x}} \right):\left( {\dfrac{1}{y}} \right):\left( {\dfrac{1}{z}} \right) = 2:3:5$, then $x:y:z = ?$

Answer 1

Hint: In order to solve the given question , we should know the important concepts related to the question that is Ratio and Proportion . We can say that the comparison or simplified form of two quantities of the same kind is referred to as ratio. This relation gives us how many times one quantity is equal to the other quantity. The sign used to denote a ratio is ‘:’. Also the Proportion is an equation which defines that the two given ratios are equivalent to each other. In other words, the proportion states the equality of the two fractions or the ratios. When two ratios are equal in value, then they are said to be in proportion. In simple words, it compares two ratios. Proportions are denoted by the symbol ‘::’ or ‘=’. Using these concepts we can get our required solution .

Complete step by step solution:
We are given in the question that $\left( {\dfrac{1}{x}} \right):\left( {\dfrac{1}{y}} \right):\left( {\dfrac{1}{z}} \right) = 2:3:5$ and we have to calculate $x:y:z$ .
So, we will consider the first two ratios $\left( {\dfrac{1}{x}} \right):\left( {\dfrac{1}{y}} \right) = 2:3$
Thereafter, we will perform cross multiplication –
$\left( {\dfrac{y}{x}} \right) = \dfrac{2}{3}$
Now we will generate a value for one variable , let suppose here for ‘y’ –
 $y = \dfrac{2}{3}x$
Similarly , we will do the same steps for the next two ratios – $\left( {\dfrac{1}{y}} \right):\left( {\dfrac{1}{z}} \right) = 3:5$
Thereafter, we will perform cross multiplication –
$\left( {\dfrac{z}{y}} \right) = \dfrac{3}{5}$
Now we will generate a value for one variable , let suppose here for ‘y’ –
 $z = \dfrac{3}{5}y$
Now , we will put the individual values if ‘y’ and ‘z’ in this equation $x:y:z$
$
   \Rightarrow x:y:z \\
   \Rightarrow x:\dfrac{2}{3}x:\dfrac{3}{5}y \\
 $
Now as we have value of y in terms of x we will again substitute the value of y , we get-
$
   \Rightarrow x:\dfrac{2}{3}x:\dfrac{3}{5}y \\
   \Rightarrow x:\dfrac{2}{3}x:\dfrac{3}{5} \times \dfrac{2}{3}x \\
 $
We will cancel out the common factors in order to simplify it .
 $ \Rightarrow x:\dfrac{2}{3}x:\dfrac{2}{5}x$
Now take that value of x that is divisible by both denominators 3 and 5 .
So , we take $x = 15$
Substituting the value of x in $ \Rightarrow x:\dfrac{2}{3}x:\dfrac{2}{5}x$, we get-
\[
   \Rightarrow 15:\dfrac{2}{3} \times 15:\dfrac{2}{5} \times 15 \\
   \Rightarrow 15:2 \times 5:2 \times 3 \\
   \Rightarrow 15:10:6 \\
 \]
Therefore, the required ratio is \[x:y:z = 15:10:6\].

Note:
> The ratio should exist between the quantities of the same kind.
> Ratio and Proportion are explained majorly based on fractions. When a fraction is represented in the form of a:b, then it is a ratio whereas a proportion states that two ratios are equal.
> While comparing two things, the units should be similar.
> There should be a significant order of terms.
> The comparison of two ratios can be performed, if the ratios are equivalent like the fractions.