Question

Find the third proportional to Rs 2.40, Rs 4.80

Answer 1

Here, we will proceed by firstly reducing the ratio of the given numbers (i.e., 2.40 and 4.80) into simplest ratio and then we will use the formula i.e., if $\dfrac{a}{b} = \dfrac{c}{d}$, then the third proportional to b, a is given by $x = b \times \left( {{c^2}} \right)$

Complete step-by-step answer :
The equality of any two ratios is defined as proportion.
Since, for any four numbers i.e., a, b, c and d if a:b :: c:d or $\dfrac{a}{b} = \dfrac{c}{d}$, then the number d is called as the fourth proportional to the numbers a, b and c.
For example, if 2:4 :: 5:3 or $\dfrac{2}{4} = \dfrac{5}{3}$, then 3 is the fourth proportional to 2,4 and 5.
Similarly, for any four numbers i.e., a, b, c and d if a:b :: c:d or $\dfrac{a}{b} = \dfrac{c}{d}$, then the number c is called as the third proportional to the numbers a and b.
For example, if 2:4 :: 5:3 or $\dfrac{2}{4} = \dfrac{5}{3}$, then 5 is the third proportional to 2 and 4.
In a continued proportion, the third proportional is the variable in a proportion with two equal terms. The third proportional of any proportion is simply defined as the second term of the mean terms.
A third proportional is equal to the square of the equal terms, divided by the unequal term.
If $\dfrac{a}{b} = \dfrac{c}{d}$, then the third proportional (denoted by x) to b, a is given by
$x = b \times \left( {{c^2}} \right)$
As, we have to find the third proportional to Rs 2.40, Rs 4.80
Here, $\dfrac{{4.80}}{{2.40}} = \dfrac{2}{1}$ or 4.80:2.40 :: 2:1
So, the required third proportional = $2.40 \times {\left( 2 \right)^2} = 2.40 \times 4 = 9.60$
Therefore, the third proportional to Rs 2.40, Rs 4.80 is Rs 9.60.

Note :Since, we can see from the above solution that the third proportional is a special case of the fourth proportional. There is also one more term known as proportional mean which can be calculated by exacting the square root from the product of the extremes. For, example for the proportion $\dfrac{2}{x} = \dfrac{x}{8}$, the proportional mean is given by $x = \pm \sqrt {16} = \pm 4$.