A started a business with Rs. $ 2,70,000 $ was joined by B three months afterwards. How much money did B invest if the profit share of A at the end of the year was three fifths of the total profit?

Hint: Here first of all we will assume the profit in the business to be “x” and will frame the equation considering the given in the form of the mathematical expression and then will take the ratios of the profit and considering the contribution of A will find the contribution of B.

Complete step-by-step answer:
Let us assume the overall profit in a business be “x” Rs.
Also given that at the end of the year, profit of A was three-fifth of the total profit $ = \dfrac{3}{5}x $
And therefore, profit for B is $ = x - \dfrac{3}{5}x = \dfrac{{5x - 3x}}{5} = \dfrac{{2x}}{5} $
Now take the ratio of the profit for A and B
 $ \dfrac{A}{B} = \dfrac{{\dfrac{{3x}}{5}}}{{\dfrac{{2x}}{5}}} $
Common factors from the numerator and the denominator cancel each other. Numerator’s denominator goes to the denominator and the denominator’s denominator goes to the numerator.
 $ \dfrac{A}{B} = \dfrac{{3 \times 5}}{{2 \times 5}} $
Again, common factors from the numerator and the denominator cancel each other.
 $ \dfrac{A}{B} = \dfrac{3}{2} $
A invested Rs. $ 2,70,000 $ ,
The contribution of B can be given by $ = \dfrac{2}{3} \times 270000 $
Find the factors for the term in the numerator of the above expression –
 $ = \dfrac{2}{3} \times 3 \times 90000 $
Common multiples from the numerator and the denominator cancel each other and therefore remove from the numerator and the denominator.
 $ = 2 \times 90000 $
Simplify the above expression finding the product of the terms –
 $ = 180000 $ Rs.
Hence, B invested Rs. $ 180000 $
So, the correct answer is “Rs. $ 180000 $ ”.

Note: Always remember that the ratio is the comparison between two like terms and it is always unitless. Always remember the ratio A to B and B to A are two different ratios and are not the same. In ratios, common factors from the numerator and the denominator cancel each other.