Question

A certain amount of money has to be divided between two persons P and Q in the ratio 3 : 5. But it was divided in the ratio of 2 : 3 and thereby Q loses Rs.10. What was the amount?
(a) Rs.250
(b) Rs.300
(c) Rs.350
(d) Rs.400

Answer 1

Hint: Take ‘x’ as the amount to be divided. Find the amount ratio to be divided in ratio 3 : 5. For P, we can take it as \[\dfrac{3x}{\left( 3+5 \right)}=\dfrac{3x}{8}\]. We will do the same for Q. Now find the dividend amount with ratio 2 : 3. Now subtract the ratio of Q from both the cases and equate to the loss of Q i.e. Rs.10. Solve it and get the value of x.

Complete step-by-step answer:
It is said that there are 2 persons P and Q. The money was to be divided in the ratio 3 : 5 but it was divided as 2 : 3.
Let us consider ‘x’ as the amount to be divided between P and Q is in the ratio 3 : 5 which becomes,
\[\dfrac{3x}{3+5}\] for person P and \[\dfrac{5x}{3+5}\] for person Q.
\[\therefore \] Original amount to be divided = \[\dfrac{3x}{8}\left( P \right)\] and \[\dfrac{5x}{8}\left( Q \right)\]- (1)
Now it is said that the amount actually was divided in 2 : 3.
\[\therefore \]Actual divided amount = \[\dfrac{2x}{2+3}\] for person P and \[\dfrac{3x}{2+3}\] for Q.
\[\therefore \]Actual divided amount = \[\dfrac{2x}{5}\left( P \right)\] and \[\dfrac{3x}{5}\left( Q \right)\] - (2)
From (1) and (2) the loss encountered by person Q = \[\dfrac{5x}{8}-\dfrac{3x}{5}\].
From the question we have been told that the person Q encountered a loss of Rs.10. Hence equating loss of Q as,
\[\Rightarrow \dfrac{5x}{8}-\dfrac{3x}{5}=10\]
Let us simplify the above expression and find the value of x.
\[\Rightarrow \dfrac{25x-24x}{40}=10\]
Apply cross multiplication property.
\[\therefore x=10\times 40=400\]
Thus the amount to be divided between P and Q was Rs.400.
\[\therefore \] Option (d) is the correct answer.

Note: It is said that Q losses Rs.10, thus subtract the ratio of Q. If you subtract taking the ratios of P, i.e. \[\dfrac{3x}{8}\] and \[\dfrac{2x}{5}\], then you might get the wrong amount as you find answer. Once students form the equation as \[\Rightarrow \dfrac{5x}{8}-\dfrac{3x}{5}=10\], then they can even substitute the options as x and check which one satisfies the equation. But, this is just a waste of time because here option (d) is the right answer and it would have taken four trials to get to the final answer.