Question

A person on tour has rupees 4200 for his expenses. If he extends his tour for 3days, he has to cut down daily expenses by rupees 70. Find the original duration of the tour.

Answer 1

Hint – In this question let the total number of days of the tour be x days, and let the daily expenses be y rupees. Use the concept that the number of days multiplied by daily expenses is equal to the total rupees, and then use the constraint when the tour is extended this means that now the number of days in the tour become (x + 3), and the daily expenses become (y – 70). This will help formulation of equations, solve them to get the answer.
Complete step-by-step answer:
Total rupees a person has = 4200.
Let the number of days of the tour = x
And let the daily expenses = y Rs.
Therefore, number of days multiplied by daily expenses is equal to the total rupees so we have,
$ \Rightarrow xy = 4200$................. (1)
Now if he extends the tour by three days he has to cut the daily expenses by rupees 70.
So the number of days in the tour become = (x + 3)
And the daily expenses become = (y – 70) Rs.
$ \Rightarrow \left( {x + 3} \right)\left( {y - 70} \right) = 4200$
Now simplify this equation we have,
$ \Rightarrow \left( {xy - 70x + 3y - 210} \right) = 4200$
Now substitute the value from equation (1) in above equation we have,
$ \Rightarrow \left( {4200 - 70x + 3y - 210} \right) = 4200$
\[ \Rightarrow 3y - 70x = 210\]
Now from equation (1) we have,
$ \Rightarrow y = \dfrac{{4200}}{x}$ so substitute this value in above equation we have,
\[ \Rightarrow 3 \times \dfrac{{4200}}{x} - 70x = 210\]
Now divide by 70 throughout we have,
\[ \Rightarrow 3 \times \dfrac{{60}}{x} - x = 3\]
\[ \Rightarrow 180 - {x^2} = 3x\]
\[ \Rightarrow {x^2} + 3x - 180 = 0\]
Now factorize the above equation we have,
$ \Rightarrow {x^2} + 15x - 12x - 180 = 0$
$ \Rightarrow x\left( {x + 15} \right) - 12\left( {x + 15} \right) = 0$
$ \Rightarrow \left( {x + 15} \right)\left( {x - 12} \right) = 0$
$ \Rightarrow x = - 15,12$
So negative value is not possible.
So the number of days of the tour = 12.
So this is the required answer.

Note – The quadratic equation formed can easily be solved by another method of Dharacharya form instead of simple factorization. In Dharacharya formula the quadratic equations of the form $a{x^2} + bx + c = 0$, has roots $x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$. This helps solving quadratic equations in very less time.