Question

Ashwin working in a bank can claim $Rs.15$ for each km which he travels by taxi and $Rs.5$ for each km when he goes in his own car. If Ashwin claims $Rs.500$ in one week for traveling $80km$, how many km did he travel by taxi?

Answer 1

Hint:
This is related to solving systems of equations in two variables and trying to get a common solution to the equations. In certain problems where we deal with 2 variables or unknowns for finding their solutions comes under the Linear equation in 2 variables.
After knowing the given conditions that are also called constraints we frame the several equations (for example) \[ax + by = c\] & \[dx + ey = f\], where x and y are variables and rest are constraints.
As we have several methods such as the Substitution Method, Cross-multiplication Method, Elimination Methods, etcetera. by which we can arrive at the required solution.
Substitution Method: In this method, we mainly value any one variable from any equation and put it to the other equation so that the Linear equation in 2 variables is converted to Linear equation in one variable and we get the solution.
Elimination Method: In this method, we mainly try to eliminate any one of the variables so that we are left only with one variable and lastly we solve that to get the solution.
Complete step by step solution-
Given:
Ashwin can travel both by taxi and in his own car to his bank.
For each km, if he travels by taxi he could claim amount from the bank \[ = {\text{ }}Rs.{\text{ }}15\;\]and,
For each km, if he travels by his own car he could claim amount from the bank \[ = {\text{ }}Rs.{\text{ }}5\]
On a week total distance travelled by him \[ = 80{\text{ }}km\]
And total money claimed by him \[ = {\text{ }}Rs.500\]
Let the distance travelled by him by taxi be x km
And the distance travelled by him by his car be y km
So it can be said that,
\[x + y = 80\] …………equation (1)
And,\[15x + 5y = 500\] …………equation (2)
Let us solve it by Substitution Method:
From equation(1),\[y = 80 - x\]
This value of y we put it to the equation (2) and we get
\[15x + 5\left( {80 - x} \right){\text{ }} = {\text{ }}500\]
\[15x + 400 - 5x = 500\]
\[10x = 500 - 400\]
\[10x = 100\]
So, $x = \dfrac{{100}}{{10}} = 10$
Hence, the distance travelled by taxi \[ = {\text{ }}{\mathbf{10}}{\text{ }}{\mathbf{km}}\]

Note: It can also be solved by cross multiplication that is,
\[x + y - 80 = 0\]
\[15x + 5y - 500 = 0\] …. from equation1 and 2
$$ $$ $\dfrac{x}{{ - 400 + 500}} = \dfrac{y}{{ - 500 + 1200}} = \dfrac{1}{{15 - 5}}$
$x = \dfrac{{100}}{{10}} = 10$ and, $y = \dfrac{{700}}{{10}} = 70$