mental ability math question

Question

A worker may claim Rs. 15 for each km if he travels by taxi and Rs. 5 for each km if he drives his own car. If in one week he claimed Rs. 500 for travelling 80 km, how many kms did he travel by taxi ?

Answer 1

Assume that this man drove only his car the entire time. That means he can claim Rs. 400. This is 100 less than the 500, so we will need to replace some of these car kilometers with taxi kilometers. Each taxi kilometer gets Rs. 10 more, so we only need 10 taxi kilometers to fill the 100. Therefore, the solutions is 10 taxi kilometers and 70 car kilometers.

Answer 2

Let's say he travelled $x$ kilometres by taxi and hence he must have travelled $80-x$ kilometres by car. Then we have $$500 = 15x + 5(80-x) \iff 100 = 10x \iff x = 10 \mathrm{\,km}.$$

Answer 3

This gives a simple system of linear equations in the kilometres driven per taxi $x_t$ and per car $x_c$: $$ x_t + x_c = 80 \\ 15 x_t + 5 x_c = 500 $$ which can be modeled as $$ A x = b $$ with $$ A = \left( \begin{matrix} 1 & 1 \\ 15 & 5 \end{matrix} \right) \quad b = (80, 500)^T $$ and $$ x = A^{-1} b $$ With $$ A^{-1} = \left( \begin{matrix} -1/2 & 1/10 \\ 3/2 & -1/10 \end{matrix} \right) $$ the solution is $x = (x_t, x_c)^T = (10, 70)^T$.