How many km does each walk per hour?

Question

In $11$ hours $C$ walks $12.5 \text{ km}$ less than what $D$ does in $12$ hours; and in $5$ hours $D$ walks $3.25 \text{ km}$ less than what $C$ does in $7$ hours. How many $\text{km}$ does each walk per hour?

Attempted solution:

Let speed of $C$ and $D$ be $S_C$ and $S_D$ respectively. Let $D$ travel $x \text{ km}$ in $12$ hours. Let $C$ travel $x \text{ km}$ in $7$ hours.

Accordingly:

$$S_C=\frac{x-12.5}{11}$$

$$S_D=\frac{x}{12}$$

$$S_C=\frac{y}{7}$$

$$S_D=\frac{y-3.25}{5}$$

Fur variables, four unknowns, we should get the answer on solving.

On solving, I get $S_C= 7.07$ and $S_D=4.25$.

However the answer is $S_C= 3.5$ and $S_D=4.25$.

I am aware of other method of solving. But I want to know where I have gone wrong in formulating these equations. Please help.

Answer 1

We know that speed is the ratio of distance and time, that is, distance = speed $\times $ time. Using your notation, we get, $$12S_D -11S_C =12.5$$ $$7S_C -5S_D =3.25$$ Hope you can take it from here by solving these two equations.

Answer 2

Let $v_c$ and $v_d$ denote the velocity of c and d respectively. We can then write:

$\begin{align} v_c &= \frac{12v_d-12.5}{11} \\ v_d &= \frac{7v_c-3.25}{5} \\ \end{align}$

So this gives us a system of two equations with two variables, which implies we can solve the system. The easiest way to to that here is to isolate one of the two and then substitute it in the other equation. For example, we isolate $v_c$:

$v_d = \frac{7v_c-3.25}{5} \implies v_c = \frac{5v_d+3.25}{7}$.

I believe you can proceed yourself from here! :)