Singapore Elementary 6 Math Question on Distance Town A and Town B are 243km apart.
At 0800, Max left Town A and cycled towards Town B. Jane left Town A and cycled towards Town B at 0830.
Max's cycling speed was 6km/h slower than Jane. At 1100, Jane caught up with Max.
When Jane was 12km from Town B, she stopped to wait for Max. How far was Max from Jane when she stopped to wait for him?
 A: Let $v_m$ and $v_j$ be Max's and Jane's velocities respectively.
$$\implies v_m = v_j - 6 \implies v_j - v_m = 6\quad\dots(1) $$
At 11:00, both have covered an equal distance $x$,
$$\implies v_m = \frac{x}{8 - 11};\quad v_j = \frac{x}{8.5 - 11}\\
\implies x = 3v_m = 2.5v_j\\
\implies 3v_m - 2.5v_j = 0\quad\dots(2) $$
Now, adding $(1)\times 3$ and $(2)$,
$$\implies 3v_j - 3v_m +3v_m -2.5v_j= 18
\\\implies 0.5v_j = 18 \\
\implies v_j = 36$$
Putting $v_j =36$ in $(1)$,
$$\implies v_m = 36 - 6 = 30$$
When Jane stops, she has covered $d = (243 - 12) \text{km}$ from Town $A$. The time she takes to do so can be found as follows:
$$v_j = \frac{d_j}{t} \implies t = \frac{231}{36} \approx 6.4167$$
We can find the distance covered by Max in the same time:
$$v_m = \frac{d_m}{t} \implies d_m = 30\times \frac{231}{36} = 192.5 $$
Now, let the distance between Max and Jane at this time be $\Delta d$,
$$\implies \Delta d = |d_m - d_j| = |192.5 - 231| = 38.5$$
$\therefore$ Max was $38.5 \text{ km}$ away from Jane when she stopped for him at 2:55 pm
A: Okay, so let's begin.
First, let's call Jane's speed $r$, and Max's speed $r -6$.
Since they both travel the same distance at 11:00,
$$d = r*2.5$$
And
$$d = (r-6)*3$$
Set them equal to one another:
$$2.5r = 3r - 18$$
$$0.5r = 18$$
$$r = \frac{36 \text{km}}{\text{hr}}$$
And Max's speed is:
$$36 - 6 = \frac{30 \text{km}}{\text{hr}}$$
Now when Jane is $12 km$ away from Town B, she has traveled $243-12 = 231$ km.
So we can find the time it took:
$$231 = 36*t, t= 6.42 \text{hrs}$$
We can use this to find the distance that Max traveled:
$$d = 30*(6.42), d = 192.6 \text{km}$$
Now we simply subtract the distances:
$$231 - 192.6 = 38.4 \text{km}$$
comment if you have any questions.
