Justin and Joe started driving eastwards from point A and point B resp

can someone explain why the answer is D) and not C), and how they solved it?

Justin starts at point A and stops after every 12km. So we can say the points at which he is stopping are 12,24,36,48,60,72,84,96,108,120,....

Joe starts at 8km east of point A that is he is already 8km ahead of Justin. Joe stops after every 10km. So the points at which he stops are 8+10=18, 18+10=28, then 38,48,58,68,78,88,98,108,118,128,....

Now, if we compare the stopping points for both Justin and Joe, we can see the common distances at which they stop are 48,108,...

This is in Arithmetic Progression where the first term is 48 and the common difference is 108-48=60.

So we can find other common points which are 108+60=168, 168+60=228. Within 230km, the points at which they both stop are thus 48,108,168,228 so 4 points. Option (D) is correct.

I hope this helps.

Another way to solve this:

Justin starts at A which is 8km behind B
Joe starts at B

Justin stops after every 12km and Joe stops after every 10 km
This means they'll stop after every LCM(12,10) km = 60km
However, the first time they stop at the same distance will be different because of the 8km gap
So all we need to calculate is the first time they'll stop together due to the 8km gap

Take point B as reference, let this be the zero of the number line
Justin is at -8
Joe at 0

- Joe reaches 10 (0+10), Justin reaches 4 (-8+12)
- Next, Joe reaches 20, Justin reaches 16
- Next, Joe reaches 30, Justin reaches 28
- Next Joe reaches 40, Justin also reaches 40 (Bingo!)

So they first meet at 40km from point B

Now, they'll meet after every 60km (remember, our reference is point B, not A)

So now they'll meet at :
- First 40 km from B
- Then 100 km from B (now after every 60km)
- Then 160 km from B
- Then 220 km from B
And we stop here, because the next point will be greater than 230 km from A (constraint given in the STEM: "what was the number of points within a distance of 230 kilometres east of point A at which both Justin and Joe stopped for a break?")

Total = 4 such points (D)

You can also solve this by taking point A as reference, in that case the first time they meet will come out to be 48 instead and every other point will also have a 8km increase

Justin and Joe started driving eastwards from point A and point B respectively, where point B was 8 kilometres east of point A.

If Justin stopped for a break after every 12 kilometres and Joe stopped for a break after every 10 kilometres, what was the number of points within a distance of 230 kilometres east of point A at which both Justin and Joe stopped for a break?

Let us take A as 0 km point
B is at 8 km east of point A.

Stopping points for Justin = {12, 24, 36, 48, 60, 72, 84, 96, 108, 120, 132, 144, 156, 168, 180, 192, 204, 216, 228}
Stopping points for Joe = {18, 28,38,48, 58, 68, 78, 88, 98, 108, 118, 128, 138, 148, 158, 168, 178, 188, 198, 208, 218, 228}

Common stopping points for Justin and Joe = {48, 108, 168, 228} : 4 points at which both Justin and Joe stopped for a break

IMO D