Winning is Not the Strategy

When Bob completes $\text{1 km}$ , he beats John by $\dfrac 1{\frac 14} \times 25 = 100 \text{ m}$ . Then John completes $1000-100=\text{900 m}$ of the race.

When John completes $\text{900 m}$ , he beats Josh by $\dfrac {\frac 9{10}}{\frac 12} \times 20 = 36 \text{ m}$ . Then Josh completes $900-36=\text{864 m}$ of the race.

Therefore, Bob beats Josh by $1000-864 = \boxed{\text{136 m}}$ .

Bob can beat John by $25\si{\meter}$ in a $250\si{\meter}$ race, so he can beat John by $100\si{\meter}$ in a $1000\si{\meter}$ race. When Bob crosses the finish line, John will be at $900\si{\meter}$ . John can beat Josh by $20\si{\meter}$ in a $500\si{\meter}$ race, so in a $900\si{\meter}$ race, John will be $20\si{\meter} \cdot \frac {900\si{\meter}} {500\si{\meter}} = 36\si{\meter}$ in front. Bob is $100\si{\meter}$ ahead of John who is $36\si{\meter}$ in front of Josh, so Bob beats Josh by $\boxed{136\si{\meter}}$ .

$\begin{aligned} v_{\text{John}} &=\frac{225}{250} v_{\text{Bob}} \\ v_{\text{Josh}} &=\frac{480}{500} v_{\text{John}} \\ &=\frac{864}{1000} v_{\text{Bob}} \\ \end{aligned}$

Therefore, Bob beat Josh by $1000-864=136$ m.

I just approximated .. Bob ends up 100 meters ahead of John by 100m and John is ahead of Josh by 40m at the end. Therefore 100 + 40 = 140m.. the closest answer was 136 meters. Was my line of reasoning correct or am I just a lucky dummy? :)

John's rate is 90% of Bob's rate, and Josh's rate is 96% 0f John's; so josh's rate is .9x.96= .864 = 86.4% of Bob's. When Bob is at 1000, josh will be at 864. Ed Gray

{ 1/2km=500m & 1/4km=250m }

John = [(250-25)/250] *bob

     = (9/10) *bob

Josh =[(500-20)/500] *john

     = (24/25) *john

=> Josh = (108/125) *bob

for 1km race , bob=1000m then-

=> josh = 864 m

So... Bob-Josh = 136m

If Bob is 25m ahead of John after Bob completes 1/4 (or 25%) of the race, then we can multiply by 4 to find that Bob will be 100m ahead of John after Bob completes 100% of the race.

We can use the similar logic to find that John will be 40m ahead of Josh after John completes 100% of the race.

However when Bob completes 100% of the race, John will NOT have completed 100% of the race because he is 100m behind Bob. John will have only completed 90% of the race. We know this because:

1000 - 100 = 900

900 ÷ 1000 = 9/10 = 90%

So the solution to our problem is:

(25m × 4) + ((20m × 2) × 9/10) = 136m

Easy peasy. Bob at 250m when John is at 225m means V-sub-John = 0.9 x V-sub-Bob. John at 500m when Josh is at 480m means V-sub-Josh = 0.98 x V-sub-John. The rates are all constant. V-sub-Josh = 0.9 x 0.98 x V-sub-Bob = 0.864 x V-sub-Bob.

Bob at 1000m puts Josh at 864m, 136m behind.

Speed Ratio = Bob : John = $\frac{1}{4}$ km : $\frac{1}{4}$ km - 25m = 250m : 225m = 10 : 9 = 250 : 225

Speed Ratio = John : Josh = $\frac{1}{2}$ km : $\frac{1}{2}$ km -20m = 500m : 480m = 25 : 24 = 225 : 216

Distance Ratio = Bob : John : Josh = 250 : 225 : 216 = 1000m : 900m : 864m

1000m - 864m = 136m

*Take note: Speed ratio = Distance Ratio

Bob speed 10% more than Jhon and Jhon speed 4% more than Josh. so at last 100+36 = 136m.

Common sense solution for those, who are lazy writing down equations. When Bob finishes, he's 4x25m ahead of John. That's 100m. When John finishes, he's 2x20m ahead of Josh. That's 40m. As John is 100m behind Bob, he has only run 900m of the km by the time Bob finishes. That's why he'll only be 9/10 (900/1000) of the final distance ahead of Josh. 40x(9/10)=36. When bob finishes, he'll be 100m ahead of John who will, at that timepoint, be 36m ahead of Josh. Final result - 136m.