A's speed is 1 ³? ₈ times of B's. In a race A gives B a start of 300 m. How long should the race course be so that both reach the winning post simultaneously?

Time taken by A to run 1 Km = 30 × 2 = 60 seconds.
Time taken by B to run 1 Km = 35 × 2 = 70 seconds.
? A can give B a start of (70 - 60) = 10 seconds.
In 35 sec B runs 500 m
? In 10 sec B runs = 500 × 10 / 35 = 1000 / 7 = 142⁶?₇ meters
So, A can give B a start of 142⁶?₇ meters in a Km race.

B covers 200 m while A 190 m
B covers 180 m while A 190 x 180 / 200 = 171 m
Hence, C covers 200 m while A = 171 m
So, C can give to A (200 ? 171) = 29 m start.

In a game of 50, If A scores 50 then B scores 44.
? A scores 65 while B 44 * 65 / 50 = 57.2 points
Now, B scores 57.2 points while C 52
? B scores 55 points while C = 52 * 55 / 57.2 = 50
Hence, in game of 55 points B can give C (55 ? 50) = 5 points

? A runs 100 m while B runs 95 m
? A runs 400 m while B runs 95 x 400 / 100 = 380 m

Again, B runs 200 m while C runs 185 m
? B runs 380 m while C runs 185 x 200 / 380 = 3515 / 10 = 351.5 m

Hence, A runs 400 m while C runs 351.5 m
So, A can beat C by (400 ? 351.5) = 48.5 m

Time taken by A in one round = 3000 / 200 = 15 min.
Time taken by B in one round = 3000 / 150 = 20 min
L.C.M. of 15 min and 20 min = 60 min = 1 hr.
Hence, they will meet first together at the starting point after 1 hour.

2nd Method to solve this
Let us assume after T minutes both A and B will meet first time together at starting points.
According to question,
Distance covered by A in T minutes = 200 x T = 200T
Distance covered by B in T minutes = 150 x T = 150T
200T - 150T = 3000 As we know (3 KM = 3000 meters)
? 50T = 3000
? 5T = 300
? T = 300/5 = 60 Minutes.
As we know ( 60 minutes = 1 hours)
? T = 1 Hours

While A runs 1000 m, B runs 1000 - 80 = 920 m and
While B runs 1000 m, C runs 1000 - 60 =940 m.
? While B runs 920 m; C runs
940 / 1000 × 920 = 4324 / 5 m
? While B runs 920 m; C runs 940 × 920 / 1000 = 4324 / 5 m
? While A runs 1000 m, C runs 4324 / 5 m
? A can beat C by 1000 - 4324 / 5 = 676 / 5 = 135¹?₅ m.

When A makes 5 rounds, B makes 4 rounds.
In order to pass each other, the difference in number of rounds made by each must be one.
Here, A passes B each time, when A makes 5 rounds.
Distance covered by A in 5 rounds = 5 * 400 / 100 × = 2 Km.
In covering 2 Km, A passes B 1 time.
? In covering 5 Km, A passes B
= 5 / 2 = 2¹?₂ times.

Let, times (in sec) taken by A and B to run a Km, be P and Q, respectively.
When B gets a start of 50 m, B runs.
1000 - 50 = 950 m while A runs 1000 m.
? 950 / 100 Q - P = 14,
i.e., 0.95Q - P = 14 ?(1)
and, when B gets a start of 22 seconds, A runs for (Q - 22) seconds, while B runs for Q seconds.
? 1000 - 1000 / P (Q - 22) = 20
i.e., 50Q - 49P = 1100. ?(2)
Multiplying Eq. (1) by 49 and subtract from Eq. (2)
3.45Q = 414
? Q= 120 sec.
? from (1)
? P = 0.95 × 120 - 14 = 100 seconds.

Let us assume A cover P meter in the time after which he meets with B.
According to question,
Time taken by A to cover the P distance = Time taken by B to cover the 4200 - P distance.
Use the distance and time formula
P / 500 = (4200 - P ) / 700
Or 700P = 500 x 4200 - 500P
Or 700P + 500P = 500 x 4200
Or 1200P = 500 x 4200
Or 12P = 5 x 4200
Or P = 5 x 4200 / 12
Or P = 5 x 350
Or P = 1750 meters
Required time taken by A to cover 1750 meters = 1750 / 500 = 175/50
Required time taken by A to cover 1750 meters = 3.5 minutes.

A scores 100 while B scores 100 - 20 = 80 and C scores 100 - 28 = 72.
? While B scores 80, C scores 72.
? While B scores 100, C scores 72 × 100 / 80 = 90.
? B can give C 100 - 90 = 10 points.