Formulas
1.In a $x_{1}$ Metre race A beats B by $x_{2}$ Metre or t seconds. then time taken by B =$\dfrac{x_1\times t}{x_2}$
Time taken by A= (time taken by B - t seconds)
2.If A can run x metre race in $t_{1}$ seconds and B in $t_{2}$ seconds, where $t_{1} < t_{2}$, then
A beats B by a distance $\dfrac{x}{t_{2}}\left(t_{2}-t_{1}\right)$ metres.
Race
A race is a contest of speed in running, riding, driving, sailing, rowing etc., over a particular distance.
Race Course
Race course is the ground or path on which contests are conducted.
Starting Point
Starting Point is the point from which a race starts.
Winning Point (or Goal)
Winning Point (or Goal) is the point where a race finishes.
Dead-heat Race
A race is said to be a dead-heat race if all the persons contesting the race reach the winning point (goal) exactly at the same time.
Winner
Winner is the person who first reaches the winning point.
Start:
Suppose A and B are two contestants in a race. If before the start of the race, A is at the starting point and B is ahead of A by 12 metres, then we say that 'A gives B, a start of 12 metres'.
To cover a race of 100 metres in this case, A will have to cover 100 metres while B will have to cover only (100 - 12) = 88 metres.
In a 100 race, 'A can give B 12 m' or 'A can give B a start of 12 m' or 'A beats B by 12 m' means that while A runs 100 m, B runs (100 - 12) = 88 m.
Game
A game of 100m means that the person among the contesting who scores 100 points first is the winner. If A scores 100 point while B scores only 80 points, then we say that A can give B 20 points.
Let A and B be two contestants in a race. Lets examine some of the general statements and their mathematical interpretations.
| statements | mathematical interpretations |
|---|---|
| A beats B by t seconds | A finishes the race t seconds before B finishes. |
| A gives B a start of t seconds | A starts t seconds after B starts from the same starting point. |
| A gives B a start of x metres | While A starts from the starting point, B starts x meters ahead from the same starting point at the same time. |
| In a game of 100, A can give B 20 points | While A scores 100 points, B scores only 100-20=80 points. |
In a $x_{1}$ Metre race A beats B by $x_{2}$ Metre or t seconds. then time taken by B =$\dfrac{x_1\times t}{x_2}$
Time taken by A= (time taken by B - t seconds)
Example:
In a 200 M race A beats B by 35 M or 7 seconds,then time taken by A?
B covers 35m in 7 seconds B take time = (200*7)/35=40
A takes time = (40-7)= 33 Sec.
Exercise:
B covers 100m in 25 seconds B take time =(4000*25)/100=1000 sec=16 min 40 sec.
A takes time =1000 sec-25sec=975 sec= 16 min 15 sec.
If A can run x metre race in $t_{1}$ seconds and B in $t_{2}$ seconds, where $t_{1} < t_{2}$, then
A beats B by a distance $\dfrac{x}{t_{2}}\left(t_{2}-t_{1}\right)$ metres.
Example:
A can run 1.5 km distance in 2 min 20 seconds, while B can run this distance in 2 min 30 sec. By how much distance can A beat B?
A takes time 2.20 min. = 140 Sec.
B takes time 2.30 min. = 150 Sec.
Diffrence = (150-140) = 10 Sec.
The distance covered by B=(1500*10)/150
=100m.
Exercise:
The difference in the timing of A and B is 5 seconds. Hence, A beats B by 5 seconds.
The distance covered by B in 5 seconds = (80 * 5) / 25 = 16m
Hence, A beats B by 16m.