If two runners P & Q are in a race, the ratio of speeds of two runners P & Q is 4:5. If P has a start of 100 m, then what is the distance where both will finish the race at the sam... If two runners P & Q are in a race, the ratio of speeds of two runners P & Q is 4:5. If P has a start of 100 m, then what is the distance where both will finish the race at the same time? Read more

Steps to Solve

1. Understanding the Race Conditions

   In a race of 1000 m, Ram beats Shyam by 50 m. This means that when Ram finishes the race (1000 m), Shyam has only run 950 m.

2. Calculating Ram's Speed

   Let ( V_r ) be Ram's speed and ( V_s ) be Shyam's speed. The time taken by Ram to finish the race is the same as the time taken by Shyam to run 950 m.

   Thus, we have: $$ \frac{1000}{V_r} = \frac{950}{V_s} $$

3. Finding Ratios of Speeds

   Rearranging the equation gives: $$ V_r = \frac{1000}{950} V_s = \frac{20}{19} V_s $$

   This means Ram runs 20 units of distance in the time Shyam runs 19 units.

4. Finding the Distance Shyam Runs Before Ram Finishes

   Now, we want to determine by how much distance Ram beats Shyam in a 1000 m race. As per the problem statement, we need to find out how far behind Shyam is when Ram finishes the race.

   The distance covered by Shyam when Ram completes 1000 m can be calculated: $$ \text{Distance covered by Shyam} = V_s \times \frac{1000}{V_r} $$

5. Substituting the Values

   Substituting ( V_r = \frac{20}{19} V_s ) into the equation: $$ \text{Distance covered by Shyam} = V_s \times \frac{1000}{\frac{20}{19} V_s} = \frac{1000 \times 19}{20} = 950 , m $$

6. Final Calculation of Distance Shyam is Behind Ram

   The distance between Shyam's finishing point and Ram's finishing point is: $$ 1000 - 950 = 50 , m $$