Tracks and Races

Harsh is $50\%$ faster than Kartik. In a race, Harsh gave Kartik a head start of $500 \mathrm{m}$. Both finished the race simultaneously, Find the length of the race.

On a circular track of length $2400 \mathrm{m}$, Seeta, Geeta and Reeta start from the same point simultaneously with speeds $18 \mathrm{km}/\mathrm{h}$, $36 \mathrm{km}/\mathrm{h}$ and $54 \mathrm{km}/\mathrm{h}$. respectively. Find the minimum time after which they will meet if they are running in the same direction.

In a $1000 \mathrm{m}$ race, $\mathrm{A}$ beats $\mathrm{B}$ by $300 \mathrm{m}$. In the same race, $\mathrm{B}$ beats $\mathrm{C}$ by $400 \mathrm{m}$. Find the distance by which $\mathrm{A}$ beats $\mathrm{C}$.

On a circular track of length $1500 \mathrm{m}$, A and B start from the start point simultaneously, with speeds $36 \mathrm{km}/\mathrm{h}$ and $45 \mathrm{km}/\mathrm{h}$. Find the minimum time after which they will meet if they are running in same direction.

On a circular track of length $1500 \mathrm{m}$, Anchal and Aishwarya start from the same point simultaneously with speeds of $36 \mathrm{km}/\mathrm{h}$ and $18 \mathrm{km}/\mathrm{h}$ respectively. Find the minimum time after which they will meet if they are running in opposite directions.

In a race of $100 m$, A can beat B by $10 m$ and C by $13 m$. In a race of $180 m$, C will beat B by:

A, B, and C start running a race from the same starting point at the same time in the same direction. A 's speed around a path which is an equilateral triangle. B 's path is a square and C 's path is a regular hexagon. One edge of the triangular path, square path, and hexagonal path completely overlaps with each other. If all of them complete one round at the same time then which of the following is true?

A man decided to run $15$ rounds of a circular track of $400 \mathrm{m}$ in certain time with certain speed. He starts running but after completing some round around the track he reduces his speed by $40\%$ due to which he takes $4 \min$ extra as scheduled. But if he reduces his speed by completing $3$ more rounds he would have reached $2 \min$ earlier than the time he actually reached.

Find the number of rounds at which he decided to reduce his speed?