A dog wants to catch a cat that is twenty-seven steps ahead of him. The cat takes eight steps to every five steps taken by the dog. Two steps of the dog are equal to five steps of the cat. How many steps of the dog will be required to catch the cat ?
We are given that 2 dog steps are equal to 5 cat steps in distance: 2 * D_d = 5 * D_c => D_d = (5/2) * D_c.
The cat starts 27 steps ahead of the dog. The initial distance between them is 27 * D_c.
In a certain time interval, the cat takes 8 steps and the dog takes 5 steps.
Distance covered by cat in this interval = 8 * D_c.
Distance covered by dog in this interval = 5 * D_d = 5 * (5/2) * D_c = 25/2 * D_c = 12.5 * D_c.
In this time interval, the dog reduces the distance between them by (Distance covered by dog) – (Distance covered by cat) = 12.5 * D_c – 8 * D_c = 4.5 * D_c.
The total distance the dog needs to close is the initial distance, which is 27 * D_c.
Number of intervals required to catch the cat = (Total distance to cover) / (Distance reduced per interval)
Number of intervals = (27 * D_c) / (4.5 * D_c) = 27 / 4.5 = 27 / (9/2) = 27 * (2/9) = 3 * 2 = 6 intervals.
In each interval, the dog takes 5 steps.
Total dog steps required = Number of intervals * Steps taken by dog per interval
Total dog steps = 6 * 5 = 30 steps.
After 30 dog steps, the dog covers 30 * D_d = 30 * (5/2) * D_c = 75 * D_c.
The dog takes 5 steps per interval, so 30 dog steps means 30/5 = 6 intervals have passed.
In 6 intervals, the cat takes 6 * 8 = 48 steps.
The cat started 27 * D_c ahead. Its final position is 27 * D_c + 48 * D_c = 75 * D_c.
The dog’s final position is 75 * D_c. The dog catches the cat when their positions are the same.