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 ?

To solve this, we establish a common unit of distance using the cat's steps (C). Given that 2 dog steps equal 5 cat steps (2D = 5C), we find that 1 dog step = 2.5C.

Next, we compare their speeds over the same duration. In the time the dog takes 5 steps, the cat takes 8 steps:

• Distance covered by dog = 5 × 2.5C = 12.5C
• Distance covered by cat = 8 × 1C = 8C
• Relative gain by dog per 5 dog steps = 12.5C - 8C = 4.5C

The initial lead of the cat is 27 cat steps. To find the number of "5-step cycles" needed to bridge this gap:

Number of cycles = 27C / 4.5C = 6 cycles

Total dog steps required = 6 cycles × 5 steps/cycle = 30 steps. Thus, the dog will catch the cat in 30 steps.