A dog takes 3 leaps for every 5 leaps of a hare. If one leap of the dog is equal in length to 3 leaps of the hare, what is the ratio of the speed of the dog to the speed of the hare?

Introduction / Context:
This question involves relative speed expressed through leap counts and leap lengths. It checks understanding that speed is distance travelled per unit time, and both frequency of leaps and the length of each leap contribute to the effective speed of an animal.

Given Data / Assumptions:

• The dog takes 3 leaps while the hare takes 5 leaps in the same time.
• One leap of the dog covers the same distance as 3 leaps of the hare.
• We must find the ratio of the speed of the dog to the speed of the hare.

Concept / Approach:
Speed is proportional to (number of leaps per unit time) multiplied by (length of each leap). If we assume a unit of time (for example one second), we can calculate distances covered by each animal in that time and compare them. Using symbolic leap length for the hare makes the comparison easy.

Step-by-Step Solution:
Step 1: Let length of one hare leap be L units. Step 2: Then length of one dog leap is 3L, because it is equal to 3 hare leaps. Step 3: In a fixed time interval, the dog takes 3 leaps. Step 4: In the same time, the hare takes 5 leaps. Step 5: Distance covered by dog in that time = number of leaps * length per leap = 3 * 3L = 9L. Step 6: Distance covered by hare in that time = 5 * L = 5L. Step 7: Since time is the same for both, speed is directly proportional to distance covered. Step 8: Therefore, speed of dog : speed of hare = 9L : 5L = 9 : 5.

Verification / Alternative check:
We can choose a concrete value for L, for example L = 1 metre. Then in one time unit, the dog covers 9 metres and the hare covers 5 metres. Their speed ratio is 9 : 5. Any other positive value of L would cancel out in the ratio, so 9 : 5 is robust and independent of actual leap length.

Why Other Options Are Wrong:
Ratios like 2 : 3, 4 : 7 or 5 : 6 ignore either the difference in leap frequency or the difference in leap length. For instance, 3 leaps versus 5 leaps alone would give 3 : 5, while leap length alone of 3L versus L would give 3 : 1. Correct speed comparison must combine both effects, leading to 9 : 5 rather than any of the distractor ratios.

Common Pitfalls:
A common mistake is to take ratio of leaps only (3 : 5) and ignore leap length, or to use only leap length and forget that the hare makes more leaps in the same time. Another error is to multiply the ratios incorrectly instead of multiplying the actual distance components. Always remember that speed depends on both frequency and distance per action.

Final Answer:
The ratio of the speed of the dog to the speed of the hare is 9 : 5.