Three cats A, B and C roam in such a way that when cat A takes 5 steps, B takes 6 steps, and C takes 7 steps during the same time. Also, 6 steps of A cover the same distance as 7 steps of B and 8 steps of C. What is the ratio of their speeds (A : B : C)?

Introduction / Context:
This question mixes step counts with step lengths to create a speed comparison problem. The speed of each cat depends on how many steps it takes per unit time and how long each step is. We know how many steps A, B and C take in the same time interval, and we are given equivalences between their step lengths. From this, we must derive the ratio of their speeds.

Given Data / Assumptions:
• In the same time, A takes 5 steps, B takes 6 steps, C takes 7 steps. • 6 steps of A = 7 steps of B = 8 steps of C (same distance). • All motions are uniform during the comparison interval.

Concept / Approach:
Speed is defined as distance per unit time. For each cat, speed = (number of steps taken per time) * (length of each step). Let step lengths be L_A, L_B and L_C. The equivalence 6L_A = 7L_B = 8L_C allows us to write each length in terms of a common constant. We then compute the product of step count and step length as a proportional measure of speed. Finally, we simplify the resulting ratio to get integers.

Step-by-Step Solution:
Step 1: Let the common distance 6L_A = 7L_B = 8L_C = K, where K is some constant distance. Step 2: Then L_A = K / 6, L_B = K / 7, L_C = K / 8. Step 3: Number of steps per time unit: A takes 5 steps, B takes 6 steps, C takes 7 steps (in the same time). Step 4: Speed of A ∝ 5 * L_A = 5 * (K / 6) = 5K / 6. Step 5: Speed of B ∝ 6 * L_B = 6 * (K / 7) = 6K / 7. Step 6: Speed of C ∝ 7 * L_C = 7 * (K / 8) = 7K / 8. Step 7: Ignore K since it is common. So speed ratio A : B : C = (5/6) : (6/7) : (7/8). Step 8: To convert these fractions into whole numbers, take the least common multiple of the denominators 6, 7 and 8, which is 168. Step 9: Multiply each fraction by 168: (5/6) * 168 = 5 * 28 = 140, (6/7) * 168 = 6 * 24 = 144, (7/8) * 168 = 7 * 21 = 147. Step 10: Therefore, the speed ratio is 140 : 144 : 147.

Verification / Alternative check:
We can quickly check if any common factor can further simplify the ratio 140 : 144 : 147. The differences between terms are small and prime factorizations show no single integer greater than 1 divides all three values simultaneously. Hence, 140 : 144 : 147 is already in simplest form and accurately represents the relative speeds as derived from both step counts and step lengths.

Why Other Options Are Wrong:
• 40 : 44 : 47 is roughly the original ratio divided by 3.5, but does not preserve integer scaling from the precise calculations. • 15 : 21 : 28 and 252 : 245 : 240 do not derive from combining 5/6, 6/7 and 7/8 with a common multiplier and thus do not respect the given conditions.

Common Pitfalls:
A common mistake is to use only the step counts (5 : 6 : 7) and ignore the step length differences. Another error is to assume step lengths are equal despite the explicit equivalence condition. Not clearing the fractions correctly when forming the ratio can also lead to arithmetic errors.

Final Answer:
The ratio of their speeds A : B : C is 140 : 144 : 147.