Steve can complete the 100 m dash in 10 seconds while Paul can run it in 12 seconds. How does Steve's time compare to Paul's?
A) Steve is $5 / 6$ as fast as Paul
B) Steve is $1 / 2$ as fast as Paul
C) Steve is $1 / 2$ slower than Paul
D) Steve is $5 / 6$ slower than Paul
Answer (1)
Calculate Steve's speed: s p ee d St e v e = 10 100 = 10 m/s .
Calculate Paul's speed: s p ee d P a u l = 12 100 = 3 25 m/s .
Find the ratio of Steve's speed to Paul's speed: s p ee d P a u l s p ee d St e v e = 25/3 10 = 5 6 .
Steve is 5 6 times as fast as Paul, so Paul is 6 5 as fast as Steve. The answer is A) Steve is 6 5 as fast as Paul.
Explanation
Problem Analysis Steve can complete the 100 m dash in 10 seconds, while Paul can run it in 12 seconds. We need to compare Steve's time to Paul's time to determine how much faster or slower Steve is compared to Paul.
Calculating Speeds and Ratio First, we calculate Steve's speed: s p ee d St e v e = t im e d i s t an ce = 10 100 = 10 m/s Next, we calculate Paul's speed: s p ee d P a u l = t im e d i s t an ce = 12 100 = 3 25 m/s Now, we find the ratio of Steve's speed to Paul's speed: s p ee d P a u l s p ee d St e v e = 25/3 10 = 25 10 × 3 = 25 30 = 5 6
Interpreting the Ratio The ratio of Steve's speed to Paul's speed is 5 6 . This means Steve is 5 6 times as fast as Paul. We can also say Paul is 6 5 as fast as Steve.
Comparing Steve's Time to Paul's The question asks how Steve's time compares to Paul's. Since Steve is 5 6 times as fast as Paul, Paul is 6 5 as fast as Steve. Therefore, Steve is faster than Paul. The options compare Steve's speed to Paul's, so we look for an option that states Steve is a certain fraction as fast as Paul. Since Paul is 6 5 as fast as Steve, Steve is 5 6 times as fast as Paul. However, the options are given in terms of how much slower Steve is. To find how much slower Paul is compared to Steve, we calculate 1 − 6 5 = 6 1 . This means Paul is 6 1 slower than Steve. Alternatively, to find how much faster Steve is compared to Paul, we calculate 5 6 − 1 = 5 1 . The question is how does Steve's time compare to Paul's, so we want to know how much faster Steve is. Since the options are in terms of how much slower, we need to find how much slower Paul is than Steve, which is 6 5 as fast as Steve.
Determining the Correct Option The correct answer is that Steve is 5 6 as fast as Paul, which means Paul is 6 5 as fast as Steve. Therefore, Steve is 6 5 as fast as Paul is not correct. However, Steve is faster, so we need to find how much faster Steve is compared to Paul. Since Paul is 6 5 as fast as Steve, Steve is 6 5 slower than Paul is not correct.
Final Answer The correct option is A) Steve is 6 5 as fast as Paul.
Examples
Understanding relative speeds is useful in many real-life scenarios. For example, if you're planning a road trip with friends, knowing the relative speeds of your cars helps you estimate arrival times and coordinate meeting points. If one car travels at 6 5 the speed of another, you can calculate how much earlier the faster car will arrive at the destination, allowing for better planning and communication during the trip.
