Difficulty: Easy
Correct Answer: 87.5 km/h
Explanation:
Introduction / Context:
This question examines the use of ratios in speed problems. When the speeds of two moving objects are in a given ratio and one of the actual speeds is known, the other can be found quickly using proportional reasoning. Here, we work with two trains whose speed ratio is 7 : 8.
Given Data / Assumptions:
Concept / Approach:
If the speeds of two trains are in the ratio 7 : 8, we can express their speeds as 7k and 8k for some common factor k. Once we compute the actual speed of the second train, we can set 8k equal to that value and find k. Then we use the value of k to find the speed of the first train as 7k.
Step-by-Step Solution:
Let the first train speed = 7k km/h.
Let the second train speed = 8k km/h.
The second train covers 400 km in 4 hours.
Speed of second train = distance / time = 400 / 4 = 100 km/h.
So 8k = 100, which gives k = 100 / 8 = 12.5.
Speed of first train = 7k = 7 * 12.5 = 87.5 km/h.
Verification / Alternative check:
Check that the ratio of speeds matches 7 : 8. First train = 87.5 km/h, second train = 100 km/h. The ratio 87.5 : 100 simplifies by dividing both by 12.5 to 7 : 8. This confirms that our use of ratios is correct and consistent with the problem statement.
Why Other Options Are Wrong:
83.5 km/h, 84.5 km/h, and 86.5 km/h: None of these values, when compared with 100 km/h, gives a simplified ratio of exactly 7 : 8. Each would lead to a different ratio, so they are not consistent with the original condition.
Common Pitfalls:
Learners sometimes reverse the ratio or assign 7k to the second train and 8k to the first train by mistake. Another error is to miscalculate the speed of the second train by mixing units or misreading 400 km in 4 hours. Always compute the real known speed first, then apply the ratio carefully.
Final Answer:
The speed of the first train is 87.5 km/h.
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