Math Problem Statement
Magan covered a certain distance at his normal speed. If he travels 20 km/h faster, it takes him 2 hours less to cover the same distance. If he travels 10 km/h slower, it takes him 1 hour 36 minutes more to cover the given distance. What is his time if he travels with normal speed Options: A. 7 hours B. 7.5 hours C. 8 hours D. 8.5 hours E. 6 hours
Solution
Let be the distance Magan travels and be his normal speed in km/h.
We are given two key conditions:
-
If he travels 20 km/h faster, it takes him 2 hours less.
- His time at normal speed is .
- His time at 20 km/h faster is .
- According to the problem, hours.
This gives us the equation:
-
If he travels 10 km/h slower, it takes him 1 hour and 36 minutes (or 1.6 hours) more.
- His time at 10 km/h slower is .
- According to the problem, hours.
This gives us the equation:
We now have two equations:
Let's solve these equations step by step.
Step 1: Solve for in terms of
From equation 1: Multiply through by to eliminate the denominators: Simplify: Now we have in terms of .
Step 2: Substitute into the second equation
Substitute into equation 2: This becomes: Multiply through by 10 to simplify: Factor out : Simplify the expression inside the parentheses: Substitute back: Cancel from both sides: Multiply both sides by : Expand both sides: Simplify:
Step 3: Calculate the time
Now that we know km/h, we can find the time it takes to travel the distance at normal speed: Substitute : Thus, the time is:
Final Answer:
The time it takes Magan to cover the distance at his normal speed is . Thus, the correct option is C.
Do you want more details or have any questions?
Here are 5 related questions to expand your understanding:
- How would the time change if Magan's normal speed were 50 km/h instead of 60 km/h?
- What if the speed increase and decrease were different values? How would that affect the solution?
- How would the distance Magan travels be calculated if the speed were not known initially?
- What other methods could be used to solve similar rate and distance problems?
- Could this type of problem be solved graphically? How would you represent it?
Tip: Always pay attention to units of time, especially when they are given in hours and minutes, and convert them appropriately.
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Math Problem Analysis
Mathematical Concepts
Algebra
Speed, Distance, and Time
Formulas
Time = Distance / Speed
Equation for faster speed: d/v - d/(v+20) = 2
Equation for slower speed: d/(v-10) - d/v = 1.6
Theorems
Solving simultaneous equations
Suitable Grade Level
Grades 10-12
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