In covering certain distance, the average speeds of X and Y are in the ratio 4 : 5. If X takes 45 minutes more than Y to reach the destination, then what is the time taken by Y to reach the destination?
180 minutes
Solving Speed, Time, and Distance Problems
This question involves the concepts of speed, time, and distance. A fundamental relationship exists between these three variables:
\( \text{Distance} = \text{Speed} \times \text{Time} \)
When the distance covered is constant, the speed and the time taken to cover that distance are inversely proportional to each other. This means that if speed increases, time decreases, and vice versa, for the same distance.
Analyzing the Given Information
We are given the following information about individuals X and Y covering the same distance:
- The ratio of their average speeds is \( \text{Speed}_X : \text{Speed}_Y = 4 : 5 \).
- X takes 45 minutes more than Y to reach the destination. This means \( \text{Time}_X - \text{Time}_Y = 45 \) minutes.
Using the Inverse Proportion Relationship
Since the distance is the same for both X and Y, their speeds and times are inversely proportional. Therefore, the ratio of the time taken by X and Y will be the inverse of their speed ratio.
\( \text{Time}_X : \text{Time}_Y = \text{Speed}_Y : \text{Speed}_X \)
Given \( \text{Speed}_X : \text{Speed}_Y = 4 : 5 \), the ratio of their times is:
\( \text{Time}_X : \text{Time}_Y = 5 : 4 \)
Setting up the Time Equation
Let the time taken by X be \( 5k \) and the time taken by Y be \( 4k \), where \( k \) is a constant.
- Time taken by X, \( T_X = 5k \)
- Time taken by Y, \( T_Y = 4k \)
We know that X takes 45 minutes more than Y. So, we can write the equation:
\( T_X - T_Y = 45 \) minutes
Substitute the values in terms of \( k \):
\( 5k - 4k = 45 \)
Solving for the Constant and Time
Simplifying the equation:
\( k = 45 \)
Now we can find the time taken by Y by substituting the value of \( k \) into the expression for \( T_Y \):
\( T_Y = 4k \)
\( T_Y = 4 \times 45 \)
\( T_Y = 180 \) minutes
To verify, we can also find the time taken by X:
\( T_X = 5k = 5 \times 45 = 225 \) minutes
The difference is \( T_X - T_Y = 225 - 180 = 45 \) minutes, which matches the information given in the question.
Step-by-Step Solution Summary
- Understand the inverse relationship between speed and time for a constant distance.
- Use the given speed ratio (4:5) to find the time ratio, which is the inverse (5:4).
- Represent the times taken by X and Y as \( 5k \) and \( 4k \) respectively.
- Set up an equation using the given time difference: \( 5k - 4k = 45 \).
- Solve the equation to find the value of \( k \).
- Substitute the value of \( k \) into the expression for the time taken by Y (\( 4k \)).
- Calculate the final time taken by Y.
The time taken by Y to reach the destination is 180 minutes.
| Concept | Relationship | Application |
|---|---|---|
| Speed, Time, Distance | Distance = Speed × Time | Constant distance implies inverse relationship between speed and time. |
| Ratio of Speeds | \( S_X : S_Y = 4 : 5 \) | Ratio of times \( T_X : T_Y = 5 : 4 \). |
| Time Difference | \( T_X - T_Y = 45 \) minutes | Set up equation using time ratio: \( 5k - 4k = 45 \). |
Revision Table: Key Concepts
| Concept | Definition/Rule | How it applies here |
|---|---|---|
| Inverse Proportion | When two quantities are inversely proportional, their product is constant. If A and B are inversely proportional, \( A \propto 1/B \) or \( AB = \text{constant} \). | For a fixed distance, Speed and Time are inversely proportional. \( \text{Speed} \times \text{Time} = \text{Distance (constant)} \). |
| Ratios | A comparison of two quantities. If \( a:b = c:d \), then \( a/b = c/d \). | Used to relate the speeds and then the times of X and Y. \( S_X/S_Y = 4/5 \) implies \( T_X/T_Y = 5/4 \). |
| Algebraic Equations | Mathematical statements with an equals sign, involving variables. | Used to solve for the unknown constant \( k \) based on the time difference. |
Additional Information: Speed, Time, and Distance Formulas
Here are the basic formulas related to speed, time, and distance:
- \( \text{Distance} = \text{Speed} \times \text{Time} \)
- \( \text{Speed} = \frac{\text{Distance}}{\text{Time}} \)
- \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \)
It's important to ensure that units are consistent when using these formulas (e.g., if speed is in km/hour, time should be in hours and distance in km). In this problem, the time difference is given in minutes, and we calculated the time in minutes, which is consistent.
Understanding the relationship between speed and time when distance is constant is crucial for solving many problems. If \( S_1 \) and \( S_2 \) are speeds and \( T_1 \) and \( T_2 \) are the times taken to cover the same distance, then \( S_1 \times T_1 = S_2 \times T_2 \), which means \( S_1/S_2 = T_2/T_1 \).
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Important Questions from Relative Speed
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