Allison Drove Home At 56 Mph, But Her Brother Austin, Who Left At The Same Time, Could Drive At Only 40 Mph. When Allison Arrived, Austin Still Had 112 Miles To Go. How Far Did Allison Drive? Miles
Introduction
In this article, we will explore a classic problem in mathematics that involves solving for distance. The problem is as follows: Allison drove home at 56 mph, but her brother Austin, who left at the same time, could drive at only 40 mph. When Allison arrived, Austin still had 112 miles to go. We will use algebraic equations to solve for the distance Allison drove.
Understanding the Problem
Let's break down the problem and understand what is given and what is asked. We are given the following information:
- Allison drove at a speed of 56 mph.
- Austin drove at a speed of 40 mph.
- Austin still had 112 miles to go when Allison arrived.
We are asked to find the distance Allison drove.
Setting Up the Equation
To solve this problem, we need to set up an equation that relates the distance Allison drove to the distance Austin still had to go. Let's denote the distance Allison drove as x. Since they left at the same time, the time it took for Allison to drive home is the same as the time it took for Austin to drive the remaining distance.
We can use the formula: distance = speed × time. Since the time is the same for both Allison and Austin, we can set up the following equation:
x / 56 = (x + 112) / 40
Simplifying the Equation
To simplify the equation, we can cross-multiply:
40x = 56(x + 112)
Expanding the equation, we get:
40x = 56x + 6272
Subtracting 56x from both sides, we get:
-16x = 6272
Dividing both sides by -16, we get:
x = -6272 / 16
x = -392
Interpreting the Result
However, the result x = -392 does not make sense in the context of the problem. The distance cannot be negative. This means that our initial assumption that the time it took for Allison to drive home is the same as the time it took for Austin to drive the remaining distance is incorrect.
Revisiting the Problem
Let's revisit the problem and try a different approach. We can use the formula: distance = speed × time. Since the time is the same for both Allison and Austin, we can set up the following equation:
x / 56 = (x + 112) / 40
We can also use the fact that the time it took for Allison to drive home is the same as the time it took for Austin to drive the remaining distance. Let's denote the time as t. Then, we can set up the following equation:
x / 56 = t
(x + 112) / 40 = t
Solving for Time
We can equate the two expressions for t:
x / 56 = (x + 112) / 40
Cross-multiplying, we get:
40x = 56(x + 112)
Expanding the equation, we get:
40x = 56x + 6272
Subtracting 56x from both sides, we get:
-16x = 6272
Dividing both sides by -16, we get:
x = -392
However, this result is still incorrect. Let's try a different approach.
Using a Different Approach
We can use the fact that the distance Allison drove is equal to the distance Austin drove plus the distance Austin still had to go. Let's denote the distance Austin drove as y. Then, we can set up the following equation:
x = y + 112
We can also use the fact that the time it took for Allison to drive home is the same as the time it took for Austin to drive the remaining distance. Let's denote the time as t. Then, we can set up the following equation:
x / 56 = t
y / 40 = t
Solving for Distance
We can equate the two expressions for t:
x / 56 = y / 40
Cross-multiplying, we get:
40x = 56y
Substituting x = y + 112, we get:
40(y + 112) = 56y
Expanding the equation, we get:
40y + 4480 = 56y
Subtracting 40y from both sides, we get:
4480 = 16y
Dividing both sides by 16, we get:
y = 280
Substituting y = 280 into the equation x = y + 112, we get:
x = 280 + 112
x = 392
Conclusion
In this article, we solved a classic problem in mathematics that involved solving for distance. We used algebraic equations to solve for the distance Allison drove. We tried different approaches and finally arrived at the correct solution: x = 392 miles.
Final Answer
Introduction
In our previous article, we solved a classic problem in mathematics that involved solving for distance. We used algebraic equations to solve for the distance Allison drove. In this article, we will answer some frequently asked questions related to the problem.
Q: What is the main concept used to solve this problem?
A: The main concept used to solve this problem is the formula: distance = speed × time. We also used algebraic equations to solve for the distance Allison drove.
Q: Why did we get an incorrect result initially?
A: We got an incorrect result initially because we assumed that the time it took for Allison to drive home is the same as the time it took for Austin to drive the remaining distance. However, this assumption was incorrect, and we had to revisit the problem and try a different approach.
Q: What is the correct approach to solve this problem?
A: The correct approach to solve this problem is to use the formula: distance = speed × time. We also need to use the fact that the distance Allison drove is equal to the distance Austin drove plus the distance Austin still had to go.
Q: How do we know that the distance Allison drove is equal to the distance Austin drove plus the distance Austin still had to go?
A: We know that the distance Allison drove is equal to the distance Austin drove plus the distance Austin still had to go because they left at the same time, and the time it took for Allison to drive home is the same as the time it took for Austin to drive the remaining distance.
Q: What is the final answer to the problem?
A: The final answer to the problem is x = 392 miles.
Q: What is the significance of this problem?
A: This problem is significant because it involves solving for distance using algebraic equations. It also involves using the formula: distance = speed × time, which is a fundamental concept in mathematics.
Q: How can this problem be applied in real-life situations?
A: This problem can be applied in real-life situations such as calculating the distance traveled by a car or a plane. It can also be used to calculate the time it takes to travel a certain distance.
Q: What are some common mistakes that people make when solving this problem?
A: Some common mistakes that people make when solving this problem include assuming that the time it took for Allison to drive home is the same as the time it took for Austin to drive the remaining distance. They may also forget to use the fact that the distance Allison drove is equal to the distance Austin drove plus the distance Austin still had to go.
Conclusion
In this article, we answered some frequently asked questions related to the problem of Allison's drive home. We discussed the main concept used to solve the problem, the correct approach to solve the problem, and the significance of the problem. We also discussed how the problem can be applied in real-life situations and some common mistakes that people make when solving the problem.
Final Answer
The final answer is: 392