Math: Find The Difference In Race Times
Let's dive into a fun math problem that involves understanding race times and how to calculate the difference using an equation! We've got two runners, Annabel and another runner, and we want to figure out just how much faster the other runner was. This is a classic word problem that's perfect for practicing our algebra skills. We'll set up an equation with a variable to represent the unknown and then solve it to find the answer. So, grab your thinking caps, and let's get ready to solve this!
Understanding the Problem: Annabel's Race Time
In this scenario, we're presented with a clear piece of information: Annabel's time for a race was 46.5 seconds. This is our starting point, the benchmark against which we'll measure the other runner's performance. The problem also states that another runner's time was 8.75 seconds faster than Annabel's. The core task here is to write and evaluate an equation with a variable to determine the precise difference between Annabel's time and this faster runner's time. It sounds straightforward, but it requires careful attention to the wording. When we say someone is 'faster,' it means their time is *less* than the other person's time. So, if the other runner was 8.75 seconds faster, it means their actual race time was Annabel's time minus 8.75 seconds. The question asks for the *difference* between their times. This is a bit of a trick! The problem *gives* us the difference directly: 8.75 seconds. However, to make it a proper equation-solving exercise, we need to construct an equation that reflects this scenario and then solve for a variable. This process helps solidify the understanding of how variables represent unknown quantities and how equations can model real-world situations. We'll be using basic arithmetic operations – subtraction in this case – and the concept of a variable to represent one of the unknown times, or perhaps the difference itself if we were to approach it differently. The key is to translate the words into mathematical symbols accurately. Let's break down how we can represent this mathematically and what the question is truly asking us to demonstrate.
Setting Up the Equation: Introducing the Variable
To find the difference between Annabel's time and the other runner's time, we first need to establish what we know and what we want to find. We know Annabel's time is 46.5 seconds. We also know that the other runner's time was 8.75 seconds faster. The question asks us to write and evaluate an equation with a variable to find the *difference* between their times. This is a crucial point: the problem *gives* us the difference directly (8.75 seconds). However, to fulfill the requirement of writing and evaluating an equation with a variable, we can set up an equation to find the *other runner's actual time* and then, if needed, calculate the difference, or we can directly represent the difference with a variable. Let's choose to represent the other runner's time with a variable, say 'r'. Since the other runner was 8.75 seconds faster than Annabel, their time 'r' can be expressed as: r = Annabel's time - 8.75 seconds. Plugging in Annabel's known time, we get: r = 46.5 - 8.75. This equation, r = 46.5 - 8.75, allows us to calculate the other runner's actual time. If the question strictly means the *difference* between their times, and we're to use a variable to *find* that difference, we could define a variable, let's call it 'd', to represent this difference. The problem states that the difference *is* 8.75 seconds. So, we could write an equation like: Annabel's time - other runner's time = d. We know Annabel's time is 46.5. We also know that the other runner's time is (46.5 - 8.75). So, substituting these into our difference equation: 46.5 - (46.5 - 8.75) = d. This simplifies to 46.5 - 46.5 + 8.75 = d, which means d = 8.75. However, this approach feels a bit circular because the difference is explicitly given. A more instructive way to use a variable here is to find the *other runner's time*. Let 't' be the other runner's time in seconds. We know Annabel's time is 46.5 seconds. The other runner was 8.75 seconds faster, meaning their time is *less* than Annabel's by 8.75 seconds. So, the equation to find the other runner's time is: t = 46.5 - 8.75. This equation uses a variable 't' to represent the unknown time of the other runner and sets up the calculation based on the information given. This is a standard way to approach such word problems in mathematics, where we translate the given scenario into a solvable algebraic expression. The equation clearly shows the relationship between Annabel's time and the other runner's time, incorporating the given difference.
Evaluating the Equation and Finding the Difference
Now that we have set up our equation, t = 46.5 - 8.75, let's evaluate it to find the other runner's time. This equation uses the variable 't' to represent the unknown time of the runner who was faster than Annabel. To solve for 't', we simply perform the subtraction. We need to subtract 8.75 from 46.5. It's important to align the decimal points when subtracting numbers with different numbers of decimal places. We can think of 46.5 as 46.50. So, the subtraction looks like this:
46.50
- 8.75
-------
Starting from the rightmost digit (the hundredths place), we can't subtract 5 from 0, so we need to borrow from the tenths place. The 5 in the tenths place becomes 4, and the 0 in the hundredths place becomes 10. So, 10 - 5 = 5.
Next, in the tenths place, we have 4 - 7. Again, we can't subtract 7 from 4, so we borrow from the ones place. The 6 in the ones place becomes 5, and the 4 in the tenths place becomes 14. So, 14 - 7 = 7.
Moving to the ones place, we have 5 - 8. We can't subtract 8 from 5, so we borrow from the tens place. The 4 in the tens place becomes 3, and the 5 in the ones place becomes 15. So, 15 - 8 = 7.
Finally, in the tens place, we have 3 - 0 (since there's no digit in the tens place for 8.75, it's considered 0), which equals 3.
Putting it all together, we get 37.75.
So, the equation evaluates to: t = 37.75 seconds. This means the other runner's actual race time was 37.75 seconds. The question specifically asks for the difference between Annabel's time and the other runner's time. We were told directly that the other runner was 8.75 seconds faster. To verify this using our calculated time, we can subtract the faster runner's time from Annabel's time: 46.5 - 37.75. Let's check this calculation:
46.50
- 37.75
-------
Following the same borrowing process as above:
- Hundredths place: 10 - 5 = 5
- Tenths place: 14 - 7 = 7
- Ones place: 15 - 7 = 8
- Tens place: 3 - 3 = 0
This gives us 8.75 seconds. Therefore, the difference between Annabel's time (46.5 seconds) and the other runner's time (37.75 seconds) is indeed 8.75 seconds. Our evaluation of the equation confirmed the difference provided in the problem statement, demonstrating how setting up and solving an equation can help us understand and verify such relationships.
Conclusion: Understanding Race Time Differences
In conclusion, we've successfully tackled a word problem involving race times by setting up and evaluating an equation with a variable. We started with Annabel's race time of 46.5 seconds and the information that another runner was 8.75 seconds faster. We defined a variable, 't', to represent the other runner's unknown time. Our equation, t = 46.5 - 8.75, allowed us to calculate this faster time. Upon evaluating the equation, we found that t = 37.75 seconds. This means the other runner completed the race in 37.75 seconds. The question asked us to find the *difference* between their times. Although the problem explicitly stated the difference was 8.75 seconds, we used our calculated time to verify this. By subtracting the faster runner's time (37.75 seconds) from Annabel's time (46.5 seconds), we confirmed that the difference is indeed 8.75 seconds. This exercise highlights the power of algebra in representing real-world scenarios and solving for unknowns. It's a fundamental skill in mathematics that helps us make sense of quantitative information. Whether it's comparing race results, calculating distances, or managing budgets, the ability to translate words into mathematical equations and solve them is invaluable. For anyone interested in delving deeper into word problems and algebraic concepts, resources like **Khan Academy** offer excellent tutorials and practice exercises. Exploring the fundamentals of algebra can unlock a deeper understanding of mathematics and its applications in everyday life. You can also find more information about calculating time differences and solving equations on websites like **Math is Fun**.
