Carrington's Exam Time How To Calculate Remaining Time

Hey guys! Ever find yourself racing against the clock during an exam? Let's break down a classic time-management problem, step by step, so you'll never feel that exam-room pressure again. In this article, we're diving into a word problem that involves fractions of time. We'll walk through Carrington's exam experience, figuring out exactly how much time he has left after tackling the first few problems. So, grab your thinking caps, and let's get started!

Understanding the Problem

Let’s start by really understanding the time problem. Word problems can feel like puzzles, but don't worry, we'll piece it together! Our main goal is to figure out how much time Carrington has left on his exam. We know a few key things: he had a set amount of time for the whole exam, he used a certain fraction of that time on the first three problems, and we need to find the remaining time. Keywords here are 'fraction of time,' 'time used,' and 'time remaining.'

First, we know Carrington has a specific amount of time for his exam, which is expressed as a fraction of an hour. Then, we are told he uses a fraction of a fraction of that time on the initial problems. That little word "of" is super important in math – it often means we need to multiply fractions. Our mission? Calculate the time spent on those problems and then subtract that from the total time to find out what's left. Easy peasy, right? Let's jump into the nitty-gritty details so you can see the entire process clearly.

Breaking Down the Time Allotment

Now, let's break down the information we have about the time allotment. The problem tells us that Carrington had a certain portion of an hour for his exam. It's crucial to identify this total time because it's our starting point. Think of it as the whole pie – we need to know the size of the pie before we can figure out how big each slice is! In this case, the total time allotted is a fraction of an hour, which means we are dealing with parts of an hour rather than a full 60 minutes. This is a typical way exam time is often presented in math problems, so it's a handy concept to get comfy with.

The next step is to understand how this total time was used. Carrington didn't spend the entire time willy-nilly; he dedicated a specific portion of it to the initial problems. The problem states that he used a fraction of this total time on the first three problems. Remember, that word “of” is waving a little multiplication flag at us! It means we're not just dealing with a fraction of an hour, but a fraction multiplied by another fraction. We're essentially finding a part of a part, which is where fraction multiplication comes into play. This is a common scenario in real-world time management too, like figuring out how much time to dedicate to specific tasks within a larger project. So, let's keep those multiplication hats on as we move forward!

Calculating Time Spent on First Problems

Alright, let's dive into the calculation! This is where we put our fraction skills to the test. We know Carrington used a fraction of a fraction of an hour for the first three problems. To find the exact amount of time, we need to multiply those fractions together. Remember, multiplying fractions is pretty straightforward: you multiply the numerators (the top numbers) and then multiply the denominators (the bottom numbers). This gives us a new fraction that represents the portion of time spent on those initial problems. It’s like finding the area of a rectangle where the sides are fractions – a visual way to think about fraction multiplication!

Once we've multiplied the fractions, we'll have a fraction of an hour representing the time Carrington used. But what does that mean in practical terms? We might want to convert this fraction of an hour into minutes to get a better sense of the actual time spent. To do this, we need to remember that there are 60 minutes in an hour. So, we'll multiply our fraction of an hour by 60 minutes. This will give us the time Carrington spent on the first three problems in good ol' minutes, which can be easier to visualize and compare. We're turning an abstract fraction into a concrete chunk of time, which is super helpful for understanding the problem and double-checking our answer.

Determining Remaining Time

Here comes the final piece of the puzzle: figuring out how much time Carrington has left. Now that we know the total time allotted for the exam and the time Carrington spent on the first three problems, we can calculate the remaining time. This is where subtraction comes into play. We'll subtract the time spent from the total time. Think of it like this: we're taking the whole pie (total exam time) and removing a slice (time spent on the first problems) to see what's left. The result will be the fraction of an hour that Carrington has remaining for the rest of the exam.

But we're not quite done yet! Just like before, we might want to convert this fraction of an hour into minutes to get a better understanding of the actual time left. We'll use the same trick we used earlier – multiplying the fraction of an hour by 60 minutes. This will give us the remaining time in minutes, which can be super helpful for Carrington (or anyone taking an exam!) to gauge how much time they have for each remaining question. It’s all about turning those fractions into practical time management skills, guys!

Step-by-Step Solution to the Problem

Okay, let's break down the actual problem and solve it step by step. This will be like a mini-math workout, but I promise, it's totally doable! Here’s the problem we’re tackling:

Carrington used $\frac{3}{4}$ of the $\frac{1}{2}$ of an hour allotted for an exam to do the first 3 problems. How much time does he have left for the remaining part of the exam?

Step 1: Calculate Time Spent on the First 3 Problems

First, we need to figure out what $\frac{3}{4}$ of $\frac{1}{2}$ of an hour actually is. Remember, "of" means multiply, so we'll multiply these fractions together:

$\frac{3}{4} \times \frac{1}{2} = \frac{3 \times 1}{4 \times 2} = \frac{3}{8}$

So, Carrington spent $\frac{3}{8}$ of an hour on the first three problems.

Step 2: Calculate Remaining Time

Now we need to find out how much time is left. The total time for the exam was $\frac{1}{2}$ of an hour, and Carrington used $\frac{3}{8}$ of an hour. To find the remaining time, we subtract the time used from the total time:

$\frac{1}{2} - \frac{3}{8}$

But we can't subtract fractions unless they have the same denominator! So, we need to find a common denominator for 2 and 8. The least common multiple of 2 and 8 is 8, so we'll convert $\frac{1}{2}$ to an equivalent fraction with a denominator of 8:

$\frac{1}{2} = \frac{1 \times 4}{2 \times 4} = \frac{4}{8}$

Now we can subtract:

$\frac{4}{8} - \frac{3}{8} = \frac{4 - 3}{8} = \frac{1}{8}$

So, Carrington has $\frac{1}{8}$ of an hour left for the rest of the exam!

Converting to Minutes (Optional)

Just for kicks, let's see how many minutes that is. To convert $\frac{1}{8}$ of an hour to minutes, we multiply by 60 (since there are 60 minutes in an hour):

$\frac{1}{8} \times 60 \text{ minutes} = \frac{60}{8} \text{ minutes}$

We can simplify this fraction by dividing both the numerator and denominator by 4:

$\frac{60 \div 4}{8 \div 4} = \frac{15}{2} \text{ minutes}$

And then convert this improper fraction to a mixed number:

$\frac{15}{2} = 7 \frac{1}{2} \text{ minutes}$

So, Carrington has 7 and a half minutes left. Talk about a nail-biter!

Final Answer

Therefore, the answer is that Carrington has $\frac{1}{8}$ of an hour left for the remaining part of the exam. Go Carrington, you got this!

Key Takeaways

Let’s recap the important bits, guys! The biggest thing to remember from this problem is how to deal with fractions of time. When a problem says “of,” it's your cue to multiply! Also, when you're adding or subtracting fractions, make sure they have the same denominator. It's like trying to add apples and oranges – you need to find a common unit (like