Calculate Remaining Training Time: Math Problem
Hey guys! Let's dive into a fun mathematical problem involving training time. Imagine a runner preparing for a race. He's got a solid plan to train for 11 hours over the three days leading up to the big event. On the first day, he clocks in $2 \frac{3}{5}$ hours, and on the second day, he pushes himself for $2 \frac{9}{10}$ hours. The burning question is: how many hours does he need to train on the final day to meet his 11-hour goal? This is a classic problem that combines mixed numbers and basic arithmetic, and we're going to break it down step by step.
Understanding the Problem
Before we start crunching numbers, let's make sure we fully understand the problem. The core question is to find the remaining training time for the third day. We know the total training time (11 hours) and the training time for the first two days ($2 \frac{3}{5}$ hours and $2 \frac{9}{10}$ hours). To solve this, we'll need to add the training times for the first two days and then subtract that sum from the total training time. This will give us the training time required for the third day.
It's essential to visualize the problem. Think of it as a pie chart where the whole pie represents 11 hours. The first slice is $2 \frac{3}{5}$ hours, the second slice is $2 \frac{9}{10}$ hours, and we need to figure out the size of the third slice. By visualizing the problem, we can better understand the steps involved in solving it.
Step 1: Converting Mixed Numbers to Improper Fractions
Okay, let's get started! The first step is to convert the mixed numbers into improper fractions. This makes it much easier to add and subtract them. Remember, a mixed number has a whole number part and a fractional part (like $2 \frac{3}{5}$), while an improper fraction has a numerator larger than or equal to its denominator (like $\frac{13}{5}$).
To convert a mixed number to an improper fraction, we multiply the whole number by the denominator and add the numerator. This becomes the new numerator, and we keep the same denominator. For $2 \frac3}{5}$, we multiply 2 by 5 (which gives us 10) and add 3, resulting in 13. So, $2 \frac{3}{5}$ becomes $\frac{13}{5}$. Let's do the same for $2 \frac{9}{10}$. We multiply 2 by 10 (which gives us 20) and add 9, resulting in 29. So, $2 \frac{9}{10}$ becomes $\frac{29}{10}$. Now we have our training times for the first two days expressed as improper fractions{5}$ hours and $\frac{29}{10}$ hours.
Step 2: Adding the Training Times for the First Two Days
Now that we have improper fractions, we can add the training times for the first two days. To add fractions, they need to have a common denominator. Looking at our fractions, $\frac{13}{5}$ and $\frac{29}{10}$, we see that the least common denominator (LCD) is 10. This is the smallest number that both 5 and 10 divide into evenly.
To get $\frac{13}{5}$ to have a denominator of 10, we need to multiply both the numerator and the denominator by 2. This gives us $\frac{13 \times 2}{5 \times 2} = \frac{26}{10}$. The fraction $\frac{29}{10}$ already has the denominator we need, so we don't need to change it.
Now we can add the fractions: $\frac{26}{10} + \frac{29}{10}$. When adding fractions with the same denominator, we simply add the numerators and keep the denominator. So, $26 + 29 = 55$, and our sum is $\frac{55}{10}$ hours. This is the total training time for the first two days.
Step 3: Subtracting the Sum from the Total Training Time
We're getting closer! We know the total training time is 11 hours, and the training time for the first two days is $\frac{55}{10}$ hours. To find the training time for the third day, we need to subtract $\frac{55}{10}$ from 11. First, we need to convert 11 into a fraction with a denominator of 10. We can do this by multiplying 11 by $\frac{10}{10}$, which gives us $\frac{11 \times 10}{10} = \frac{110}{10}$.
Now we can subtract: $\frac{110}{10} - \frac{55}{10}$. Subtracting fractions with the same denominator is similar to adding them – we subtract the numerators and keep the denominator. So, $110 - 55 = 55$, and our difference is $\frac{55}{10}$ hours. This is the training time required for the third day.
Step 4: Simplifying the Answer
We have the training time for the third day as $\frac{55}{10}$ hours, but we can simplify this fraction. Both 55 and 10 are divisible by 5. Dividing both the numerator and the denominator by 5, we get $\frac{55 \div 5}{10 \div 5} = \frac{11}{2}$. This is the simplified improper fraction.
We can also convert this improper fraction back into a mixed number. To do this, we divide the numerator (11) by the denominator (2). 11 divided by 2 is 5 with a remainder of 1. So, the whole number part is 5, the numerator of the fractional part is 1, and the denominator remains 2. Therefore, $\frac{11}{2}$ is equal to $5 \frac{1}{2}$.
The Final Answer
So, there you have it! Our runner needs to train for $5 \frac{1}{2}$ hours on the third day to meet his 11-hour goal. We solved this problem by converting mixed numbers to improper fractions, adding fractions, subtracting fractions, and simplifying the answer. This problem is a great example of how we use math in everyday situations, even in sports and training!
Practice Makes Perfect
To really master these skills, try some similar problems. You can change the total training time, the training times for the first two days, or even add another day to the training schedule. The more you practice, the more comfortable you'll become with working with fractions and mixed numbers.
Remember, the key to solving mathematical problems is to break them down into smaller, manageable steps. Understand the problem, identify the necessary operations, and work through each step carefully. And most importantly, don't be afraid to ask for help if you get stuck. Math can be challenging, but it's also incredibly rewarding when you finally solve a problem. Keep practicing, and you'll be a math whiz in no time!