Combined Work Hours: Ann, Mary & John

Combined Work Hours: Ann, Mary & John

Hey guys! Ever wondered how much time your team really puts in? Today, we're diving into a classic math problem that's all about combining efforts, just like in a real-world project. We've got Ann, Mary, and John, and we need to figure out their total combined work hours. This isn't just about adding numbers; it's about understanding how individual contributions add up to a significant whole. So, grab your calculators (or your trusty brains!) and let's crunch these numbers together. We'll break down each person's hours, find a common ground for our fractions, and sum it all up to get that sweet, sweet total.

Ann's Work Hours: A Solid Start

First up, let's talk about Ann. She's put in a solid 5 rac{1}{2} hours of work. That's five full hours and then half of another hour. To make things easier later when we combine everyone's hours, it's often helpful to convert these mixed numbers into improper fractions. For Ann, 5 rac{1}{2} becomes $(5 imes 2 + 1) / 2$, which is $11/2$ hours. Ann's contribution is significant, and it's the first piece of our puzzle. We need to keep this fraction handy as we move on to Mary and John. Thinking about work hours in fractions might seem a bit abstract, but it's super important for accuracy, especially when dealing with parts of an hour. A half-hour is 30 minutes, so Ann has worked for 5 hours and 30 minutes. This initial figure is crucial, and understanding it helps us appreciate the effort each person brings to the table. It’s the foundation upon which we'll build the total.

Mary's Contribution: A Little More Time

Next, we have Mary. She's been working for 6 rac{1}{3} hours. That's a bit more than Ann, clocking in six full hours plus a third of another hour. To convert Mary's hours into an improper fraction, we do $(6 imes 3 + 1) / 3$, which gives us $19/3$ hours. Mary's effort adds another substantial chunk to our total. A third of an hour is 20 minutes, so Mary has worked for 6 hours and 20 minutes. This detail is important for visualizing the time spent, but for calculation, we stick with the fraction $19/3$. As we gather each person's hours, remember that precision is key. These aren't just random numbers; they represent actual time invested, and accurately summing them up gives us a true picture of the collective effort. Mary's $19/3$ hours are now ready to be combined with Ann's and John's.

John's Effort: Precise and Steady

Finally, let's look at John. He's worked for 4 rac{1}{4} hours. This means four full hours and a quarter of an hour. To turn this into an improper fraction, we calculate $(4 imes 4 + 1) / 4$, resulting in $17/4$ hours. John's contribution, while seemingly less than Ann's or Mary's in whole hours, is still a vital part of the total. A quarter of an hour is 15 minutes, so John has worked for 4 hours and 15 minutes. Even though we can easily picture this in minutes, keeping it as the fraction $17/4$ is essential for our next step: adding everything together. Each fraction represents a specific duration, and when we add them, we get the complete picture of their combined productivity. John's $17/4$ hours are the final piece we need.

Combining the Hours: The Grand Total

Now for the main event, guys: combining all the hours! We need to add Ann's $11/2$ hours, Mary's $19/3$ hours, and John's $17/4$ hours. To add fractions, they must have a common denominator. Let's find the least common multiple (LCM) of our denominators: 2, 3, and 4. The LCM of 2, 3, and 4 is 12.

Now, we convert each fraction to have a denominator of 12:

• Ann: $11/2 = (11 imes 6) / (2 imes 6) = 66/12$
• Mary: $19/3 = (19 imes 4) / (3 imes 4) = 76/12$
• John: $17/4 = (17 imes 3) / (4 imes 3) = 51/12$

Now we can add the numerators: $66/12 + 76/12 + 51/12 = (66 + 76 + 51) / 12 = 193/12$ hours.

Converting Back to Mixed Numbers: The Final Answer

We have our total as an improper fraction: $193/12$. To make this easier to understand, let's convert it back into a mixed number. We divide 193 by 12:

$193 ilde{A} ext{ extdivide} 12 = 16$ with a remainder of $1$.

So, $193/12$ is equal to 16 rac{1}{12} hours.

This means that Ann, Mary, and John have worked a combined total of 16 rac{1}{12} hours. It's amazing how adding up individual efforts leads to such a substantial total! This result helps us understand the collective output and appreciate the teamwork involved. Whether it's for a school project or a real-world job, adding up hours like this is a fundamental skill. So, the correct option is B) 16 rac{1}{12}.

Why This Matters: Real-World Applications

Understanding how to combine hours, especially with fractions, is super useful, guys! Think about project management, where you need to estimate how long a task will take based on how long each team member spends on it. Or maybe you're splitting costs for something where everyone contributed a different amount of time – figuring out the total time helps make things fair. Even if you're just tracking your own hobbies or chores, adding up time spent accurately can give you a real sense of accomplishment. For example, if you spent 2 rac{1}{2} hours coding one day and 3 rac{3}{4} hours the next, you'd want to know your total coding time to see your progress. The math we did here—converting mixed numbers to improper fractions, finding a common denominator, adding fractions, and converting back—is a core skill that pops up everywhere. It’s not just about getting the right answer on a test; it’s about building a practical skill set that makes you more capable in planning, managing, and understanding the value of time. So next time you see mixed numbers, don't shy away – embrace the challenge, and you'll find the results incredibly rewarding. This problem, while simple, is a great reminder of the power of basic arithmetic in solving everyday challenges and understanding contributions.