Solving Tini's Ribbon Problem A Fraction Word Problem Explained

by ADMIN 64 views

Hey guys! Have you ever encountered a word problem that seemed like a tangled mess of numbers? Today, we're going to unravel one of those problems together, focusing on fractions and how to solve them. This isn't just about getting the right answer; it's about understanding the process, so you can tackle any similar problem with confidence. Let's dive in!

Understanding the Problem: Tini's Ribbon

So, our problem revolves around Tini, who loves crafting and uses ribbons for her projects. Tini initially has a ribbon that measures 5 1/2 meters. Picture that in your mind – five and a half meters of colorful ribbon, ready to be transformed into something beautiful. But that's not all! Tini, being the avid crafter she is, decides to buy some more ribbon, adding another 1 1/3 meters to her collection. Now, we're starting to see the total amount of ribbon Tini has to work with.

But wait, there's more to the story. Tini doesn't just hoard ribbons; she puts them to good use. She uses 2 3/4 meters of ribbon to create lovely floral decorations. Imagine the intricate designs and vibrant colors! And then, she uses another 2 1/6 meters to wrap gifts, adding a personal touch to her presents. Now, the big question looms: How much ribbon does Tini have left after all her crafting adventures?

This problem might seem daunting at first, with all the mixed numbers and different operations involved. But don't worry, we're going to break it down step by step, making it super easy to follow. We'll be using some fundamental math concepts, but we'll explain everything clearly, so you can understand not just the "how" but also the "why" behind each step. Think of it as a journey, where each step brings us closer to the final answer. So, let's get started and figure out how much ribbon Tini has left!

Step 1: Finding the Total Ribbon Length

The first thing we need to figure out is the total amount of ribbon Tini had before she started using it for her projects. Remember, she started with 5 1/2 meters and then bought an additional 1 1/3 meters. To find the total, we need to add these two amounts together. Sounds simple, right? But adding mixed numbers can be a little tricky if we don't follow the right steps. That's where our understanding of fractions comes into play. Before we can add these mixed numbers, we need to convert them into improper fractions. This makes the addition process much smoother. So, let's tackle that conversion first.

  • Converting Mixed Numbers to Improper Fractions:

    A mixed number, like 5 1/2, is a combination of a whole number (5) and a fraction (1/2). To convert it into an improper fraction, we multiply the whole number by the denominator of the fraction and then add the numerator. This result becomes our new numerator, and we keep the same denominator. Let's apply this to our numbers:

    • For 5 1/2: (5 * 2) + 1 = 11. So, 5 1/2 becomes 11/2.
    • For 1 1/3: (1 * 3) + 1 = 4. So, 1 1/3 becomes 4/3.

    Now we have two improper fractions: 11/2 and 4/3. We're one step closer to finding the total ribbon length!

  • Adding the Improper Fractions:

    Now that we have our improper fractions, we can add them together. But here's a little catch: we can only add fractions if they have the same denominator. Right now, our denominators are 2 and 3. So, we need to find the least common multiple (LCM) of 2 and 3. The LCM is the smallest number that both 2 and 3 can divide into evenly. In this case, the LCM of 2 and 3 is 6.

    To get a denominator of 6 for both fractions, we need to multiply the numerator and denominator of each fraction by the appropriate number:

    • For 11/2: We multiply both the numerator and denominator by 3 (because 2 * 3 = 6). So, 11/2 becomes 33/6.
    • For 4/3: We multiply both the numerator and denominator by 2 (because 3 * 2 = 6). So, 4/3 becomes 8/6.

    Now we can finally add our fractions: 33/6 + 8/6. To add fractions with the same denominator, we simply add the numerators and keep the denominator the same. So, 33/6 + 8/6 = 41/6.

    Therefore, the total length of ribbon Tini had is 41/6 meters. But let's not forget that fractions can also be expressed as mixed numbers. So, let's convert 41/6 back into a mixed number to get a better sense of the length.

  • Converting Back to a Mixed Number:

    To convert an improper fraction back into a mixed number, we divide the numerator by the denominator. The quotient becomes our whole number, the remainder becomes our new numerator, and we keep the same denominator. So, let's divide 41 by 6:

    • 41 divided by 6 is 6 with a remainder of 5.

    This means that 41/6 is equal to 6 5/6. So, Tini had a total of 6 5/6 meters of ribbon. We've completed the first major step in solving our problem! Now we know the total amount of ribbon Tini started with. Let's move on to the next step: figuring out how much ribbon she used.

Step 2: Calculating the Total Ribbon Used

Now that we know Tini started with a total of 6 5/6 meters of ribbon, we need to figure out how much she used for her crafting projects. Remember, she used 2 3/4 meters for floral decorations and 2 1/6 meters for wrapping gifts. To find the total amount of ribbon used, we need to add these two amounts together. Just like before, we're dealing with mixed numbers, so we'll need to convert them into improper fractions first. Are you ready to put your fraction conversion skills to the test again?

  • Converting Mixed Numbers to Improper Fractions (Again!):

    Let's quickly convert our mixed numbers into improper fractions using the same method we used before:

    • For 2 3/4: (2 * 4) + 3 = 11. So, 2 3/4 becomes 11/4.
    • For 2 1/6: (2 * 6) + 1 = 13. So, 2 1/6 becomes 13/6.

    Great! We now have two improper fractions: 11/4 and 13/6. We're on our way to finding the total ribbon used.

  • Adding the Improper Fractions (Finding a Common Denominator):

    Just like before, we can only add these fractions if they have the same denominator. Our denominators are 4 and 6, so we need to find the least common multiple (LCM) of 4 and 6. What's the smallest number that both 4 and 6 can divide into evenly? That's right, it's 12.

    Now, let's get a denominator of 12 for both fractions:

    • For 11/4: We multiply both the numerator and denominator by 3 (because 4 * 3 = 12). So, 11/4 becomes 33/12.
    • For 13/6: We multiply both the numerator and denominator by 2 (because 6 * 2 = 12). So, 13/6 becomes 26/12.

    Now we can add our fractions: 33/12 + 26/12. Remember, we add the numerators and keep the denominator the same. So, 33/12 + 26/12 = 59/12.

    Therefore, Tini used a total of 59/12 meters of ribbon. Let's convert this improper fraction back into a mixed number to get a better sense of the amount.

  • Converting Back to a Mixed Number (One More Time!):

    Let's divide 59 by 12:

    • 59 divided by 12 is 4 with a remainder of 11.

    This means that 59/12 is equal to 4 11/12. So, Tini used a total of 4 11/12 meters of ribbon. We're making great progress! We now know the total amount of ribbon Tini started with and the total amount she used. What's the final step?

Step 3: Calculating the Remaining Ribbon

We've reached the final stage of our problem! We know that Tini started with 6 5/6 meters of ribbon and used 4 11/12 meters. To find out how much ribbon she has left, we need to subtract the amount she used from the amount she started with. This is where our fraction skills will shine once again.

  • Subtracting the Mixed Numbers (Converting to Improper Fractions... Again!):

    You guessed it! We need to convert our mixed numbers into improper fractions before we can subtract them. We've already converted these numbers once, but let's do it again to refresh our memory:

    • 6 5/6 was 41/6
    • 4 11/12 was 59/12

  • Subtracting the Improper Fractions (Finding a Common Denominator... You Know the Drill!):

    To subtract fractions, they need to have the same denominator. Our denominators are 6 and 12. What's the least common multiple of 6 and 12? It's 12!

    Let's get a denominator of 12 for both fractions:

    • For 41/6: We multiply both the numerator and denominator by 2 (because 6 * 2 = 12). So, 41/6 becomes 82/12.
    • 59/12 already has a denominator of 12, so we don't need to change it.

    Now we can subtract our fractions: 82/12 - 59/12. We subtract the numerators and keep the denominator the same. So, 82/12 - 59/12 = 23/12.

    Therefore, Tini has 23/12 meters of ribbon left. But, as always, let's convert this improper fraction back into a mixed number to get a more intuitive understanding of the amount.

  • Converting Back to a Mixed Number (The Final Conversion!):

    Let's divide 23 by 12:

    • 23 divided by 12 is 1 with a remainder of 11.

    This means that 23/12 is equal to 1 11/12. So, Tini has 1 11/12 meters of ribbon left!

Conclusion: Tini's Ribbon Remnants

Phew! We made it! We successfully navigated through this fraction-filled problem and found the answer. Tini has 1 11/12 meters of ribbon left after her crafting adventures. That's quite a bit of ribbon still available for her next project! This problem might have seemed complicated at first, but by breaking it down into smaller, manageable steps, we were able to solve it with ease. Remember, the key to tackling word problems is to understand the problem, break it down, and solve it step by step. And don't be afraid of fractions – they're just numbers like any other! So, the next time you encounter a similar problem, remember Tini and her ribbons, and you'll be crafting solutions in no time!