Cake Fractions: How Much Did They Eat?

Aug 12, 2025 by Kenji Nakamura 39 views

**Unraveling the Cake Conundrum: A Delicious Dive into Fractions!**

Hey there, math enthusiasts and cake lovers! Ever found yourself staring at leftover slices, wondering exactly how much of that delicious dessert actually disappeared? Well, Carolina and her friends faced a similar sugary situation, and we're here to break down the fractional feast. So, grab a fork (or a pencil!), and let's dive into this tasty mathematical challenge.

The Cake Calamity: Deciphering the Delicious Data

Our cake conundrum centers around Carolina and her pals devouring multiple cakes, each sliced into a different number of portions. We're given three fractions representing the amount of cake consumed from each: 0.7, 9/12, and 6/8. Now, at first glance, these numbers might seem like a jumbled mess, but fear not! We're going to transform them into a common language – fractions with the same denominator – so we can easily add them up and discover the grand total of cake consumption.

First things first, let's tackle that decimal, 0.7. Remember, decimals are just another way of representing fractions, specifically fractions with denominators that are powers of ten. In this case, 0.7 is equivalent to 7 tenths, which we can write as 7/10. Now we have three fractions: 7/10, 9/12, and 6/8. Our next step is to find the least common multiple (LCM) of the denominators (10, 12, and 8). This LCM will become our common denominator, allowing us to compare and add the fractions seamlessly. The LCM of 10, 12, and 8 is 120. So, we'll convert each fraction to an equivalent fraction with a denominator of 120.

Let's start with 7/10. To get a denominator of 120, we need to multiply both the numerator and denominator by 12 (since 10 x 12 = 120). This gives us (7 x 12) / (10 x 12) = 84/120. Next up is 9/12. To get a denominator of 120, we multiply both the numerator and denominator by 10 (since 12 x 10 = 120). This results in (9 x 10) / (12 x 10) = 90/120. Lastly, we have 6/8. To get a denominator of 120, we multiply both the numerator and denominator by 15 (since 8 x 15 = 120). This gives us (6 x 15) / (8 x 15) = 90/120.

Now we've successfully transformed our original fractions into equivalent fractions with a common denominator: 84/120, 90/120, and 90/120. The final step is to add these fractions together. We simply add the numerators while keeping the denominator the same: 84/120 + 90/120 + 90/120 = (84 + 90 + 90) / 120 = 264/120. So, Carolina and her friends ate 264/120 of a cake. But wait, this fraction looks a bit unwieldy! Let's simplify it to get a better understanding of how much cake they devoured.

Simplifying the Sweet Sum: Reducing the Fraction to its Core

We've arrived at the fraction 264/120, representing the total amount of cake consumed. However, this fraction isn't in its simplest form. To make it easier to grasp, we need to reduce it by finding the greatest common divisor (GCD) of the numerator (264) and the denominator (120) and dividing both by it.

Let's find the GCD of 264 and 120. One way to do this is by listing the factors of each number and identifying the largest one they share. The factors of 120 are: 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, and 120. The factors of 264 are: 1, 2, 3, 4, 6, 8, 11, 12, 22, 24, 33, 44, 66, 88, 132, and 264. The greatest common factor they share is 24.

Now, we divide both the numerator and the denominator by 24: 264 / 24 = 11 and 120 / 24 = 5. This gives us the simplified fraction 11/5. This fraction is an improper fraction (the numerator is greater than the denominator), which means the group ate more than one whole cake. To express this in a more understandable way, we can convert it to a mixed number. To do this, we divide 11 by 5. 5 goes into 11 twice (2 x 5 = 10), with a remainder of 1. So, 11/5 is equal to 2 whole cakes and 1/5 of another cake. Carolina and her friends certainly enjoyed their sugary feast!

Alternatively, you can simplify the fraction 264/120 by dividing both the numerator and denominator by smaller common factors successively. For example, you can first divide both by 2, then by 2 again, then by 2 again, and finally by 3. This step-by-step approach can be helpful if you're not immediately sure of the greatest common divisor. No matter the method, arriving at the simplest form, 11/5 or 2 1/5 cakes, helps us clearly understand the total amount consumed.

The Sweet Conclusion: More Than Just a Slice!

So, there you have it! By converting decimals to fractions, finding a common denominator, adding the fractions, and simplifying the result, we've successfully calculated that Carolina and her friends devoured a total of 2 and 1/5 cakes. That's quite a feast! This exercise highlights the practical application of fractions in everyday life, even in situations as delicious as sharing cake. It shows us that even seemingly complex problems can be broken down into smaller, manageable steps. Plus, who knew math could be so mouthwatering?

This journey through cake fractions not only reinforced our understanding of mathematical principles but also served as a reminder of the joy of problem-solving. From decimals to fractions, common denominators to simplification, each step revealed a clearer picture of the total cake consumption. This type of hands-on approach to learning, where abstract concepts like fractions are applied to tangible scenarios like sharing cake, helps to solidify understanding and make math less intimidating and more engaging. So, the next time you're faced with a mathematical challenge, remember Carolina's cake conundrum – and know that with a little bit of fraction finesse, you can conquer any equation!

In conclusion, the process of figuring out how much cake Carolina and her friends ate demonstrates the power of fractions in representing parts of a whole and the importance of being able to manipulate fractions to solve real-world problems. Whether it's dividing a pizza, measuring ingredients for a recipe, or, in this case, calculating the total amount of cake consumed, fractions are an essential tool in our daily lives. Understanding how to work with them, as we've done in this example, opens up a whole world of mathematical possibilities and allows us to confidently tackle a variety of challenges, both in the classroom and beyond. And remember, even if the math gets a little tricky, there's always a sweet reward at the end – like a slice of cake!

Keywords Extracted & Clarified

Original Question: ¿en total cuanta torta se comieron Carolina y sus amigos?
Improved Question: If all cakes have the same shape and size, and each cake was divided into different portions, how much cake did Carolina and her friends eat in total?

Here, I've rephrased the original question to be more explicit and easier to understand. It clarifies the context (all cakes are the same size) and directly asks for the total amount of cake eaten.