Solving Fraction Problems How Much Cake Did They Eat

Fractions are a fundamental concept in mathematics, representing parts of a whole. In this article, we will delve into a fraction-based problem involving three girls—Nana, Hana, and Cecil—who share a cake at a party. This problem will help us understand how to add fractions with different denominators and arrive at the correct solution. Understanding fractions is not just essential for academic success but also for everyday life, from cooking and baking to managing finances and planning projects. This article will break down the problem step-by-step, ensuring that readers of all levels can grasp the concepts involved and apply them to similar scenarios. The goal is to make fractions less daunting and more accessible, highlighting their relevance in practical situations. We aim to provide a clear and comprehensive explanation, making the process of solving fraction problems straightforward and enjoyable. By the end of this article, you should feel confident in your ability to tackle similar problems and understand the logic behind adding fractions with different denominators.

Problem Statement

The problem we're tackling involves calculating the total fraction of cake eaten by Nana, Hana, and Cecil at a party. Nana ate $\frac{1}{8}$ of the cake, Hana ate $\frac{2}{5}$, and Cecil consumed $\frac{3}{10}$. The question is: what fraction of the cake did the three girls eat in total? This problem is a classic example of adding fractions, a concept that is crucial in various fields, including cooking, engineering, and finance. To solve this, we need to find a common denominator for the fractions, which will allow us to add them together easily. A common denominator is a number that all the denominators (the bottom numbers of the fractions) can divide into evenly. Once we have a common denominator, we can add the numerators (the top numbers of the fractions) to find the total fraction of the cake that was eaten. This problem not only tests our ability to add fractions but also our understanding of equivalent fractions and the importance of finding a common ground when dealing with different fractional parts. By working through this problem, we reinforce our skills in fraction manipulation and enhance our problem-solving abilities in mathematics.

Breaking Down the Fractions

Before we jump into solving the problem, let's break down the fractions individually to better understand what they represent. Nana ate $\frac{1}{8}$ of the cake. This means the cake was divided into 8 equal parts, and Nana ate 1 of those parts. The fraction $\frac{1}{8}$ is a relatively small portion, representing one-eighth of the whole cake. Hana, on the other hand, ate $\frac{2}{5}$ of the cake. This implies the cake was divided into 5 equal parts, and Hana ate 2 of those parts. The fraction $\frac{2}{5}$ is larger than $\frac{1}{8}$, representing a more significant portion of the cake. Finally, Cecil ate $\frac{3}{10}$ of the cake. This means the cake was divided into 10 equal parts, and Cecil ate 3 of those parts. The fraction $\frac{3}{10}$ is smaller than $\frac{2}{5}$ but larger than $\frac{1}{8}$. Understanding the relative sizes of these fractions is crucial for estimating the final answer. We can visualize these fractions as slices of a cake, each representing a different portion. By comparing these portions, we get a better sense of how much cake each girl consumed and how much they ate collectively. This initial analysis sets the stage for the next step, which involves adding these fractions together.

Finding the Common Denominator

To add fractions, they must have a common denominator. This is a crucial step because it allows us to add the numerators directly. In our problem, we need to find a common denominator for the fractions $\frac{1}{8}$, $\frac{2}{5}$, and $\frac{3}{10}$. The denominators are 8, 5, and 10. To find the common denominator, we need to identify the least common multiple (LCM) of these numbers. The LCM is the smallest number that is a multiple of all the given numbers. We can find the LCM by listing the multiples of each number and identifying the smallest one they have in common. Multiples of 8 are: 8, 16, 24, 32, 40, 48, ... Multiples of 5 are: 5, 10, 15, 20, 25, 30, 35, 40, 45, ... Multiples of 10 are: 10, 20, 30, 40, 50, ... By examining these multiples, we can see that the least common multiple of 8, 5, and 10 is 40. Therefore, 40 is our common denominator. This means we need to convert each fraction into an equivalent fraction with a denominator of 40. Finding the common denominator is a foundational step in fraction arithmetic, and mastering this skill is essential for solving more complex problems.

Converting to Equivalent Fractions

Now that we have our common denominator of 40, the next step is to convert each fraction into an equivalent fraction with this denominator. An equivalent fraction represents the same value but has a different numerator and denominator. To convert $\frac{1}{8}$ to an equivalent fraction with a denominator of 40, we need to multiply both the numerator and the denominator by the same number. In this case, we need to multiply 8 by 5 to get 40. So, we also multiply the numerator 1 by 5. This gives us $\frac{1 \times 5}{8 \times 5} = \frac{5}{40}$. Next, we convert $\frac{2}{5}$ to an equivalent fraction with a denominator of 40. We need to multiply 5 by 8 to get 40. So, we also multiply the numerator 2 by 8. This gives us $\frac{2 \times 8}{5 \times 8} = \frac{16}{40}$. Finally, we convert $\frac{3}{10}$ to an equivalent fraction with a denominator of 40. We need to multiply 10 by 4 to get 40. So, we also multiply the numerator 3 by 4. This gives us $\frac{3 \times 4}{10 \times 4} = \frac{12}{40}$. Now we have three equivalent fractions: $\frac{5}{40}$, $\frac{16}{40}$, and $\frac{12}{40}$. These fractions represent the same portions of the cake as the original fractions but now have a common denominator, making it easy to add them together. Converting fractions to equivalent forms is a fundamental skill in fraction arithmetic and is crucial for performing operations like addition and subtraction.

Adding the Fractions

With our fractions now having a common denominator, we can proceed to add them together. We have the equivalent fractions: $\frac{5}{40}$ (Nana's share), $\frac{16}{40}$ (Hana's share), and $\frac{12}{40}$ (Cecil's share). To add these fractions, we simply add the numerators while keeping the denominator the same. So, we have: $\frac{5}{40} + \frac{16}{40} + \frac{12}{40} = \frac{5 + 16 + 12}{40}$. Adding the numerators, we get $5 + 16 + 12 = 33$. Therefore, the sum of the fractions is $\frac{33}{40}$. This means that the three girls, Nana, Hana, and Cecil, ate a total of $\frac{33}{40}$ of the cake. This fraction represents a significant portion of the cake, indicating that the girls ate most of it. The process of adding fractions with a common denominator is straightforward: add the numerators and keep the denominator. This step is a critical part of solving fraction problems and demonstrates the power of using equivalent fractions to simplify calculations. Now that we have the final fraction, we can confidently answer the original question.

Simplifying the Result

In many mathematical problems, it's important to simplify the final result, especially when dealing with fractions. Simplifying a fraction means reducing it to its lowest terms, where the numerator and the denominator have no common factors other than 1. In our case, the fraction we obtained after adding the individual portions of cake eaten by Nana, Hana, and Cecil is $\frac{33}{40}$. To determine if this fraction can be simplified, we need to find the greatest common divisor (GCD) of the numerator (33) and the denominator (40). The GCD is the largest positive integer that divides both numbers without leaving a remainder. The factors of 33 are 1, 3, 11, and 33. The factors of 40 are 1, 2, 4, 5, 8, 10, 20, and 40. By comparing the factors, we can see that the only common factor between 33 and 40 is 1. Since the GCD is 1, the fraction $\frac{33}{40}$ is already in its simplest form. This means we cannot reduce it further without changing its value. Therefore, the final answer remains $\frac{33}{40}$. Understanding how to simplify fractions is crucial for presenting answers in their most concise and understandable form. In this case, the fraction is already simplified, so we know we have our final answer.

The Answer

After carefully calculating and simplifying, we have arrived at the answer to our problem. The question was: what fraction of the cake did the three girls, Nana, Hana, and Cecil, eat in total? We found that Nana ate $\frac{1}{8}$ of the cake, Hana ate $\frac{2}{5}$, and Cecil ate $\frac{3}{10}$. By adding these fractions together, we obtained a total fraction of $\frac{33}{40}$. We also confirmed that this fraction is in its simplest form, meaning it cannot be reduced any further. Therefore, the final answer is $\frac{33}{40}$. This means that the three girls ate 33 out of 40 parts of the cake. This problem demonstrates the importance of understanding fractions and how to perform basic operations like addition. By following the steps of finding a common denominator, converting fractions to equivalent forms, adding the numerators, and simplifying the result, we were able to solve the problem accurately. This exercise not only reinforces our math skills but also highlights how fractions are used in everyday situations, such as sharing a cake at a party.

Selecting the Correct Option

Now that we've calculated the total fraction of cake eaten by the three girls, let's look at the given options and select the correct one. The options provided are:

A. $\frac{33}{40}$ B. $\frac{23}{40}$ C. $\frac{13}{40}$ D. $\frac{3}{40}$

Our calculated answer is $\frac{33}{40}$. Comparing this to the options, we can see that option A, $\frac{33}{40}$, matches our calculated result. Therefore, option A is the correct answer. The other options represent different fractions and do not match the total fraction of cake eaten by the girls. This step is crucial in problem-solving as it ensures that we select the correct answer from the given choices. By carefully reviewing our calculations and comparing them to the options, we can confidently identify the solution that aligns with our work. This final step solidifies our understanding of the problem and our ability to apply mathematical concepts to real-world scenarios.

Conclusion

In conclusion, we have successfully solved the problem of determining the fraction of cake eaten by Nana, Hana, and Cecil at the party. We started by understanding the individual fractions each girl ate: $\frac{1}{8}$, $\frac{2}{5}$, and $\frac{3}{10}$. The key to solving this problem was finding a common denominator, which allowed us to add the fractions together. We identified 40 as the least common multiple of 8, 5, and 10. Next, we converted each fraction to an equivalent fraction with a denominator of 40, resulting in $\frac{5}{40}$, $\frac{16}{40}$, and $\frac{12}{40}$. Adding these fractions, we found that the three girls ate a total of $\frac{33}{40}$ of the cake. We then simplified the fraction to ensure it was in its lowest terms, confirming that $\frac{33}{40}$ is indeed the simplest form. Finally, we compared our calculated answer to the given options and correctly selected option A, $\frac{33}{40}$, as the answer. This problem highlights the importance of understanding fractions and the steps involved in adding them. Mastering these concepts is crucial for success in mathematics and for applying math skills to everyday situations. By breaking down the problem into manageable steps, we were able to solve it effectively and accurately.