Mabijo's Wall Painting Project A Fraction-by-Fraction Analysis
What fraction of the wall was painted on the first day? What fraction of the wall was painted after the second day? What fraction of the wall remains to be painted?
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Mabijo's Wall Painting Project A Fraction-by-Fraction Breakdown
Mabijo has embarked on a wall-painting project, tackling it in stages. On the first day, Mabijo successfully painted $\frac{2}{5}$ of the wall. The following day, Mabijo continued the endeavor, completing an additional $\frac{1}{3}$ of the wall. This article delves into the fractions of the wall painted, providing a detailed analysis of Mabijo's progress. We will explore the fractions painted on each day and the cumulative fraction painted after two days. Understanding fractions is crucial in various real-life applications, including home improvement projects like this one.
Before diving into the calculations, let's recap the basics of fractions. A fraction represents a part of a whole. It consists of two parts: the numerator (the top number) and the denominator (the bottom number). The denominator indicates the total number of equal parts the whole is divided into, and the numerator indicates how many of those parts are being considered. In this case, the 'whole' is the entire wall that Mabijo is painting. When adding fractions, it's essential that they have a common denominator. This means that the fractions must represent parts of the same 'whole' divided into the same number of pieces. If the fractions don't have a common denominator, we need to find the least common multiple (LCM) of the denominators and convert the fractions accordingly. This ensures that we are adding like terms, just as we do in algebra when combining variables with the same exponent. In our scenario, we'll need to find a common denominator for $\frac{2}{5}$ and $\frac{1}{3}$ before we can determine the total fraction of the wall painted. This process is fundamental to accurately assessing Mabijo's progress and understanding the remaining work.
(a) Fraction of the Wall Painted After the First Day
On the first day, the fraction of the wall painted by Mabijo was $\frac{2}{5}$. This means that out of the entire wall, which can be conceptually divided into 5 equal parts, Mabijo managed to paint 2 of those parts. This fraction, $\frac{2}{5}$, directly represents the proportion of the wall that was covered in paint by the end of the first day. There's no further calculation needed for this part; the given fraction itself is the answer. It's important to note that this fraction serves as the starting point for understanding the overall progress of the painting project. It sets the stage for calculating the cumulative progress after subsequent days. For instance, to determine the total fraction of the wall painted after the second day, we'll need to add the fraction painted on the first day, $\frac{2}{5}$, to the fraction painted on the second day. This will involve finding a common denominator and performing the addition, which we will address in the following sections. Understanding the fraction painted on the first day is crucial for tracking the overall progress and planning for the completion of the project. It provides a clear benchmark against which to measure future progress and make informed decisions about the remaining work.
(b) Fraction of the Wall Painted After the Second Day
To determine the fraction of the wall painted after the second day, we need to add the fractions painted on both the first and second days. Mabijo painted $\frac2}{5}$ of the wall on the first day and $\frac{1}{3}$ of the wall on the second day. Therefore, we need to calculate $\frac{2}{5} + \frac{1}{3}$. To add these fractions, we first need to find a common denominator. The least common multiple (LCM) of 5 and 3 is 15. We convert both fractions to have this denominator. To convert $\frac{2}{5}$ to a fraction with a denominator of 15, we multiply both the numerator and the denominator by 35 \times 3} = \frac{6}{15}$. Similarly, to convert $\frac{1}{3}$ to a fraction with a denominator of 15, we multiply both the numerator and the denominator by 53 \times 5} = \frac{5}{15}$. Now that both fractions have the same denominator, we can add them{15} + \frac{5}{15} = \frac{6 + 5}{15} = \frac{11}{15}$. Thus, after the second day, Mabijo has painted $\frac{11}{15}$ of the wall. This fraction represents the total portion of the wall that has been painted after two days of work. It provides a clear indication of Mabijo's progress and can be used to calculate the remaining fraction of the wall that needs to be painted. Understanding this cumulative fraction is essential for project planning and assessing the overall progress towards completion.
(c) Fraction of the Wall Remaining to Be Painted
To find the fraction of the wall remaining to be painted, we need to subtract the fraction of the wall that has already been painted from the whole wall, which can be represented as 1. We know that after the second day, Mabijo has painted $\frac11}{15}$ of the wall. Therefore, we need to calculate $1 - \frac{11}{15}$. To subtract a fraction from a whole number, we can rewrite the whole number as a fraction with the same denominator as the fraction being subtracted. In this case, we can rewrite 1 as $\frac{15}{15}$. Now we can perform the subtraction{15} - \frac{11}{15} = \frac{15 - 11}{15} = \frac{4}{15}$. Therefore, the fraction of the wall remaining to be painted is $\frac{4}{15}$. This fraction represents the portion of the wall that still needs to be covered with paint to complete the project. It's important to understand this fraction to plan the remaining work and estimate the time and resources required to finish the job. Knowing the fraction remaining allows for effective project management and ensures that the painting project can be completed efficiently. In practical terms, this means Mabijo has a little less than a third of the wall left to paint. This information can help Mabijo gauge how much more time and effort will be needed to finish the task, allowing for better planning and resource allocation.
In conclusion, Mabijo's wall painting project provides a practical example of how fractions are used in everyday situations. After the first day, Mabijo painted $\frac{2}{5}$ of the wall. By the end of the second day, the cumulative progress reached $\frac{11}{15}$ of the wall. This was calculated by adding the fractions painted on both days, requiring the identification of a common denominator and the subsequent addition of the numerators. Finally, to determine the remaining work, we subtracted the painted fraction from the whole, revealing that $\frac{4}{15}$ of the wall still needs to be painted. This exercise not only demonstrates the application of fraction arithmetic but also highlights the importance of fractions in project planning and progress tracking. Understanding fractions allows for accurate measurement and estimation, crucial skills in various fields, from home improvement to professional project management. The ability to work with fractions empowers individuals to break down complex tasks into manageable parts, track progress effectively, and make informed decisions about resource allocation and time management. Mabijo's project serves as a tangible illustration of these concepts, reinforcing the practical significance of fraction calculations in real-world scenarios.