Calculate Area Of Framed Photograph A Step-by-Step Guide

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A family photograph has a length of 14 2/5 cm and a breadth of 10 2/5 cm. It has a border of uniform width 2 3/5 cm. Find the area of the framed photograph.

Understanding the Problem

This is a problem involving area calculation, specifically dealing with a rectangle and a border around it. We need to determine the total area occupied by the photograph including the frame. To do this, we will first calculate the dimensions of the photograph with the frame, and then calculate the area.

The core challenge here lies in handling mixed fractions. The length and breadth of the photo, as well as the width of the border, are given in mixed fractions. To perform calculations easily, we will need to convert these mixed fractions into improper fractions. This will allow us to add and multiply these values effectively.

The problem also tests our understanding of how a border affects the overall dimensions of a shape. The border adds to each side of the photograph, so we need to consider this addition when calculating the framed dimensions. A common mistake is to only add the border width once, but it needs to be added twice – once for each side.

Finally, the problem requires a clear understanding of the formula for the area of a rectangle, which is length multiplied by breadth. Once we have the dimensions of the framed photograph, applying this formula will give us the answer.

Solution

  1. Convert Mixed Fractions to Improper Fractions

    Before we can perform any calculations, we need to convert the mixed fractions into improper fractions. This makes the arithmetic operations much simpler.

    • Length: 14 2/5 cm = (14 * 5 + 2) / 5 = 72/5 cm
    • Breadth: 10 2/5 cm = (10 * 5 + 2) / 5 = 52/5 cm
    • Border Width: 2 3/5 cm = (2 * 5 + 3) / 5 = 13/5 cm

  2. Calculate the Dimensions of the Framed Photograph

    The border adds width to both sides of the photograph, so we need to add the border width twice to both the length and the breadth.

    • Framed Length: 72/5 cm + 2 * (13/5 cm) = 72/5 cm + 26/5 cm = 98/5 cm
    • Framed Breadth: 52/5 cm + 2 * (13/5 cm) = 52/5 cm + 26/5 cm = 78/5 cm

  3. Calculate the Area of the Framed Photograph

    Now that we have the length and breadth of the framed photograph, we can calculate the area using the formula: Area = Length * Breadth

    • Area: (98/5 cm) * (78/5 cm) = 7644/25 sq cm

  4. Convert the Improper Fraction to a Mixed Fraction (Optional)

    While 7644/25 sq cm is a correct answer, it might be more intuitive to express it as a mixed fraction. This involves dividing 7644 by 25.

    • 7644 ÷ 25 = 305 with a remainder of 19
    • Therefore, 7644/25 sq cm = 305 19/25 sq cm

Final Answer

The area of the framed photograph is 7644/25 sq cm, or 305 19/25 sq cm.

In this comprehensive guide, we will delve into the step-by-step process of calculating the area of a framed family photograph. The photograph itself has dimensions given in mixed fractions, and it's surrounded by a border of uniform width, also expressed as a mixed fraction. Our goal is to determine the total area occupied by the framed photograph. This involves several key steps, including converting mixed fractions to improper fractions, calculating the dimensions of the framed photograph, and finally, applying the formula for the area of a rectangle. Understanding each of these steps is crucial for solving similar problems in geometry and mensuration.

Understanding the Problem Statement

To effectively tackle the area calculation, let’s first break down the problem statement. We have a family photograph, which essentially forms a rectangle. The length of this photograph is 14 2/5 cm, and the breadth is 10 2/5 cm. Now, this photograph has a border around it, a frame, which has a uniform width of 2 3/5 cm. This uniformity is crucial because it means the border's width is consistent on all sides. Our ultimate task is to find the total area of the photograph including this frame. This means we need to account for the extra space the frame adds to both the length and the breadth of the photograph.

The uniform width of the border is a key piece of information. It implies that the frame extends equally on all sides of the photograph. Therefore, the frame's width will need to be added twice – once for each side – to both the length and the breadth of the photograph. Failing to account for this double addition is a common mistake that can lead to an incorrect answer. Another important aspect is dealing with the mixed fractions. Mixed fractions can make calculations cumbersome, so converting them to improper fractions is a necessary step to simplify the process. This conversion allows us to perform arithmetic operations like addition and multiplication more easily and accurately.

Finally, let's highlight the core geometric concept at play: the area of a rectangle. The area of a rectangle is simply the product of its length and breadth. Once we have determined the overall dimensions of the framed photograph (including the frame), we can apply this formula to calculate the total area. This problem essentially combines fraction arithmetic with basic geometric principles, making it a good test of mathematical proficiency.

Converting Mixed Fractions to Improper Fractions

The initial hurdle in calculating the area of the framed photograph lies in handling mixed fractions. Mixed fractions, consisting of a whole number and a proper fraction, can be cumbersome to work with directly. Therefore, our first step is to convert these mixed fractions into improper fractions. An improper fraction is one where the numerator (the top number) is greater than or equal to the denominator (the bottom number). Converting to improper fractions simplifies arithmetic operations such as addition, subtraction, multiplication, and division.

Let's start with the length of the photograph, which is 14 2/5 cm. To convert this mixed fraction to an improper fraction, we follow a simple procedure. We multiply the whole number part (14) by the denominator of the fractional part (5), and then add the numerator of the fractional part (2). This result becomes the new numerator, and we keep the same denominator (5). So, the calculation is (14 * 5) + 2 = 70 + 2 = 72. Therefore, the improper fraction equivalent of 14 2/5 is 72/5.

Next, we'll convert the breadth of the photograph, which is 10 2/5 cm, using the same method. We multiply the whole number part (10) by the denominator (5), and then add the numerator (2). This gives us (10 * 5) + 2 = 50 + 2 = 52. So, the improper fraction equivalent of 10 2/5 is 52/5.

Lastly, we need to convert the width of the border, which is 2 3/5 cm. Again, we apply the same procedure: multiply the whole number (2) by the denominator (5), and then add the numerator (3). This gives us (2 * 5) + 3 = 10 + 3 = 13. Therefore, the improper fraction equivalent of 2 3/5 is 13/5. By converting all the mixed fractions to improper fractions, we now have the length as 72/5 cm, the breadth as 52/5 cm, and the border width as 13/5 cm. These improper fractions will make the subsequent calculations much easier and more straightforward.

Calculating the Dimensions of the Framed Photograph

With the mixed fractions now converted to improper fractions, we can proceed to calculate the dimensions of the framed photograph. This is a crucial step as it determines the overall size of the photograph including the border. Remember, the border adds to the dimensions on all sides of the photograph. Therefore, we need to account for the border width on both sides of both the length and the breadth.

Let's start with the length of the framed photograph. We know the original length of the photograph is 72/5 cm, and the border width is 13/5 cm. Since the border adds to both ends of the length, we need to add the border width twice to the original length. This means we need to add 13/5 cm + 13/5 cm, or 2 * (13/5 cm), to the original length. The calculation is as follows: Framed Length = 72/5 cm + 2 * (13/5 cm) = 72/5 cm + 26/5 cm. To add these fractions, since they have the same denominator, we simply add the numerators: 72 + 26 = 98. Therefore, the framed length is 98/5 cm.

Now, let's calculate the breadth of the framed photograph. Similarly, we know the original breadth is 52/5 cm, and the border width is 13/5 cm. Again, the border adds to both sides of the breadth, so we need to add the border width twice. The calculation is as follows: Framed Breadth = 52/5 cm + 2 * (13/5 cm) = 52/5 cm + 26/5 cm. Adding the numerators, we get 52 + 26 = 78. Therefore, the framed breadth is 78/5 cm. We now have the dimensions of the framed photograph: a length of 98/5 cm and a breadth of 78/5 cm. These dimensions, expressed as improper fractions, are ready for the final step of calculating the area.

Calculating the Area of the Framed Photograph

Now that we have determined the dimensions of the framed photograph, we can proceed to calculate its area. The area of a rectangle is given by the simple formula: Area = Length * Breadth. We have the framed length as 98/5 cm and the framed breadth as 78/5 cm. So, we simply need to multiply these two fractions together.

The calculation is as follows: Area = (98/5 cm) * (78/5 cm). To multiply fractions, we multiply the numerators together and the denominators together. The product of the numerators is 98 * 78 = 7644, and the product of the denominators is 5 * 5 = 25. Therefore, the area of the framed photograph is 7644/25 sq cm.

This result, 7644/25 sq cm, is a correct answer, but it is expressed as an improper fraction. While improper fractions are perfectly valid, it is often more intuitive to express the area as a mixed fraction. To convert 7644/25 to a mixed fraction, we need to divide the numerator (7644) by the denominator (25). This division will give us a whole number quotient and a remainder. The quotient will be the whole number part of the mixed fraction, and the remainder will be the numerator of the fractional part, with the same denominator (25).

Performing the division, 7644 ÷ 25 gives us a quotient of 305 and a remainder of 19. Therefore, the mixed fraction equivalent of 7644/25 is 305 19/25. So, the area of the framed photograph can also be expressed as 305 19/25 sq cm. Both 7644/25 sq cm and 305 19/25 sq cm represent the same area, but the mixed fraction form might be easier to visualize and understand in practical terms.

Final Answer and Conclusion

After meticulously working through each step, we have arrived at the solution. The area of the framed photograph is 7644/25 square centimeters, or equivalently, 305 19/25 square centimeters. This comprehensive calculation involved several key steps, each requiring careful attention to detail and a solid understanding of mathematical principles.

First, we tackled the challenge of dealing with mixed fractions by converting them into improper fractions. This conversion was crucial for simplifying the arithmetic operations and ensuring accuracy in subsequent calculations. We then moved on to calculating the dimensions of the framed photograph, which required us to consider the uniform width of the border and how it extends on all sides of the photograph. This step highlighted the importance of understanding how a border affects the overall dimensions of a shape, as the border width needed to be added twice to both the length and the breadth.

Next, we applied the fundamental formula for the area of a rectangle, which is the product of its length and breadth. This calculation gave us the area of the framed photograph in the form of an improper fraction. Finally, we converted the improper fraction into a mixed fraction, providing an alternative representation of the area that is often more intuitive and easier to interpret.

This problem serves as an excellent example of how different mathematical concepts, such as fraction arithmetic and geometric principles, can be combined to solve real-world problems. It also underscores the importance of breaking down complex problems into smaller, manageable steps, and carefully considering each step to arrive at the correct solution. The final answer, expressed in both improper and mixed fraction forms, provides a complete and thorough understanding of the area occupied by the framed family photograph.

In conclusion, the area of the framed photograph, incorporating the border, is calculated to be either 7644/25 sq cm or the equivalent mixed fraction representation of 305 19/25 sq cm. This problem showcases a practical application of basic geometric principles combined with arithmetic operations involving fractions.