Area Of Framed Photograph Calculation With Border Width

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Introduction

In this article, we will delve into a practical geometry problem involving a family photograph and its frame. We'll explore how to calculate the area of the framed photograph, considering the dimensions of the photograph itself and the uniform width of the border. This problem combines basic arithmetic operations with the application of area formulas, making it an excellent exercise in mathematical reasoning and problem-solving.

Problem Statement: Unveiling the Dimensions and the Frame

Let's begin by carefully examining the problem statement. We have a family photograph with specific dimensions: a length of 14 2/5 cm and a breadth of 10 2/5 cm. These mixed fractions need to be converted into improper fractions for easier calculations. The photograph is encased in a frame with a uniform width of 2 3/5 cm. Our primary goal is to determine the area of the entire framed photograph, encompassing both the photograph itself and the frame surrounding it.

Converting Mixed Fractions to Improper Fractions: A Crucial First Step

Before we proceed with area calculations, it's essential to convert the mixed fractions representing the dimensions into improper fractions. This conversion simplifies the multiplication and addition operations required later in the solution. Let's break down the conversion process:

  • Length: 14 2/5 cm

    • Multiply the whole number (14) by the denominator (5): 14 * 5 = 70
    • Add the numerator (2) to the result: 70 + 2 = 72
    • Place the sum (72) over the original denominator (5): 72/5 cm

  • Breadth: 10 2/5 cm

    • Multiply the whole number (10) by the denominator (5): 10 * 5 = 50
    • Add the numerator (2) to the result: 50 + 2 = 52
    • Place the sum (52) over the original denominator (5): 52/5 cm

  • Frame Width: 2 3/5 cm

    • Multiply the whole number (2) by the denominator (5): 2 * 5 = 10
    • Add the numerator (3) to the result: 10 + 3 = 13
    • Place the sum (13) over the original denominator (5): 13/5 cm

Now that we've converted all the mixed fractions into improper fractions, we have the following:

  • Photograph Length: 72/5 cm
  • Photograph Breadth: 52/5 cm
  • Frame Width: 13/5 cm

This conversion is a foundational step, ensuring accuracy and ease in the subsequent calculations.

Determining the Dimensions of the Framed Photograph: Incorporating the Border

Now that we have the dimensions of the photograph and the width of the frame, we can calculate the overall dimensions of the framed photograph. The frame adds to the length and breadth on both sides, so we need to account for this additional width. Let's break down the calculation:

Calculating the Framed Length

The framed length is the sum of the photograph's length and twice the frame width (since the frame extends on both sides). Mathematically, this can be represented as:

Framed Length = Photograph Length + 2 * Frame Width

Substituting the values we have:

Framed Length = (72/5) cm + 2 * (13/5) cm

First, we multiply the frame width by 2:

2 * (13/5) cm = 26/5 cm

Now, we add this to the photograph's length:

Framed Length = (72/5) cm + (26/5) cm

Since the denominators are the same, we can simply add the numerators:

Framed Length = (72 + 26) / 5 cm

Framed Length = 98/5 cm

Calculating the Framed Breadth

Similarly, the framed breadth is the sum of the photograph's breadth and twice the frame width. The formula is:

Framed Breadth = Photograph Breadth + 2 * Frame Width

Substituting the known values:

Framed Breadth = (52/5) cm + 2 * (13/5) cm

Again, we first multiply the frame width by 2:

2 * (13/5) cm = 26/5 cm

Then, we add this to the photograph's breadth:

Framed Breadth = (52/5) cm + (26/5) cm

Adding the numerators since the denominators are the same:

Framed Breadth = (52 + 26) / 5 cm

Framed Breadth = 78/5 cm

Now, we have the dimensions of the framed photograph:

  • Framed Length: 98/5 cm
  • Framed Breadth: 78/5 cm

With these dimensions, we are well-prepared to calculate the area of the framed photograph.

Calculating the Area of the Framed Photograph: Applying the Area Formula

Now that we have determined the dimensions of the framed photograph, we can proceed to calculate its area. The area of a rectangle (which is the shape of our framed photograph) is given by the formula:

Area = Length * Breadth

In our case, the length is the framed length (98/5 cm) and the breadth is the framed breadth (78/5 cm). Substituting these values into the formula, we get:

Area = (98/5) cm * (78/5) cm

To multiply fractions, we multiply the numerators together and the denominators together:

Area = (98 * 78) / (5 * 5) cm²

Now, let's perform the multiplication:

98 * 78 = 7644

5 * 5 = 25

So, the area becomes:

Area = 7644 / 25 cm²

This result is an improper fraction, and while it is a correct representation of the area, it's often more practical to express it as a mixed fraction or a decimal. Let's convert it into a mixed fraction first.

To convert 7644/25 into a mixed fraction, we perform division:

7644 ÷ 25 = 305 with a remainder of 19

So, the mixed fraction is 305 19/25 cm².

Alternatively, we can convert the improper fraction to a decimal by dividing 7644 by 25:

7644 ÷ 25 = 305.76

Therefore, the area of the framed photograph is 305.76 cm².

Summary of the Area Calculation

We calculated the area of the framed photograph by multiplying its framed length and framed breadth. The framed dimensions were obtained by adding twice the frame width to the original photograph's dimensions. The area calculation yielded a result of 7644/25 cm², which can be expressed as a mixed fraction (305 19/25 cm²) or as a decimal (305.76 cm²).

Final Answer: Area of the Framed Photograph

In conclusion, the area of the framed photograph is 7644/25 cm², which is equivalent to 305 19/25 cm² or 305.76 cm². This result provides us with the total surface area occupied by the framed photograph, including both the photograph itself and the surrounding frame.

Step-by-Step Solution: A Recap of the Process

To solidify our understanding, let's recap the step-by-step solution we followed to calculate the area of the framed photograph:

  1. Convert Mixed Fractions to Improper Fractions:
    • We converted the mixed fractions representing the photograph's dimensions (length and breadth) and the frame width into improper fractions. This conversion simplified subsequent calculations.
  2. Calculate the Framed Length:
    • We added twice the frame width to the photograph's length to find the framed length.
  3. Calculate the Framed Breadth:
    • Similarly, we added twice the frame width to the photograph's breadth to find the framed breadth.
  4. Calculate the Area of the Framed Photograph:
    • We multiplied the framed length and framed breadth to obtain the area of the framed photograph.
  5. Express the Area in Different Forms:
    • We expressed the area as an improper fraction, a mixed fraction, and a decimal to provide a comprehensive understanding of the result.

By following these steps, we successfully calculated the area of the framed photograph, demonstrating the application of mathematical concepts in a practical scenario.

Real-World Applications: Beyond the Photograph

The problem we've solved here isn't just limited to calculating the area of a framed photograph. The underlying principles and techniques have broader applications in various real-world scenarios. Let's explore a few examples:

  • Construction and Interior Design: When planning renovations or decorating a space, calculating areas is crucial for determining the amount of materials needed, such as flooring, paint, wallpaper, or tiles. Understanding how borders or frames affect the overall dimensions helps in accurate estimation and planning.
  • Gardening and Landscaping: Determining the area of a garden bed or a lawn is essential for calculating the amount of soil, fertilizer, or grass seed required. Similarly, when designing a patio or a pathway with borders, understanding how the borders impact the overall area helps in material estimation and layout planning.
  • Picture Framing and Art Presentation: In the context of art, understanding how frames affect the dimensions and area of a piece is crucial for presentation and display. It helps in selecting appropriate frame sizes and calculating the amount of matting or mounting materials needed.
  • Manufacturing and Product Design: Many products involve frames or borders, and calculating their dimensions and areas is essential for manufacturing processes and material estimation. This applies to products ranging from electronic displays to furniture.
  • Mapping and Surveying: Calculating areas is a fundamental aspect of mapping and surveying. Understanding how to account for borders or margins is important in creating accurate maps and land surveys.

These examples illustrate how the concepts and techniques used in solving our photograph problem have wider applicability in various fields. By mastering these skills, we can tackle a range of practical problems involving dimensions, areas, and borders.

Practice Problems: Sharpening Your Skills

To further enhance your understanding and problem-solving abilities, let's explore a few practice problems related to calculating areas with borders. These problems will provide you with an opportunity to apply the concepts we've discussed and refine your skills.

  1. Rectangular Garden with a Path: A rectangular garden measures 15 meters in length and 10 meters in breadth. A path of uniform width 2 meters surrounds the garden. Calculate the area of the path.
  2. Painting with a Border: A rectangular painting has dimensions 60 cm by 40 cm. It is mounted on a rectangular board with a 5 cm wide border around the painting. Find the area of the board.
  3. Swimming Pool with a Deck: A rectangular swimming pool is 20 meters long and 12 meters wide. A deck of uniform width 3 meters surrounds the pool. Determine the area of the deck.
  4. Carpet with a Border: A rectangular room is 8 meters long and 6 meters wide. A carpet is placed in the room, leaving a 1-meter border of uncovered floor around the carpet. Calculate the area of the carpet.

By attempting these practice problems, you'll gain confidence in your ability to solve similar problems and apply these skills in real-world scenarios. Remember to follow the step-by-step approach we discussed earlier, including converting mixed fractions to improper fractions (if applicable), calculating the framed dimensions, and then determining the area.

Conclusion: Mastering the Art of Area Calculation

In this comprehensive exploration, we've tackled the problem of calculating the area of a framed family photograph. We've walked through the process step by step, from converting mixed fractions to improper fractions to calculating the framed dimensions and finally determining the area. We've also discussed the broader applications of these concepts in various real-world scenarios and provided practice problems to further enhance your skills.

By mastering the art of area calculation, you'll be well-equipped to solve a wide range of practical problems involving dimensions, borders, and surface areas. This skill is not only valuable in mathematical contexts but also in various fields such as construction, design, gardening, and manufacturing.

Continue practicing and applying these concepts, and you'll find yourself confidently tackling geometric challenges in your daily life and professional endeavors. Remember, mathematics is not just about formulas and equations; it's about developing problem-solving skills and applying them to understand and interact with the world around us.