Calculating Perimeter Of Rectangle Paper Sheet A Math Guide

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Hey guys! Ever wondered how to figure out the perimeter of something, especially when the measurements are a bit… quirky? Let's dive into a fun math problem involving a rectangular sheet of paper. We'll break it down step by step, so you can ace similar questions in the future. Get ready to put on your math hats!

Understanding Perimeter and Rectangles

Before we jump into the problem, let's quickly recap what perimeter means and the properties of a rectangle. The perimeter, in simple terms, is the total distance around the outside of a shape. Imagine walking along the edges of a park; the total distance you walk is the perimeter of the park. For a rectangle, this means adding up the lengths of all four sides.

Rectangles have some cool features that make calculating their perimeters easier. They have two pairs of sides that are equal in length. The longer sides are usually called the length, and the shorter sides are called the width. So, if you know the length and width of a rectangle, you can easily find the perimeter. The formula for the perimeter (P) of a rectangle is:

P = 2 * (length + width)

This formula works because you have two sides with the length and two sides with the width. We're just adding them all up!

The Problem: A Quirky Rectangle

Okay, let's get to the exciting part! We have a rectangular sheet of paper, but the dimensions are given in mixed numbers, which might seem a bit tricky at first. Don't worry, we'll tackle it together.

The problem states:

A rectangle sheet of paper is 15 3/4 cm long and 12 1/2 cm wide. Find its perimeter.

So, we know:

  • Length: 15 3/4 cm
  • Width: 12 1/2 cm

Our mission is to find the total distance around this paper, which is its perimeter. Let's get started!

Step-by-Step Solution

Step 1: Convert Mixed Numbers to Improper Fractions

Mixed numbers are a combination of a whole number and a fraction (like 15 3/4). To make our calculations easier, we need to convert these mixed numbers into improper fractions. An improper fraction is where the numerator (the top number) is greater than or equal to the denominator (the bottom number).

Converting 15 3/4:

  1. Multiply the whole number (15) by the denominator (4): 15 * 4 = 60
  2. Add the numerator (3) to the result: 60 + 3 = 63
  3. Keep the same denominator (4).

So, 15 3/4 becomes 63/4.

Converting 12 1/2:

  1. Multiply the whole number (12) by the denominator (2): 12 * 2 = 24
  2. Add the numerator (1) to the result: 24 + 1 = 25
  3. Keep the same denominator (2).

So, 12 1/2 becomes 25/2.

Now we have:

  • Length: 63/4 cm
  • Width: 25/2 cm

Step 2: Apply the Perimeter Formula

Remember the formula for the perimeter of a rectangle? It's:

P = 2 * (length + width)

Let's plug in our values:

P = 2 * (63/4 + 25/2)

Step 3: Add the Fractions

Before we can multiply, we need to add the fractions inside the parentheses. To add fractions, they need to have the same denominator (the bottom number). The least common denominator (LCD) of 4 and 2 is 4. So, we need to convert 25/2 into an equivalent fraction with a denominator of 4.

To do this, we multiply both the numerator and the denominator of 25/2 by 2:

(25 * 2) / (2 * 2) = 50/4

Now we can add the fractions:

63/4 + 50/4 = (63 + 50) / 4 = 113/4

So, our equation now looks like this:

P = 2 * (113/4)

Step 4: Multiply by 2

Now we just need to multiply the fraction by 2. We can think of 2 as 2/1:

P = (2/1) * (113/4)

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

P = (2 * 113) / (1 * 4) = 226/4

Step 5: Simplify the Fraction

We have an improper fraction, 226/4. Let's simplify it. First, we can divide both the numerator and the denominator by their greatest common factor, which is 2:

226/4 = (226 ÷ 2) / (4 ÷ 2) = 113/2

Step 6: Convert Back to a Mixed Number (Optional)

Sometimes, it's helpful to convert the improper fraction back into a mixed number. To do this, we divide the numerator (113) by the denominator (2):

113 ÷ 2 = 56 with a remainder of 1

This means we have 56 whole parts and 1 remaining part out of 2. So, the mixed number is 56 1/2.

The Answer

The perimeter of the rectangular sheet of paper is 113/2 cm, or 56 1/2 cm. Awesome! We solved it!

Why This Matters: Real-World Applications

You might be wondering,