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Chapter 9: Problem 14

Find the perimeter of each square. $$ 3 \frac{1}{8} \text { yd on a side } $$

Short Answer

Expert verified
The perimeter of the square is 12.5 yards.

Step by step solution

01

Convert Mixed Fraction to Improper Fraction

Convert the mixed fraction for the side length of the square to an improper fraction.Given: \(3 \frac{1}{8}\) yards Convert this to an improper fraction: \(3 \frac{1}{8} = \frac{3 \times 8 + 1}{8} = \frac{25}{8}\)
02

Calculate the Perimeter of the Square

A square has 4 equal sides. The perimeter \(P\) of a square is given by the sum of all its sides.\[P = 4 \times \text{side length}\]Using the side length: \[P = 4 \times \frac{25}{8}\]
03

Simplify the Expression

Simplify the expression to find the perimeter in yards.\[P = 4 \times \frac{25}{8} = \frac{4 \times 25}{8} = \frac{100}{8} = 12.5\] yards

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Mixed Fraction
A mixed fraction, also known as a mixed number, combines a whole number and a fraction. For example, in the exercise given, the side of the square is expressed as a mixed fraction: 3 1/8 yards. This means you have three whole yards and an additional fraction of a yard. Understanding mixed fractions is crucial as they often appear in measurements and real-world scenarios.

To solve mathematical problems, mixed fractions usually need to be converted into improper fractions.
Improper Fraction
An improper fraction is where the numerator (the top part of the fraction) is greater than or equal to the denominator (the bottom part). To convert a mixed fraction to an improper fraction, you can follow these steps:
  • Multiply the whole number by the denominator.
  • Add the result to the numerator.
  • Place the sum over the original denominator.
For example, to convert 3 1/8 into an improper fraction, multiply 3 by 8 (24), then add 1 (25), resulting in 25/8. This step makes calculations easier in upcoming processes like finding perimeters or areas.
Simplifying Fractions
Simplifying fractions means reducing them to their simplest form. In our exercise, the final solution was a fraction: \(\frac{100}{8}\). To simplify this, divide both the numerator and the denominator by their greatest common divisor (GCD).

Here鈥檚 how you can do it:
  • Find the GCD of 100 and 8, which is 4.
  • Divide both 100 and 8 by 4.
  • We get \(\frac{100/4}{8/4} = \frac{25}{2}\), which can also be written as a decimal (12.5).
This step makes the result more understandable and easier to work with.
Basic Geometry
Geometry is the branch of mathematics dealing with shapes, sizes, and properties of space. A square is a basic geometric shape with four equal sides. To find the perimeter (the total length around the square), you multiply the length of one side by four.

For instance, if each side is 3 1/8 yards:
  • First, convert 3 1/8 yards to an improper fraction: 25/8 yards.
  • Then, multiply by 4 (the number of sides in a square): \(P = 4 \times \frac{25}{8} = \frac{100}{8} = 12.5\) yards.
This method applies to any square, providing a consistent way to calculate perimeters and understand basic geometric principles.

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