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Chapter 9: Problem 19

Find the area of the square. \(5 \frac{3}{8}\) yd on a side

Short Answer

Expert verified
\(\frac{1849}{64}\) square yards

Step by step solution

01

Convert Mixed Number to Improper Fraction

First, convert the mixed number \(5 \frac{3}{8}\) yd into an improper fraction. Multiply the whole number 5 by the denominator 8 and add the numerator 3: \(5 \times 8 + 3 = 40 + 3 = 43\) Therefore, \(5 \frac{3}{8}\) becomes \(\frac{43}{8}\).
02

Calculate the Area

The area of a square is found by squaring the length of one side. Using \(\frac{43}{8}\) yd, the area is: \[\left(\frac{43}{8}\right)^2 = \frac{43 \times 43}{8 \times 8} = \frac{1849}{64}\] So, the area of the square is \(\frac{1849}{64}\) square yards.
03

Simplify the Fraction

If possible, simplify the fraction \(\frac{1849}{64}\). Since 1849 and 64 have no common factors besides 1, the fraction \(\frac{1849}{64}\) is already in its simplest form.

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Key Concepts

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

convert mixed number to improper fraction
To solve problems involving mixed numbers, we often need to convert them to improper fractions. Mixed numbers have both a whole number and a fractional part. For example, let's consider the mixed number \(5 \frac{3}{8}\).
Here's how to convert it into an improper fraction:
  • Multiply the whole number (5) by the denominator of the fraction (8).
  • Add the result to the numerator (3).
  • Place this result over the original denominator (8).

So, for \(5 \frac{3}{8}\), you do \(5 \times 8 + 3 = 40 + 3 = 43\).
This means \(5 \frac{3}{8}\) becomes \(\frac{43}{8}\) as an improper fraction.
square of a fraction
When finding the area of a square, you need to square the length of one side. If the side length is a fraction, you will be squaring that fraction.
Let's say the side length is \(\frac{43}{8}\). To square a fraction, follow these steps:
  • Square the numerator: \(43 \times 43 = 1849\).
  • Square the denominator: \(8 \times 8 = 64\).
  • Place the squared numerator over the squared denominator: \(\frac{1849}{64}\).

Therefore, squaring \(\frac{43}{8}\) results in \(\frac{1849}{64}\). This fraction represents the area of the square in square yards.
simplify fraction
To simplify fractions, you divide the numerator and the denominator by their greatest common divisor (GCD). However, sometimes the fraction is already in its simplest form.
In this case, we have \(\frac{1849}{64}\) from squaring the side of the square earlier. We need to check if 1849 and 64 have any common factors.
In this case, 1849 and 64 share no common factors other than 1, so the fraction cannot be simplified further.
Hence, \(\frac{1849}{64}\) is already in its simplest form. It鈥檚 good practice to always verify whether a fraction can be simplified for a clearer and more concise answer.

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Most popular questions from this chapter

Find the area of the square. \(45.5 \mathrm{~m}\) on a side

Evaluate. $$ 10^{5} $$

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

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