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Chapter 9: Problem 14

Find the area of each rectangle. $$ 10 \frac{1}{3} \mathrm{mi} \text { by } 20 \frac{2}{3} \mathrm{mi} $$

Short Answer

Expert verified
The area is 213 \( \frac{5}{9} \) square miles.

Step by step solution

01

Convert Mixed Numbers to Improper Fractions

Convert 10 \( \frac{1}{3} \) mi and 20 \( \frac{2}{3} \) mi into improper fractions.For 10 \( \frac{1}{3} \):\[ 10 \frac{1}{3} = \frac{10 \times 3 + 1}{3} = \frac{31}{3} \ \]For 20 \( \frac{2}{3} \):\[ 20 \frac{2}{3} = \frac{20 \times 3 + 2}{3} = \frac{62}{3} \ \]
02

Multiply the Lengths

Multiply the improper fractions to find the area of the rectangle.\[ \text{Area} = \frac{31}{3} \times \frac{62}{3} \ = \frac{31 \times 62}{3 \times 3} \ = \frac{1922}{9} \ \]
03

Simplify the Fraction as Needed

Convert the improper fraction \( \frac{1922}{9} \) to a mixed number if necessary. Divide 1922 by 9 to get the quotient and the remainder.\[ 1922 \text{ divided by } 9 = 213 \text{ R } 5 \ \]So,\[ \frac{1922}{9} = 213 \frac{5}{9} \ \text{Area} = 213 \frac{5}{9} \text{ square miles} \ \]

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Key Concepts

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

mixed numbers
Mixed numbers are numbers that combine a whole number and a fraction. For example, in the problem, we have 10 \(\frac{1}{3}\) and 20 \(\frac{2}{3}\). These are both mixed numbers.
Mixed numbers are useful for representing quantities that are more than a whole but not quite another whole number. To work with these in calculations, we often need to convert them to improper fractions.
For instance, the mixed number 10 \(\frac{1}{3}\) can be turned into an improper fraction by:
  • Multiplying the whole number 10 by the denominator 3, giving us 30.
  • Adding the numerator 1 to this result, which gives us 31.
  • Putting this result over the original denominator, resulting in \(\frac{31}{3}\).
Similarly, for 20 \(\frac{2}{3}\), convert by:
  • Multiplying 20 by 3 to get 60.
  • Adding the numerator 2 to get 62.
  • Putting this over the denominator to get \(\frac{62}{3}\).
improper fractions
An improper fraction has a numerator larger than its denominator. For example, \(\frac{31}{3}\) and \(\frac{62}{3}\) are improper fractions.
When calculating things like areas, improper fractions make multiplication simpler because we avoid dealing with mixed numbers directly in the multiplication process.
To multiply two improper fractions:
  • Multiply the numerators together.
  • Then multiply the denominators together.
For our exercise, the area calculation involves multiplying \(\frac{31}{3}\) by \(\frac{62}{3}\):
  • Numerators: 31 x 62 = 1922.
  • Denominators: 3 x 3 = 9.
  • So the product is \(\frac{1922}{9}\).
multiplying fractions
Multiplying fractions involves a straightforward process: numerator multiplied by numerator and denominator by denominator. For example:
Given \(\frac{31}{3}\) and \(\frac{62}{3}\), you multiply the two numerators (31 and 62) and the two denominators (3 and 3).
Steps involved are:
  • Multiply the numerators: 31 x 62 = 1922.
  • Multiply the denominators: 3 x 3 = 9.
  • The resulting fraction is \(\frac{1922}{9}\).
This process ensures accuracy and is more manageable, especially when dealing with large numbers. Multiplying fractions directly also helps in minimizing mistakes.
simplifying fractions
Simplifying fractions means making the numbers as simple as possible to read. For improper fractions, you might want to convert them back to mixed numbers.
For \(\frac{1922}{9}\), you perform the following steps:
  • Divide 1922 by 9, which gives you a quotient of 213.
  • The remainder is 5.
  • Write this as 213 \(\frac{5}{9}\).
This gives a much easier-to-understand result. By simplifying, not only does the fraction look cleaner, but it also makes it easier for others to comprehend the value represented.

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