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Chapter 4: Problem 455

In the following exercises, find the difference. $$5 \frac{2}{9}-3 \frac{4}{9}$$

Short Answer

Expert verified
The difference is \(1 \frac{7}{9}\).

Step by step solution

01

Convert Mixed Numbers to Improper Fractions

First, we need to convert the mixed numbers into improper fractions. For \(5 \frac{2}{9}\), multiply the whole number 5 by the denominator 9 and add the numerator 2. This gives: \[5 \frac{2}{9} = \frac{5 \times 9 + 2}{9} = \frac{45 + 2}{9} = \frac{47}{9}\]. For \(3 \frac{4}{9}\), multiply the whole number 3 by the denominator 9 and add the numerator 4. This gives: \[3 \frac{4}{9} = \frac{3 \times 9 + 4}{9} = \frac{27 + 4}{9} = \frac{31}{9}\].
02

Subtract the Improper Fractions

Now, subtract the two improper fractions. Since the denominators are the same, subtract the numerators directly: \[\frac{47}{9} - \frac{31}{9} = \frac{47 - 31}{9} = \frac{16}{9}\].
03

Convert the Result Back to a Mixed Number

Convert the improper fraction \(\frac{16}{9}\) back to a mixed number by dividing 16 by 9. The quotient is 1 and the remainder is 7, so: \[\frac{16}{9} = 1 \frac{7}{9}\].

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Key Concepts

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

Improper Fractions
Improper fractions are a type of fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number). For example, \( \frac{47}{9} \) and \( \frac{31}{9} \) from our exercise are improper fractions.
To convert a mixed number into an improper fraction, you follow these steps:
  • Multiply the whole number part by the denominator.
  • Add the result to the numerator.
  • Write the result over the original denominator.
So for \( 5 \frac{2}{9} \), we calculate \( 5 \times 9 + 2\), resulting in \( 47 \), making the improper fraction \( \frac{47}{9} \).
Mixed Numbers
Mixed numbers consist of a whole number and a proper fraction. For instance, \( 5 \frac{2}{9} \) and \( 3 \frac{4}{9} \) are examples of mixed numbers. These are often easier to understand and work with as they clearly show the whole number part alongside the fractional part.
When converting a mixed number to an improper fraction, you combine the whole and fractional parts together into a single fraction.
Here鈥檚 how:
  • Multiply the whole number by the denominator of the fractional part.
  • Add the numerator of the fraction part to the result.
  • The sum is written as the numerator with the original denominator.
This process helps when performing operations like addition or subtraction of fractions, as it allows you to work with simpler numerical forms.
Fraction Subtraction
Subtracting fractions involves a few key steps. In our example, we started with mixed numbers, so we converted them to improper fractions first.
Here鈥檚 a quick guide on subtracting fractions:
  • Ensure the fractions have a common denominator.
  • If they don鈥檛, find a common denominator and adjust the fractions accordingly.
  • Subtract the numerators while keeping the denominator the same.
  • Simplify the resulting fraction if necessary.
In our problem, both fractions \( \frac{47}{9} \) and \( \frac{31}{9} \) had the same denominator. Subtraction was straightforward with \( \frac{47}{9} - \frac{31}{9} = \frac{16}{9} \). Converting this result back to a mixed number \(1 \frac{7}{9} \) makes it easier to understand.
Prealgebra
Prealgebra introduces basic mathematical concepts that are foundational for algebra. Understanding how to work with mixed numbers and improper fractions is a significant part of prealgebra.
You will learn to:
  • Convert between mixed numbers and improper fractions.
  • Add, subtract, multiply, and divide fractions.
  • Simplify fractions and reduce them to their simplest form.
  • Understand and apply the processes for finding common denominators.
Grasping these concepts in prealgebra will help you progress to more advanced topics in algebra and other areas of mathematics. It lays the groundwork for solving more complicated equations and understanding the relationships between numbers.

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Most popular questions from this chapter

In the following exercises, perform the indicated operations and simplify. $$-\frac{22}{25}+\frac{9}{40}$$

In the following exercises, perform the indicated operations and simplify. $$-\frac{9}{10}-\frac{3}{4}$$

In the following exercises, evaluate the given expression. Express your answers in simplified form, using improper fractions if necessary. 2\(x^{2} y^{3}\) when \(x=-\frac{2}{3}\) and \(y=-\frac{1}{2}\)

In the following exercises, perform the indicated operations and simplify. $$\frac{11}{12}-\frac{2}{3}$$

In the following exercises, translate and solve. The difference of \(q\) and one-tenth is \(\frac{1}{2}\) .
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