Chapter 2: Problem 64
Divide the following fractions and mixed numbers. Reduce to lowest terms. \(\frac{7}{8} \div 2 \frac{1}{4}=\) ______
Short Answer
Expert verified
\(\frac{7}{18}\)
Step by step solution
01
Convert the Mixed Number to an Improper Fraction
To perform division with fractions, we first need to convert the mixed number \(2 \frac{1}{4}\) into an improper fraction. A mixed number consists of a whole part and a fraction part. The mixed number \(2 \frac{1}{4}\) is equal to \(\frac{9}{4}\), because \(2 \times 4 + 1 = 9\).
02
Set Up the Division as Multiplication by the Reciprocal
In order to divide by a fraction, we multiply by its reciprocal. The reciprocal of \(\frac{9}{4}\) is \(\frac{4}{9}\). Therefore, the problem \(\frac{7}{8} \div \frac{9}{4}\) can be rewritten as \(\frac{7}{8} \times \frac{4}{9}\).
03
Multiply the Fractions
Multiply the numerators and the denominators. The product of the numerators is \(7 \times 4 = 28\), and the product of the denominators is \(8 \times 9 = 72\). This gives us \(\frac{28}{72}\).
04
Reduce the Fraction to Lowest Terms
To reduce \(\frac{28}{72}\) to its lowest terms, we need to divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 28 and 72 is 4. Thus, dividing the numerator and the denominator by their GCD gives us \(\frac{28 \div 4}{72 \div 4} = \frac{7}{18}\).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Improper Fractions
When working with improper fractions, remember that these are fractions where the numerator (top number) is larger than or equal to the denominator (bottom number). Improper fractions can be helpful in various mathematical operations, such as division and multiplication. They provide a unified way of dealing with quantities that might otherwise be given as mixed numbers.
To convert a mixed number into an improper fraction, multiply the whole number part by the denominator of the fractional part and add the result to the numerator of the fractional part. This new numerator sits over the original denominator.
For example, in the mixed number \(2 \frac{1}{4}\), we multiply \(2 \times 4\) and add \(1\) to get \(9\). Thus, \(2 \frac{1}{4}\) becomes \(\frac{9}{4}\). This clear numerical form makes further calculations easier to handle.
To convert a mixed number into an improper fraction, multiply the whole number part by the denominator of the fractional part and add the result to the numerator of the fractional part. This new numerator sits over the original denominator.
For example, in the mixed number \(2 \frac{1}{4}\), we multiply \(2 \times 4\) and add \(1\) to get \(9\). Thus, \(2 \frac{1}{4}\) becomes \(\frac{9}{4}\). This clear numerical form makes further calculations easier to handle.
Reciprocals
Understanding reciprocals is essential for dividing fractions. The reciprocal of a fraction is simply another fraction where the numerator and denominator are swapped. For example, the reciprocal of \(\frac{9}{4}\) is \(\frac{4}{9}\).
When dividing fractions, you multiply by the reciprocal of the divisor. This is because division is the inverse of multiplication, and thus turning a division problem into a multiplication one with the reciprocal simplifies the process.
For instance, to divide \(\frac{7}{8}\) by \(\frac{9}{4}\), you convert the operation to \(\frac{7}{8} \times \frac{4}{9}\). This method efficiently navigates the division of fractions by leveraging the intuitive multiplication process.
When dividing fractions, you multiply by the reciprocal of the divisor. This is because division is the inverse of multiplication, and thus turning a division problem into a multiplication one with the reciprocal simplifies the process.
For instance, to divide \(\frac{7}{8}\) by \(\frac{9}{4}\), you convert the operation to \(\frac{7}{8} \times \frac{4}{9}\). This method efficiently navigates the division of fractions by leveraging the intuitive multiplication process.
Reducing Fractions
Reducing fractions to their simplest form is crucial for presenting neat and standard answers. A fraction is in its lowest terms when the numerator and the denominator share no common factors other than 1.
To reduce a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD). Finding the GCD involves identifying the largest integer that evenly divides both numbers.
For example, the fraction \(\frac{28}{72}\) can be reduced by identifying the GCD, which is 4. Dividing both 28 and 72 by 4, we get \(\frac{7}{18}\). This process simplifies the fraction, making it cleaner and easier to interpret.
To reduce a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD). Finding the GCD involves identifying the largest integer that evenly divides both numbers.
For example, the fraction \(\frac{28}{72}\) can be reduced by identifying the GCD, which is 4. Dividing both 28 and 72 by 4, we get \(\frac{7}{18}\). This process simplifies the fraction, making it cleaner and easier to interpret.
Mixed Numbers
Mixed numbers are numbers combining a whole number and a proper fraction, like \(2 \frac{1}{4}\). Mixed numbers are often used in everyday scenarios, such as when measuring ingredients or describing parts of an hour.
Converting mixed numbers to improper fractions is often necessary for algebraic operations like addition, subtraction, multiplication, and especially division.
This conversion simplifies arithmetic operations. For instance, changing \(2 \frac{1}{4}\) into \(\frac{9}{4}\) allows you to utilize fraction rules without dealing with separate whole and fractional parts. Once calculations are completed, if required, you can convert back to a mixed number for easy interpretation in certain real-world contexts.
Converting mixed numbers to improper fractions is often necessary for algebraic operations like addition, subtraction, multiplication, and especially division.
This conversion simplifies arithmetic operations. For instance, changing \(2 \frac{1}{4}\) into \(\frac{9}{4}\) allows you to utilize fraction rules without dealing with separate whole and fractional parts. Once calculations are completed, if required, you can convert back to a mixed number for easy interpretation in certain real-world contexts.
