Problem 1
Question
Find the quotient in each case by replacing the divisor by its reciprocal and multiplying. $$\frac{3}{4} \div \frac{1}{5}$$
Step-by-Step Solution
Verified Answer
The quotient is \( \frac{15}{4} \) or \( 3\frac{3}{4} \).
1Step 1: Identify the Divisor's Reciprocal
To divide by a fraction, we multiply by its reciprocal. Identify the reciprocal of \( \frac{1}{5} \). The reciprocal of a fraction is found by swapping its numerator and denominator. Therefore, the reciprocal of \( \frac{1}{5} \) is \( \frac{5}{1} \) or simply 5.
2Step 2: Replace Division with Multiplication
Replace the division operation with multiplication by the reciprocal identified in Step 1. Thus, the original problem \( \frac{3}{4} \div \frac{1}{5} \) becomes \( \frac{3}{4} \times \frac{5}{1} \).
3Step 3: Multiply the Fractions
Multiply the numerators and the denominators. For the numerators: \( 3 \times 5 = 15 \). For the denominators: \( 4 \times 1 = 4 \). Therefore, \( \frac{3}{4} \times \frac{5}{1} = \frac{15}{4} \).
4Step 4: Simplify the Result
The fraction \( \frac{15}{4} \) is already in its simplest form since 15 and 4 have no common factors other than 1. Additionally, as an improper fraction, it can be expressed as a mixed number. Divide 15 by 4 to get 3 with a remainder of 3. Therefore, \( \frac{15}{4} \) is equivalent to the mixed number \( 3\frac{3}{4} \).
Key Concepts
FractionsReciprocalsImproper FractionsMixed Numbers
Fractions
Fractions represent parts of a whole and are composed of two numbers: the numerator and the denominator. The numerator is the top number, indicating how many parts are being considered, while the denominator, the bottom number, shows the total number of equal parts in a whole.
When dealing with fractions, the primary objective is to understand their value relative to a complete entity. For example, in the fraction \( \frac{3}{4} \), there are 3 parts taken out of a total 4 parts. The smaller the denominator, the larger each part is, which is why \( \frac{1}{4} \) is larger than \( \frac{1}{5} \), even though both fractions signify one part of a whole divided into four or five parts, respectively.
Understanding fractions is crucial for performing operations such as addition, subtraction, multiplication, and division with them. Though it may seem complex at first, using fractions becomes simpler with practice.
When dealing with fractions, the primary objective is to understand their value relative to a complete entity. For example, in the fraction \( \frac{3}{4} \), there are 3 parts taken out of a total 4 parts. The smaller the denominator, the larger each part is, which is why \( \frac{1}{4} \) is larger than \( \frac{1}{5} \), even though both fractions signify one part of a whole divided into four or five parts, respectively.
Understanding fractions is crucial for performing operations such as addition, subtraction, multiplication, and division with them. Though it may seem complex at first, using fractions becomes simpler with practice.
Reciprocals
Reciprocals are essential when it comes to dividing fractions. By definition, the reciprocal of a fraction is obtained by flipping that fraction. This means you exchange the positions of the numerator and the denominator. For instance, the reciprocal of \( \frac{1}{5} \) is \( \frac{5}{1} \) or 5. Similarly, the reciprocal of \( \frac{3}{4} \) would be \( \frac{4}{3} \).
When we divide by a fraction, we actually multiply by its reciprocal. This method simplifies the division process significantly. In the exercise, instead of dividing \( \frac{3}{4} \) by \( \frac{1}{5} \), we multiply \( \frac{3}{4} \) by 5, using the reciprocal of \( \frac{1}{5} \). This approach prevents the complexity of direct division with fractions.
When we divide by a fraction, we actually multiply by its reciprocal. This method simplifies the division process significantly. In the exercise, instead of dividing \( \frac{3}{4} \) by \( \frac{1}{5} \), we multiply \( \frac{3}{4} \) by 5, using the reciprocal of \( \frac{1}{5} \). This approach prevents the complexity of direct division with fractions.
Improper Fractions
Improper fractions occur when the numerator is greater than or equal to the denominator, indicating the fraction is greater than or equal to one whole. For instance, \( \frac{15}{4} \) is an improper fraction as 15 exceeds 4.
These fractions are vital in expressing numbers greater than one using a single fraction. After performing operations on fractions, you might end up with an improper fraction. While improper fractions are mathematically correct, converting them into mixed numbers often provides a clearer interpretation of their value.
In our example, \( \frac{15}{4} \) is the result of the multiplication step and can be converted to a mixed number. This flexibility in representation allows for better understanding and use in various mathematical contexts.
These fractions are vital in expressing numbers greater than one using a single fraction. After performing operations on fractions, you might end up with an improper fraction. While improper fractions are mathematically correct, converting them into mixed numbers often provides a clearer interpretation of their value.
In our example, \( \frac{15}{4} \) is the result of the multiplication step and can be converted to a mixed number. This flexibility in representation allows for better understanding and use in various mathematical contexts.
Mixed Numbers
A mixed number combines a whole number with a fractional part. It represents the same value as an improper fraction but offers a viewpoint more aligned with everyday understanding. For example, the improper fraction \( \frac{15}{4} \) can be expressed as the mixed number \( 3\frac{3}{4} \). This indicates there are 3 whole units and an additional \( \frac{3}{4} \) of another unit.
To convert an improper fraction to a mixed number, divide the numerator by the denominator to find the whole number. The remainder becomes the numerator of the fractional part, with the original denominator remaining unchanged. This mixed number method is especially useful in measuring and cooking, where understanding the tangible amount is necessary.
Grasping mixed numbers is a key skill, and practice can aid in making these conversions intuitively.
To convert an improper fraction to a mixed number, divide the numerator by the denominator to find the whole number. The remainder becomes the numerator of the fractional part, with the original denominator remaining unchanged. This mixed number method is especially useful in measuring and cooking, where understanding the tangible amount is necessary.
Grasping mixed numbers is a key skill, and practice can aid in making these conversions intuitively.
Other exercises in this chapter
Problem 1
Find each of the following products. (Multiply.) $$\frac{2}{3} \cdot \frac{4}{5}$$
View solution Problem 1
Find the following sums and differences, and reduce to lowest terms. (Add or subtract as indicated.) $$\frac{3}{6}+\frac{1}{6}$$
View solution Problem 1
Identify each of the numbers below as either a prime number or a composite number. For those that are composite, give at least one divisor (factor) other than t
View solution Problem 1
Name the numerator of each fraction. $$\frac{1}{3}$$
View solution