Question

Divide the following fractions and mixed numbers. Reduce to lowest terms. \(3 \div \frac{2}{5}=\) ______

Short answer

7 and half

Step-by-step solution

Convert Whole Number to Fraction

The first step in solving this problem is to express the whole number 3 as a fraction. We do this by writing it as \( \frac{3}{1} \).

Rewrite Division as Multiplication

Division of fractions can be tricky, so it's easier to convert the division operation into multiplication. According to the reciprocal rule, dividing by a fraction \( \frac{a}{b} \) is the same as multiplying by its reciprocal \( \frac{b}{a} \). In this case, we convert \( 3 \div \frac{2}{5} \) to \( \frac{3}{1} \times \frac{5}{2} \).

Multiply the Fractions

Now, multiply the two fractions. To multiply fractions, multiply the numerators together and the denominators together. This gives: \[ \frac{3}{1} \times \frac{5}{2} = \frac{3 \times 5}{1 \times 2} = \frac{15}{2} \]

Convert Improper Fraction to Mixed Number

The result \( \frac{15}{2} \) is an improper fraction. To convert it to a mixed number, divide 15 by 2. The quotient is 7, and the remainder is 1. This means \( \frac{15}{2} = 7 \frac{1}{2} \).

Confirm Reduction to Lowest Terms

Verify that the fraction part of the mixed number \( \frac{1}{2} \) is indeed in its lowest terms. Since the greatest common divisor of 1 and 2 is 1, \( \frac{1}{2} \) is already reduced.

Key concepts

Converting Whole Numbers

Whenever we deal with division involving fractions and whole numbers, it's essential to convert the whole number to a fraction. This step makes operations straightforward. For any whole number like 3, we can write it as a fraction by putting it over 1. This means 3 becomes \( \frac{3}{1} \). By doing this, we're effectively setting up the number for any fraction operation, such as multiplication or division.

This conversion is crucial because it allows us to easily apply rules that are inherent to fraction operations. Moreover, understanding how to convert whole numbers to fractions helps in maintaining consistency across various mathematical expressions and operations.

Reciprocal Rule

In fraction operations, when dividing by a fraction, you can simplify the process by using the reciprocal. The reciprocal of a fraction \( \frac{a}{b} \) is simply flipping it to become \( \frac{b}{a} \). This is crucial because dividing by a fraction is equivalent to multiplying by its reciprocal.

For example, in our exercise, instead of directly dividing \( 3 \div \frac{2}{5} \), we convert it into a multiplication problem: \( \frac{3}{1} \times \frac{5}{2} \). This step is fundamental in making complex division tasks easier and helps avoid confusion. Mastering the reciprocal rule is a pivotal skill in solving division problems involving fractions seamlessly.

Improper Fractions

Improper fractions are those where the numerator (top number) is larger than the denominator (bottom number). They frequently appear in mathematical problems and often require simplification or conversion into more understandable forms.

In the provided exercise, after multiplying the fractions, we end up with the result \( \frac{15}{2} \), which is an improper fraction. Working with improper fractions can be convenient in calculations, but sometimes it's necessary to convert them into mixed numbers for clarity or further operations. Understanding improper fractions and how to manipulate them ensures proficiency in fraction mathematics.

Mixed Numbers

A mixed number combines a whole number and a fraction. It is commonly used to express values that go beyond simple fractions, providing a clearer picture of the magnitude of a number.

Converting an improper fraction like \( \frac{15}{2} \) into a mixed number involves dividing the numerator by the denominator. For \( 15 \div 2 \), the quotient is 7, and the remainder is 1. This converts the fraction into the mixed number 7 \( \frac{1}{2} \).

• The first part of the mixed number (7) reflects the whole part.
• The fraction part (\( \frac{1}{2} \)) represents the remaining part of the division.

Mixed numbers thus provide an alternative way to represent improper fractions, making the results more intuitive for real-world applications.