Question

Divide the following fractions and mixed numbers. Reduce to lowest terms. \(1 \frac{1}{2} \div \frac{3}{4}=\) ______

Short answer

2

Step-by-step solution

Convert Mixed Number to Improper Fraction

First, convert the mixed number \(1 \frac{1}{2}\) into an improper fraction. A mixed number is composed of a whole number and a fraction. Multiply the denominator \(2\) by the whole number \(1\) and add the numerator \(1\) to get \(3\). Therefore, \(1 \frac{1}{2}\) equals \(\frac{3}{2}\).

Divide by Reciprocal

To divide fractions, multiply by the reciprocal of the second fraction. The reciprocal of \(\frac{3}{4}\) is \(\frac{4}{3}\). Thus, \(\frac{3}{2} \div \frac{3}{4}\) becomes \(\frac{3}{2} \times \frac{4}{3}\).

Multiply the Fractions

Multiply the numerators together and the denominators together: \(3 \times 4 = 12\) and \(2 \times 3 = 6\). This results in the fraction \(\frac{12}{6}\).

Simplify the Fraction

Simplify \(\frac{12}{6}\) by dividing both the numerator and the denominator by their greatest common divisor, which is \(6\). \(\frac{12 \div 6}{6 \div 6} = \frac{2}{1}\), which simplifies further to \(2\).

Key concepts

Mixed Numbers

A mixed number is a combination of a whole number and a fraction. For instance, \(1 \frac{1}{2}\) is a mixed number where \(1\) is the whole number and \(\frac{1}{2}\) is the fraction. Mixed numbers are particularly handy because they give a straightforward way to represent numbers that are greater than or equal to one but less than two, and so on.

• To work with mixed numbers in arithmetic operations, converting them to improper fractions can simplify the process.
• An improper fraction has a numerator larger than its denominator.
• To convert a mixed number to an improper fraction: multiply the whole number by the denominator, add the numerator, and place this sum over the original denominator.

Simplifying Fractions

Simplifying fractions is all about finding the simplest form of a fraction, where the numerator and denominator have no common divisors other than 1.

• Start by identifying the greatest common divisor (GCD) of the numerator and the denominator.
• An efficient method to find the GCD is by listing the factors of each number and choosing the largest one they have in common.
• Divide both the numerator and the denominator by this GCD.

For instance, when you have the fraction \(\frac{12}{6}\), both 12 and 6 can be divided by 6, which gives you \(\frac{2}{1}\). The simplified form provides the clearest understanding of the ratio between the numbers.

Reciprocal

Finding the reciprocal of a number is easy and crucial for operations like division when dealing with fractions. A reciprocal is essentially created by swapping the numerator and the denominator of a fraction.

• A fraction such as \(\frac{a}{b}\) has a reciprocal of \(\frac{b}{a}\).
• This concept is important because division of fractions is typically performed by multiplication with the reciprocal.
• When you divide by a fraction, you multiply by its reciprocal instead.

Consider dividing \(\frac{3}{2}\) by \(\frac{3}{4}\). By multiplying \(\frac{3}{2}\) with the reciprocal of \(\frac{3}{4}\), which is \(\frac{4}{3}\), we convert the division into multiplication; thus, simplifying the process considerably.