Question

Divide the following fractions and mixed numbers. Reduce to lowest terms. \(2 \frac{1}{3} \div 4 \frac{1}{6}=\) ______

Short answer

\(2 \frac{1}{3} \div 4 \frac{1}{6} = \frac{14}{25}\)

Step-by-step solution

Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers to improper fractions. For the first mixed number, \(2 \frac{1}{3}\), multiply the whole number 2 by the denominator 3 and add the numerator 1, which results in \(\frac{7}{3}\). Similarly, for the second mixed number, \(4 \frac{1}{6}\), multiply the whole number 4 by the denominator 6 and add the numerator 1, resulting in \(\frac{25}{6}\). Thus, the division problem is \(\frac{7}{3} \div \frac{25}{6}\).

Divide Fractions by Multiplying by the Reciprocal

To divide fractions, multiply by the reciprocal of the second fraction. The reciprocal of \(\frac{25}{6}\) is \(\frac{6}{25}\). Multiply \(\frac{7}{3}\) by \(\frac{6}{25}\) to proceed with the division: \(\frac{7}{3} \times \frac{6}{25}\).

Multiply the Fractions

Multiply the numerators and multiply the denominators: \(7 \times 6 = 42\) for the numerator, and \(3 \times 25 = 75\) for the denominator. So the result is \(\frac{42}{75}\).

Simplify the Fraction

To simplify \(\frac{42}{75}\), find the greatest common factor (GCF) of 42 and 75. The GCF is 3. Divide both the numerator and the denominator by 3 to simplify: \(\frac{42 \div 3}{75 \div 3} = \frac{14}{25}\). The simplified fraction is \(\frac{14}{25}\).

Key concepts

Mixed Numbers

Mixed numbers are a combination of a whole number and a proper fraction. They represent numbers that fall between whole numbers on the number line. For instance, the mixed number \(2 \frac{1}{3}\) indicates "2 and one-third." To work with mixed numbers, especially in operations like division, converting them into improper fractions is essential. This means transforming the mixed number into a fraction where the numerator exceeds the denominator. - Multiply the whole number by the denominator.- Add the result to the numerator.- Place this result over the original denominator.For example, with \(2 \frac{1}{3}\), you multiply 2 by 3, add 1 to get 7, and express it as \(\frac{7}{3}\). This conversion simplifies subsequent operations like division or multiplication.

Improper Fractions

An improper fraction is a type of fraction where the numerator is greater than or equal to the denominator. They often appear after converting mixed numbers or in calculations such as multiplications or divisions. An improper fraction signifies that the value represented is greater than or equal to 1. For example, \(\frac{7}{3}\) translates to 7 parts of size 1/3 each, which is the same as saying \(2\frac{1}{3}\) when expressed as a mixed number.Improper fractions are easier to work with in calculations than mixed numbers, offering a smoother process for arithmetic operations like addition, subtraction, multiplication, and division. This is why they are often used in problem-solving and equations where mixed numbers or fractions are manipulated.

Reciprocal

A reciprocal is what you multiply a number by to get the product of 1. For fractions, this means flipping the numerator and denominator. For instance, the reciprocal of \(\frac{25}{6}\) is \(\frac{6}{25}\).The concept of reciprocals comes into play when dividing fractions. Instead of dividing directly, you multiply by the reciprocal of the divisor. This trick simplifies the operation considerably:- Take \(\frac{y}{z}\) as the divisor- Replace division \((\div)\) with multiplication \((\times)\)- Flip the fraction to get the reciprocal \(\frac{z}{y}\)In our example, \(\frac{7}{3} \div \frac{25}{6}\) is turned into multiplication with the reciprocal: \(\frac{7}{3} \times \frac{6}{25}\). This is a pivotal concept in fraction division.

Simplifying Fractions

Simplifying fractions involves reducing them to their lowest possible terms. The goal is to make the fraction as simple as possible while retaining its value. This is done by dividing the numerator and the denominator by their greatest common factor (GCF).Here's how it works:- Find the GCF of both the numerator and the denominator.- Divide both the numerator and the denominator by this GCF.In our case, after dividing, we got \(\frac{42}{75}\). The GCF of 42 and 75 is 3. So, divide both parts of the fraction by 3: \(\frac{42 \div 3}{75 \div 3}\) resulting in the simplified fraction \(\frac{14}{25}\).By simplifying fractions, calculations are often more manageable and solutions clearer for practical interpretation.