Question

Change the following improper fractions to mixed numbers, and reduce to lowest terms. \(\frac{30}{4}=\) ______

Short answer

The mixed number is \(7\frac{1}{2}\).

Step-by-step solution

Divide the Numerator by the Denominator

To convert \(\frac{30}{4}\) into a mixed number, perform the division of 30 by 4. When you divide 30 by 4, the quotient is 7 and the remainder is 2. So, \[30 \div 4 = 7 \quad R2\]The whole number part of our mixed number is 7.

Form the Mixed Number

Use the quotient as the whole number and the remainder as the numerator of the fractional part, keeping the original denominator. So, \(\frac{30}{4} = 7\frac{2}{4}\).This represents the original improper fraction as a mixed number.

Simplify the Fraction Part

Simplify the fractional part by dividing both the numerator and the denominator by their greatest common divisor (GCD). Here, the GCD of 2 and 4 is 2. Divide both the numerator and the denominator by 2: \[ \frac{2}{4} \rightarrow \frac{2 \div 2}{4 \div 2} = \frac{1}{2} \] Thus, \(7\frac{2}{4}\) simplifies to \(7\frac{1}{2}\).

Key concepts

Mixed Numbers

An improper fraction is a fraction where the numerator (the top number) is larger than the denominator (the bottom number), like \(\frac{30}{4}\). A mixed number, on the other hand, is a combination of a whole number and a proper fraction. Here's how you can convert an improper fraction into a mixed number:

• First, divide the numerator by the denominator. In our case, divide 30 by 4.
• The quotient becomes the whole number part, while the remainder becomes the numerator of the new fractional part.
• Keep the same denominator, and you'll have your mixed number.

For \(\frac{30}{4}\), dividing 30 by 4, we get 7 with a remainder of 2. Therefore, the mixed number is \(7\frac{2}{4}\). This concept helps in expressing fractions in a more tangible way, making them easier to understand and use in calculations.

Greatest Common Divisor

To simplify a fraction, we often need to find the greatest common divisor (GCD), which is the largest number that divides both the numerator and the denominator without leaving a remainder.

• Finding the GCD is essential for fraction simplification because it helps reduce fractions to their simplest form.
• You can determine the GCD by listing the factors of each number and finding the largest one they share.

In the example \(7\frac{2}{4}\), we need to simplify \(\frac{2}{4}\). The factors of 2 are 1 and 2, and the factors of 4 are 1, 2, and 4. The largest common factor is 2, making it the GCD. With this information, we divide both the numerator and the denominator by 2, simplifying \(\frac{2}{4}\) to \(\frac{1}{2}\). This process ensures our mixed number \(7\frac{2}{4}\) reduces correctly to \(7\frac{1}{2}\).

Fraction Simplification

Simplification is the process of reducing a fraction to its simplest form, where the numerator and denominator are as small as possible but still retain the same value. Here's how to simplify:

• First, identify the GCD of the fraction's numerator and denominator.
• Divide both by the GCD to obtain a simpler fraction.

For example, in \(7\frac{2}{4}\), the fraction \(\frac{2}{4}\) can be simplified. The GCD of 2 and 4 is 2, so divide both 2 and 4 by 2, resulting in \(\frac{1}{2}\). Thus, the mixed number becomes \(7\frac{1}{2}\) after simplification.
Simplifying fractions is important because it makes mathematical operations, like addition, subtraction, and comparison, easier by eliminating unnecessary complexity in calculations. It also presents the fraction in its most refined and universally understandable form.