Question

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

Short answer

The answer is 1.

Step-by-step solution

Convert Mixed Numbers to Improper Fractions

The fractions given are \( \frac{6}{4} \) and \( \frac{1}{2} \), which are already in improper fractions or a simple fraction form. We proceed with them directly.

Find a Common Denominator

To subtract the fractions, they must have a common denominator. The least common denominator (LCD) of 4 and 2 is 4.

Adjust the Fractions

Adjust \( \frac{1}{2} \) to have the denominator of 4. Multiply both the numerator and denominator by 2: \( \frac{1}{2} = \frac{2}{4} \).

Subtract the Fractions

With the common denominator of 4, subtract the numerators: \( \frac{6}{4} - \frac{2}{4} = \frac{4}{4} \).

Simplify the Fraction

Simplify \( \frac{4}{4} \) to its lowest terms. Since the numerator equals the denominator, \( \frac{4}{4} = 1 \).

Key concepts

Improper Fractions

An improper fraction is a type of fraction where the numerator, or top number, is greater than or equal to the denominator, or bottom number. This means that the fraction is equal to or greater than 1. Examples include \( \frac{6}{4} \) and \( \frac{5}{3} \). Improper fractions can initially look complicated but they are really useful in operations like addition, subtraction, multiplication, and division because you don't have to separate whole numbers from fractional parts. Improper fractions are often result from converting mixed numbers into fractions or when operations between fractions result in values greater than the denominator. Unlike proper fractions, improper fractions can be seen as mixed numbers. For instance, \( \frac{6}{4} \) can be expressed as 1 whole and \( \frac{1}{2} \). When dealing with subtraction, like in the original exercise, it's beneficial to use improper fractions as it straightforwardly lines up the parts to subtract, without worrying about whole numbers.

Least Common Denominator

The least common denominator (LCD) is the smallest number that two or more numbers can all divide into without leaving a remainder. It is central to performing operations such as addition and subtraction with fractions that have different denominators. For example, if you are subtracting \( \frac{6}{4} - \frac{1}{2} \), you need a common ground to work on, which in this case, is the same base or denominator for both fractions. Since the denominators here are 4 and 2, you find the LCD by determining the smallest number that both 4 and 2 divide into evenly, which is 4. Finding the least common denominator helps to transform fractions into equivalent fractions with a shared base, simplifying the arithmetic process. Without it, comparing or performing operations on fractions could be actually more complex. Here’s a simple tip:

• Always identify the largest number among the denominators.
• See if the smaller denominators divide into it.
• If not, multiply larger numbers until you find a common multiple everyone can fit into.

Simplifying Fractions

Simplifying a fraction means reducing it to its smallest or simplest form, where the numerator and the denominator have no common factors other than 1. This means turning the fraction into one that is equivalent but has the smallest possible integers. In step 5 of the original solution, we encountered the fraction \( \frac{4}{4} \). Simplifying it was straightforward because both numbers are the same and divide evenly, yielding 1. This is an example of a fraction that can be simplified directly to a whole number. When simplifying more complex fractions:

• Determine the greatest common divisor (GCD) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCD.

This reduction doesn’t change the value of the fraction, it only makes it easier to read and understand.Simplifying is not just about "making it look nicer," but also features heavily in further math operations, as it simplifies subsequent calculations and comparisons.