Question

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

Short answer

The result of the subtraction is 1.

Step-by-step solution

Convert Improper Fractions if Any

Since \(\frac{6}{4}\) is an improper fraction (the numerator is greater than the denominator), we need to simplify it first. Divide the numerator by the denominator: \(6 \div 4 = 1\) with a remainder of \(2\). So, \(\frac{6}{4}\) can be expressed as a mixed number: \(1\frac{2}{4}\). Simplify \(\frac{2}{4}\) to \(\frac{1}{2}\). Thus, \(\frac{6}{4} = 1\frac{1}{2}\).

Subtract the Whole Number and Fraction Separately

Subtract the fractions \(\frac{1}{2} - \frac{1}{2} = 0\). Since there is nothing left to subtract from the whole number \(1\), the result of the subtraction is \(1 - 0 = 1\).

Check for Simplification

The result \(1\) is already a whole number, and there is no common fraction to simplify further.

Key concepts

Improper Fractions

An improper fraction is simply a fraction where the numerator (the top number) is larger than the denominator (the bottom number). This means the fraction represents a value greater than one. Improper fractions can be troublesome to work with when subtracting because they often require simplification.
To simplify or convert an improper fraction, divide the numerator by the denominator. The quotient becomes the whole number, and the remainder becomes the new numerator over the original denominator. For example:

• Given: \(\frac{6}{4}\)
• Divide 6 by 4 to get 1 with a remainder of 2.
• The fraction becomes a mixed number: \(1 \frac{2}{4}\).

This mixed number can still be simplified further if the fraction part can be reduced. Converting to mixed numbers often makes subtraction easier later on.

Mixed Numbers

Mixed numbers combine whole numbers and proper fractions. They offer a practical way to express numbers, especially when the improper fractions are confusing or cumbersome.
Let's take a closer look at how they are used:

• Mixed numbers can simplify operations like subtraction when each part is handled separately (whole numbers apart from the fractions).
• For example, with \(1 \frac{2}{4}\), subtract the fractional parts separately from the whole parts.

Mixed numbers not only make numbers look neater but also clarify processes like borrowing in subtraction especially when dealing with unlike denominators.

Simplification in Fractions

Simplification is reducing a fraction to its lowest terms, which means simplifying the fraction as much as possible so that the numerator and the denominator have no common factors other than 1. This is crucial for ensuring the result is easily understood.
Simplification steps include:

• Identify the greatest common divisor (GCD) of the numerator and denominator.
• Divide both by the GCD.
• For \(\frac{2}{4}\), the GCD is 2, so divide both 2 and 4 by 2 to get \(\frac{1}{2}\).

This step is essential for proper presentation of the result, ensuring clarity and precision in mathematical expressions.

Whole Numbers in Fractions

Whole numbers can also participate in subtraction problems involving fractions. It helps to see whole numbers as fractions themselves to simplify operations.
Consider these points:

• Whole numbers can be expressed as fractions by placing them over 1 (e.g., 1 as \(\frac{1}{1}\)).
• This transformation allows consistent processes to be applied, such as finding a common denominator if necessary.
• In subtraction like \( \frac{6}{4} - \frac{1}{2}\), converting to mixed numbers and handling the whole number part separately streamlines the arithmetic.

Using whole numbers expressed as fractions often acts as a bridging tool in fraction problems, especially in subtraction, making calculations flow smoothly.