Problem 35 - Prealgebra

Question

The problems below will allow you to review subtraction of fractions and mixed numbers. $$\frac{3}{4}-\frac{1}{4}$$

Step-by-Step Solution

Answer

\(\frac{3}{4} - \frac{1}{4} = \frac{1}{2}\).

Step 1: Understand Fractions

Fractions are part of a whole. In this problem, both fractions have the same denominator of 4, which means they are parts of the same whole.

Step 2: Subtract the Numerators

Since the denominators are the same, you simply subtract the numerators (the top numbers of the fractions). This means: \[\frac{3}{4} - \frac{1}{4} = \frac{3-1}{4} = \frac{2}{4}.\]

Step 3: Simplify the Fraction

To simplify \(\frac{2}{4}\), find the greatest common divisor (GCD) of 2 and 4, which is 2, and divide both the numerator and the denominator by the GCD. \[\frac{2 \div 2}{4 \div 2} = \frac{1}{2}.\] Thus, \(\frac{2}{4}\) simplifies to \(\frac{1}{2}\).

Key Concepts

Fractions

Fractions represent parts of a whole and are used in mathematics to denote division of quantities. Each fraction consists of two parts:

• The **numerator**: This is the number on top of the fraction and indicates how many parts of the whole are being considered.
• The **denominator**: This is the number below the line and indicates how many equal parts the whole is divided into.

Understanding that fractions are a way of expressing division helps to simplify problems involving fractions. In the subtraction problem \(\frac{3}{4} - \frac{1}{4}\), both fractions have the same denominator (4), so they represent parts of the same whole. This makes subtraction straightforward as only the numerators are subtracted. Always ensure that the denominators are the same before performing arithmetic operations.

Mixed Numbers

A mixed number combines a whole number with a fraction. For example, 1\(\frac{1}{2}\) consists of the whole number 1 and the fraction \(\frac{1}{2}\). Mixed numbers represent quantities larger than a whole and are useful in various arithmetic operations.

When dealing with mixed numbers, it's sometimes easier to convert them to improper fractions before performing calculations:

• **Convert the whole number** into a fraction with the same denominator.
• **Add the fractions** together to create an improper fraction.

This can simplify the subtraction or addition of fractions, especially when one or both of the numbers in a problem are mixed numbers.

Simplifying Fractions

Simplifying fractions means reducing the fraction to its simplest form, where the numerator and the denominator have no common factors other than 1. Starting with a fraction like \(\frac{2}{4}\), the simplest form is found by dividing both parts by their greatest common divisor (GCD). For example:

• Determine the GCD of 2 and 4, which is 2.
• Divide the numerator and the denominator by the GCD: \(\frac{2 \div 2}{4 \div 2} = \frac{1}{2}\).

Simplifying fractions is an essential skill in mathematics as it helps to express results in the most concise form. This concept not only helps clarify the solution but also makes it easier to compare, add, or subtract fractions in future steps.