Problem 8 - Prealgebra

Question

Find the value of each expression. $$\frac{10-4}{1+2}$$

Step-by-Step Solution

Answer

The value of the expression is 2.

Step 1: Evaluate the Numerator

The numerator of the fraction is \(10 - 4\). Calculate this by subtracting 4 from 10, which gives \(6\).

Step 2: Evaluate the Denominator

The denominator of the fraction is \(1 + 2\). Calculate this by adding 1 to 2, which gives \(3\).

Step 3: Divide the Numerator by the Denominator

Now that we have the simplified forms of the numerator (\(6\)) and the denominator (\(3\)), divide 6 by 3. \(\frac{6}{3} = 2\).

Key Concepts

Fractions

In mathematics, a fraction represents a part of a whole. It consists of two main parts: the numerator and the denominator. These two parts are written as a ratio, with the numerator above a horizontal or slanting line, and the denominator below it. For example, in the fraction \(\frac{a}{b}\), \(a\) is the numerator, and \(b\) is the denominator. Fractions are used to express values that are not whole numbers, allowing us to represent quantities that fall between whole number values, such as half of a pizza or three-quarters of a cup of sugar.
Fractions can be further classified into different types:

• Proper Fractions: The numerator is smaller than the denominator, such as \(\frac{3}{4}\).
• Improper Fractions: The numerator is greater than or equal to the denominator, such as \(\frac{5}{3}\) or \(\frac{4}{4}\).
• Mixed Numbers: A combination of a whole number and a proper fraction, such as \(1\frac{2}{3}\).

Understanding how to manipulate fractions is fundamental, as it opens the door to more complex mathematical concepts. Always perform operations on the numerator and denominator separately when simplifying or solving fractions.

Numerator and Denominator

The numerator and denominator are essential parts of a fraction, each playing a distinct role in determining the fraction's value. Let's explore their functions and importance:

• Numerator: This is the top part of the fraction and represents how many parts of a whole or a set you have. In the expression \(\frac{10-4}{1+2}\), the numerator is \(10 - 4\). When you calculate it, the result is \(6\).
• Denominator: This is the bottom part of the fraction and indicates the total number of equal parts into which the whole is divided. In our example, the denominator is \(1 + 2\). After performing the addition, it equals \(3\).

The relationship between the numerator and the denominator defines the value of the fraction. When the numerator is larger than the denominator, you have more parts than whole, which is crucial for understanding and solving fraction-based problems. By manipulating the numerator or the denominator, we change the fraction's value, which is often essential in both simple and complex mathematical operations.

Division

Division is one of the basic arithmetic operations and is crucial when working with fractions, especially in solving expressions like \(\frac{a}{b}\). In the context of fractions, division directly relates to the concept of how many times the denominator can "fit" into the numerator.
When you divide the numerator by the denominator, you're essentially asking: "How many groups of the denominator's size can be made from the numerator?"

• In our fraction example, the calculation \(\frac{6}{3}\) asks how many times \(3\) can "fit" into \(6\). The answer is \(2\), meaning \(3\times2 = 6\).

To perform division correctly:

• First, ensure the fraction is as simplified as possible before dividing. This makes calculations easier and reduces errors.
• Recognize that division is essentially repeated subtraction or the inverse of multiplication. This helps in understanding problems conceptually.

Division gives meaning to fractions and helps provide a concrete answer or solution to how numbers relate or compare to one another through proportions.