Problem 63 - Prealgebra

Question

Multiply. $$2 \cdot \frac{1}{2}$$

Step-by-Step Solution

Answer

The product of \( 2 \times \frac{1}{2} \) is 1.

Step 1: Understand the Operation

The problem asks us to multiply two numbers: the integer 2 and the fraction \( \frac{1}{2} \). Multiplication involves combining these two numbers in a way that accounts for both.

Step 2: Multiply the Integer by the Numerator

When multiplying an integer by a fraction, we multiply the integer by the numerator of the fraction. The numerator of \( \frac{1}{2} \) is 1. So, we calculate: \( 2 \times 1 = 2 \).

Step 3: Divide by the Denominator

After multiplying by the numerator, we divide the result by the denominator of the fraction. The denominator of \( \frac{1}{2} \) is 2, so we calculate \( \frac{2}{2} = 1 \).

Step 4: Simplify the Result

The result \( \frac{2}{2} \) simplifies to 1 because both the numerator and the denominator are the same.

Key Concepts

Integer Multiplication

Integer multiplication is a fundamental arithmetic operation that involves combining two whole numbers. When you multiply integers, you are essentially adding a number to itself a certain number of times. For instance, if you multiply 2 by 3, you can think of it as adding 2 three times: 2 + 2 + 2 = 6.
Here's a simple process for multiplying integers:

• Consider two numbers, such as 4 and 5.
• Calculate the total: 4 added 5 times (or vice versa) gives 20.
• This is also represented as 4 times 5 or 4 x 5, which equals 20.

When one of the integers is 1, the result is always the other number because multiplying by 1 doesn’t change the value of a number. Multiplying any integer by 0 results in 0, emphasizing the element of nullity in multiplication.

Fractions

Fractions represent parts of a whole and are expressed as one number over another, like \( \frac{a}{b} \), where \(a\) is the numerator and \(b\) is the denominator. When dealing with fractions, it's crucial to understand how they behave when combined with integers or other fractions.
Here are some basics:

• The numerator indicates how many parts we have.
• The denominator indicates how many parts make up a whole.
• Fractions are a way of expressing division or a ratio between numbers.

Fractions can be either proper or improper. A proper fraction means the numerator is smaller than the denominator, while an improper fraction has a numerator larger than or equal to the denominator. Fractions can also be converted to decimals by performing the division, and this is handy for quick calculations or comparisons.

Simplification

Simplification is about making a mathematical expression easier to work with without changing its value. When we simplify fractions, we find an equivalent fraction where the numerator and denominator have no common factors other than 1.
To simplify a fraction:

• Identify the greatest common factor (GCF) of the numerator and the denominator.
• Divide both the numerator and the denominator by this GCF.
• The resulting fraction is the simplest form.

For example, with \( \frac{6}{8} \), the GCF of 6 and 8 is 2. Dividing both by 2, we get \( \frac{3}{4} \).Simplification extends to expressions involving both integers and fractional components. It ensures that results are as straightforward as possible, avoiding unnecessary complication in calculations.