Problem 24 - Basic College Mathematics

Question

Multiply and simplify. $$ 16 \cdot \frac{1}{2} $$

Step-by-Step Solution

Answer

8

Step 1: Understand the Problem

We need to multiply the whole number 16 by the fraction \( \frac{1}{2} \).

Step 2: Multiply the Numerator

To multiply a whole number by a fraction, multiply the numerator (top number) of the fraction by the whole number. Here we get: \( 16 \times 1 = 16 \).

Step 3: Keep the Denominator

The denominator (bottom number) remains the same. So, our fraction becomes: \( \frac{16}{2} \).

Step 4: Simplify the Fraction

Divide the numerator by the denominator to simplify: \[ \frac{16}{2} = 8 \].

Key Concepts

whole number

A whole number is a number without any fractional or decimal part. In basic arithmetic, whole numbers are used to count objects and represent complete quantities. Numbers like 0, 1, 2, 3, and so forth are all whole numbers. Unlike fractions, whole numbers are straightforward and easy to understand and use in calculations. In our exercise, 16 is the whole number we need to work with.
Whole numbers are essential when learning fractions too. The interaction between whole numbers and fractions was highlighted when we multiplied 16 by \(\frac{1}{2}\).

fraction

A fraction represents a part of a whole. It consists of two parts: the numerator (top number) and the denominator (bottom number). For example, in \(\frac{1}{2}\), 1 is the numerator and 2 is the denominator. The numerator indicates how many parts we have, while the denominator shows the total number of equal parts.
Working with fractions involves performing operations like addition, subtraction, multiplication, and division. In our exercise, we multiplied the whole number 16 by the fraction \(\frac{1}{2}\). This required us to multiply the numerator of the fraction by the whole number, resulting in \(\frac{16}{2}\).
Despite fractions seeming complicated, they follow simple rules that make calculations straightforward. Multiplying a fraction by a whole number is one of these applications.

simplification

Simplification is the process of reducing a mathematical expression to its simplest form without changing its value. When dealing with fractions, simplification involves dividing both the numerator and denominator by their greatest common divisor (GCD). This ensures the fraction is expressed in its most reduced form.
In our example, once we multiplied 16 by \(\frac{1}{2}\) and obtained \(\frac{16}{2}\), we simplified the fraction by dividing the numerator by the denominator. This gave us a result of 8.
It's important to simplify fractions whenever possible, as it makes them easier to understand, compare, and use in further calculations.