Problem 77 - Fundamentals of Mathematics

Question

For the following problems, find the products. Be sure to reduce. $$16 \cdot \frac{1}{4}$$

Step-by-Step Solution

Answer

The product is 4.

Step 1: Write the Whole Number as a Fraction

Rewrite 16 as a fraction. A whole number can be written as a fraction with 1 as its denominator. So, \(16 = \frac{16}{1}\).

Step 2: Set Up the Multiplication of Fractions

Now set up the multiplication of the two fractions. Multiply \(\frac{16}{1}\) by \(\frac{1}{4}\) like this: \(\frac{16}{1} \times \frac{1}{4}\).

Step 3: Multiply the Numerators and Denominators

Multiply the numerators (the top numbers) and the denominators (the bottom numbers) separately. For the numerators: \(16 \times 1 = 16\).For the denominators: \(1 \times 4 = 4\).So, the result of the multiplication is \(\frac{16}{4}\).

Step 4: Simplify the Fraction

Simplify \(\frac{16}{4}\) by dividing both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 16 and 4 is 4.Divide the numerator by the GCD: \(16 \div 4 = 4\).Divide the denominator by the GCD: \(4 \div 4 = 1\).Thus, the simplified fraction is \(4\).

Key Concepts

Whole Numbers as Fractions

When you're multiplying fractions by whole numbers, the first step is to convert the whole number into a fraction itself. This might sound tricky, but it’s actually very simple! You use the number 1 as the denominator.
This means that any whole number, like 16 in our exercise, can be written as a fraction by placing it over 1.
Imagine it like this:

• The number 16 becomes \( \frac{16}{1} \).

This is a crucial step, as it sets up the problem correctly for fraction multiplication. It doesn't change the value, because any number divided by 1 is the number itself.Once you have both numbers in fraction form, you are ready to move to the next step, which is performing the multiplication.

Greatest Common Divisor

The Greatest Common Divisor (GCD), also known as the greatest common factor, is a key tool in simplifying fractions. Think of the GCD as the largest number that can divide both the numerator and the denominator evenly, without leaving a remainder.
In our exercise, when we arrived at the fraction \( \frac{16}{4} \), we needed to simplify it.Here's how you find the GCD:

• List the factors of the numerator (16), which are 1, 2, 4, 8, 16.
• List the factors of the denominator (4), which are 1, 2, 4.
• The largest number in both lists is 4. So, the GCD of 16 and 4 is 4.

Once you've identified the GCD, you can simplify the fraction by dividing both the numerator and the denominator by this number. This process of dividing by the GCD brings us to the simplest form of the fraction.

Simplifying Fractions

Simplifying fractions means making the fraction as simple as possible. This involves reducing the fraction to its lowest terms by dividing both the numerator and the denominator by their GCD.
In the exercise, we simplified \( \frac{16}{4} \) by using the GCD, which is 4.The steps to simplify are:

• Divide the numerator by the GCD: \( 16 \div 4 = 4 \).
• Divide the denominator by the GCD: \( 4 \div 4 = 1 \).

So, \( \frac{16}{4} \) simplifies to \( \frac{4}{1} \). Since \( \frac{4}{1} \) equals 4, we find that the simplest form of the fraction is actually a whole number 4.This simplification doesn't change the value, it just presents it in the simplest form possible, making it easier to understand and use in future calculations.