Problem 85 - Prealgebra

Question

Divide. Round the answers to the nearest thousandth, if necessary. $$\frac{21}{42}$$

Step-by-Step Solution

Answer

The simplified and rounded result is 0.5.

Step 1: Understand the Fraction

The fraction \( \frac{21}{42} \) requires division of the numerator by the denominator. This means we need to perform the division of 21 by 42.

Step 2: Simplify the Fraction

First, we check if \( \frac{21}{42} \) can be simplified by finding the greatest common divisor (GCD) of 21 and 42. The GCD is 21. Divide both the numerator and the denominator by 21. \( \frac{21 \div 21}{42 \div 21} = \frac{1}{2} \)

Step 3: Divide Numerically

Now divide 1 by 2. \( 1 \div 2 = 0.5 \).

Step 4: Round to the Nearest Thousandth

Check if the decimal result needs rounding. The answer 0.5 is already rounded to the thousandth place as it has only one decimal digit after the point.

Key Concepts

Numerator

In any given fraction, the numerator is the number above the line or bar, and it represents how many parts of a whole or collection are being considered. In the exercise with the fraction \( \frac{21}{42} \), the numerator is 21.
The numerator plays a crucial role in the process of division, as it shows the quantity we are starting with. In the example, if we think of the number 21, it represents 21 units of something that we will proportion in groups specified by the denominator.

• It is the top number in a fraction.
• Acts as the dividend in division.
• Specifies how many parts there are in consideration.

Understanding the numerator makes it easier to follow fraction operations, such as simplifying or performing arithmetic.

Denominator

The denominator of a fraction is the number below the line. In our example \( \frac{21}{42} \), the denominator is 42.
This number is highly significant as it indicates the total number of equal parts the whole is divided into. It dictates the scale or size of each part in the fraction.

• It is the bottom number in a fraction.
• Acts as the divider in division.
• Shows how many equal parts make up a whole.

The denominator's role becomes clear in division, as it tells us how many we need to group together to form one complete unit. In \( \frac{21}{42} \), 42 shows us that the entity is divided into 42 parts.

Simplifying Fractions

Simplifying fractions means reducing a fraction to its simplest form where the numerator and the denominator no longer have any common divisor except for 1.
In our example \( \frac{21}{42} \), the fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 21.
This results in \( \frac{1}{2} \).

• Determine the common factors of the numerator and denominator.
• Divide both by their greatest common factor.
• Your simplest form has a numerator and a denominator of coprime numbers.

Simplifying fractions makes them easier to work with, especially in arithmetic operations and comparisons between fractions.

Greatest Common Divisor

The concept of the greatest common divisor (GCD) is central to simplifying fractions.
The GCD of two numbers is the largest number that divides both without leaving a remainder.
For the fraction \( \frac{21}{42} \), the greatest common divisor of 21 and 42 is found to be 21.

• Check divisibility among the numbers involved.
• The GCD is the highest possible number that can divide both numbers fully.
• Applying GCD helps simplify fractions efficiently.

Finding the GCD helps in reducing fractions to their simplest forms swiftly and accurately, enhancing our ease in handling fractions during various calculations.