Problem 35 - Elementary Algebra

Question

Find the value of each of the following expressions. $$ \frac{21}{7} $$

Step-by-Step Solution

Answer

Answer: The value of the expression \(\frac{21}{7}\) is 3.

Step 1: Identify the Numerator and Denominator

In the given expression \(\frac{21}{7}\), the numerator is 21 and the denominator is 7.

Step 2: Divide the Numerator by the Denominator

Now, divide the numerator (21) by the denominator (7): \(21\div7\)

Step 3: Simplify the Expression

The division of 21 by 7 results in the integer value 3, so and the expression simplifies to: $$ \frac{21}{7}=3 $$ The value of the expression \(\frac{21}{7}\) is 3.

Key Concepts

Understanding the Numerator

In any fraction, the numerator is the number found on the top. It shows how many parts of a whole are being considered. For example, in the fraction \( \frac{21}{7} \), the numerator is 21. This means we are considering 21 parts out of a possible number of parts indicated by the denominator below. - The numerator can be any integer, but its role is always to represent the number of pieces we have. - It's vital to locate the numerator before performing any operations on a fraction. In our exercise, identifying the numerator (21) helped us understand how many parts we were working with before dividing by the denominator.

Defining the Denominator

The denominator is the number located at the bottom of a fraction. It denotes the total number of equal parts the whole is divided into. In the fraction \( \frac{21}{7} \), the denominator is 7. It tells us that the whole is divided into 7 equal parts. - The denominator cannot be zero, as division by zero is undefined. - It plays a crucial role in determining the value of each part. With our fraction \( \frac{21}{7} \), knowing the denominator was essential to perform the division and simplify the fraction to find the final value.

Simplifying Fractions

Simplification of fractions involves reducing them to their simplest form. A fraction is considered simplified when the numerator and denominator are the smallest possible integers that keep the same value of the fraction.- To simplify a fraction, divide both the numerator and the denominator by their greatest common divisor (GCD). - If the numerator is divisible evenly by the denominator, the fraction simplifies to an integer value. In our example, \( \frac{21}{7} \), we simplified by dividing 21 by 7, which gave us the integer 3. This shows that \( \frac{21}{7} = 3 \) is the simplest form, where the fraction represents a whole number without any remaining parts.