Problem 53 - Fundamentals of Mathematics

Question

Reduce, if possible, each fraction. $$\frac{21}{35}$$

Step-by-Step Solution

Answer

\(\frac{21}{35}\) reduces to \(\frac{3}{5}\).

Step 1: Identify the Greatest Common Divisor (GCD)

To reduce a fraction, we must identify the greatest common divisor (GCD) of the numerator and the denominator. The numerator is 21 and the denominator is 35. Let's determine the divisors for each number and find the largest common divisor.

Step 2: List the Divisors of 21 and 35

The divisors of 21 are 1, 3, 7, and 21. The divisors of 35 are 1, 5, 7, and 35. By comparing these lists, we can see that the largest common divisor is 7.

Step 3: Divide by the GCD

Divide both the numerator and the denominator by their greatest common divisor to simplify the fraction. We will divide both 21 and 35 by 7.\[ \frac{21}{35} = \frac{21 \div 7}{35 \div 7} = \frac{3}{5} \]

Step 4: Verify the Simplified Fraction

Ensure that the simplified fraction \(\frac{3}{5}\) cannot be further reduced by checking if 3 and 5 have any common divisors other than 1. Since they don't, \(\frac{3}{5}\) is the simplest form of the fraction.

Key Concepts

Greatest Common Divisor

The greatest common divisor (GCD) is an essential concept in math that helps us simplify fractions. It's the largest number that divides two numbers without leaving a remainder. For instance, when simplifying the fraction \(\frac{21}{35}\), finding the GCD is crucial. Here, we need to identify the largest common number that can divide both 21 and 35. Knowing how to find the GCD simplifies many mathematical processes, making calculations quicker and reducing errors.

• Start by listing all possible divisors of each number.
• Identify the largest number that appears in both lists.

Understanding the GCD doesn't just help with fractions; it also applies in many other math problems, involving ratios, factoring, and divisor calculations.

Simplifying Fractions

Simplifying fractions means expressing them in their simplest form. It involves reducing a fraction by dividing both the numerator and the denominator by their greatest common divisor. For example, with the fraction \(\frac{21}{35}\), once you find that the GCD is 7, you divide both terms by 7 to simplify.

• The original fraction is \(\frac{21}{35}\).
• Divide both the numerator and the denominator by 7.
• This gives you the simplified fraction \(\frac{3}{5}\).

A simplified fraction is easier to interpret and compute with. It's often used in mathematics to make expressions more manageable and elegant.

Numerator and Denominator

Understanding the roles of the numerator and the denominator is key to mastering fractions. The numerator is the top part of the fraction, and the denominator is the bottom part. In \(\frac{21}{35}\), 21 is the numerator, representing the part of the whole, while 35 is the denominator, showing how many equal parts make the whole.

• A higher numerator means a larger number of portions or segments.
• The denominator indicates into how many parts the whole is divided.

Recognizing these roles helps in operations like simplifying because knowing which number is which guides how we handle the fractions in calculations.

Divisors of a Number

Divisors of a number are integers that divide that number completely without leaving a remainder. They are important when finding the greatest common divisor, particularly in simplifying fractions. Take the numbers 21 and 35: identifying their divisors makes it easier to reduce \(\frac{21}{35}\) to its simplest form by finding their GCD.

• For 21, consider divisors like 1, 3, 7, and 21.
• For 35, divisors include 1, 5, 7, and 35.

By determining which divisors are shared between two numbers, we can find the GCD, crucial in reducing fractions.