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Chapter 0: Problem 2

For exercises 1-12, simplify. $$ \frac{21}{56} $$

Short Answer

Expert verified
The simplified form of \(\frac{21}{56}\) is \(\frac{3}{8}\).

Step by step solution

01

Identify the Greatest Common Divisor (GCD) of the Numerator and Denominator

To simplify the fraction, find the greatest common divisor (GCD) of 21 and 56. The GCD is the largest number that divides both numbers without leaving a remainder.
02

Determine the GCD

List the factors of each number:- Factors of 21: 1, 3, 7, 21- Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56The largest common factor is 7, so the GCD of 21 and 56 is 7.
03

Divide Both Numerator and Denominator by the GCD

Divide both the numerator and the denominator by their GCD (7):\(\frac{21 \div 7}{56 \div 7} = \frac{3}{8}\)

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Greatest Common Divisor
The Greatest Common Divisor (GCD) is essential when simplifying fractions. It is the largest number that can divide both the numerator and the denominator without leaving a remainder. To find the GCD of two numbers, list out all the factors of each number and identify the largest one they have in common. In our example, the fraction is \(\frac{21}{56}\). We listed the factors of 21 and 56 and found that the greatest common factor they share is 7. Hence, the GCD of 21 and 56 is 7. This will help us simplify the fraction by dividing both the numerator and the denominator by this number.
Numerator
The numerator is the top number in a fraction. It represents the number of parts we have out of a whole. For instance, in the fraction \(\frac{21}{56}\), 21 is the numerator. In the process of simplifying fractions, you divide the numerator by the Greatest Common Divisor (GCD) to reduce the fraction to its simplest form. Using our example, the GCD of 21 and 56 is 7. We divide 21 by 7, resulting in 3, making the new numerator in our simplified fraction \(\frac{3}{8}\).
Denominator
The denominator is the bottom number in a fraction. It indicates the total number of equal parts the whole is divided into. In our fraction \(\frac{21}{56}\), 56 is the denominator. To simplify the fraction, you also divide the denominator by the Greatest Common Divisor (GCD). Since our GCD is 7, we divide 56 by 7, which gives us 8. Hence, the simplified fraction becomes \(\frac{3}{8}\). This process helps in making fractions easier to understand and work with.
Factors
Factors are the numbers you multiply together to get another number. For example, the factors of 21 are 1, 3, 7, and 21 because 21 can be evenly divided by these numbers. Similarly, the factors of 56 are 1, 2, 4, 7, 8, 14, 28, and 56. To simplify a fraction, you need to find the greatest common factor of both the numerator and the denominator. This common factor is the Greatest Common Divisor (GCD). By dividing both the numerator and the denominator by this GCD, you get the simplest form of the fraction. For \(\frac{21}{56}\), the GCD is 7, giving us the simplified fraction \(\frac{3}{8}\) after division.

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