/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 21 \(\frac{45}{21}\)... [FREE SOLUTION] | 91影视

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Chapter 9: Problem 21

\(\frac{45}{21}\)

Short Answer

Expert verified
The simplified fraction is \(\frac{15}{7}\).

Step by step solution

01

Identify the Greatest Common Divisor (GCD)

To simplify the fraction \(\frac{45}{21}\), first, identify the greatest common divisor of the numerator (45) and the denominator (21). List the divisors of each number: 45 can be divided by 1, 3, 5, 9, 15, and 45, while 21 can be divided by 1, 3, 7, and 21. The largest common divisor is 3.
02

Divide the Numerator and Denominator by the GCD

Now that we have determined the GCD to be 3, divide both the numerator and the denominator of the fraction by this number. Divide 45 by 3 to get 15 and 21 by 3 to get 7.
03

Write the Simplified Fraction

Having divided both parts of the fraction by their GCD, the simplified fraction is \(\frac{15}{7}\).

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Key Concepts

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

Greatest Common Divisor
In the process of simplifying fractions, the concept of the Greatest Common Divisor (GCD) is invaluable. The GCD is the largest number that can exactly divide both the numerator and the denominator without leaving a remainder.
To find the GCD, list the divisors (factors) of both numbers involved. For example, in the fraction \( \frac{45}{21} \), the divisors of 45 are 1, 3, 5, 9, 15, and 45. Likewise, the divisors of 21 are 1, 3, 7, and 21.
The largest common number found in both lists is the GCD. Thus, the GCD of 45 and 21 is 3, which will be used to simplify the fraction.
Numerator and Denominator
Understanding what the numerator and denominator represent is crucial in fraction simplification. The numerator is the top number in a fraction, indicating how many parts we are considering. The denominator is the bottom number, showing the total number of equal parts the whole is divided into.
In \( \frac{45}{21} \), 45 is the numerator, and 21 is the denominator.
  • You need a thorough understanding of these parts to apply operations correctly, such as finding a common divisor.
  • This is necessary because operations like simplification involve altering both numbers while keeping their ratio the same.
When simplifying, both the numerator and denominator are divided by their GCD to keep the fraction's value unchanged but in a simpler form.
Fraction Division
Fraction division is an essential step in simplifying fractions. The crux of fraction division in this context is about dividing both the numerator and the denominator by their GCD.
For the fraction \( \frac{45}{21} \), once the GCD of 3 is determined, both numbers are divided by this common factor. Therefore, \( 45 \div 3 = 15 \) and \( 21 \div 3 = 7 \).
  • This step results in a new fraction \( \frac{15}{7} \), which is much simpler than the original but equal in value.
  • Through division by the GCD, the fractional value remains unchanged, just in an easier-to-use form.
This process ensures the original quantity represented by the fraction is maintained accurately after simplification.

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Most popular questions from this chapter

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