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Chapter 5: Problem 80

Simplify each fraction. $$\frac{16}{28}$$

Short Answer

Expert verified
The simplified form of \( \frac{16}{28} \) is \( \frac{4}{7} \).

Step by step solution

01

Identify the Greatest Common Divisor (GCD)

To simplify the fraction \( \frac{16}{28} \), find the greatest common divisor (GCD) of the numerator and the denominator. The factors of 16 are 1, 2, 4, 8, 16 and the factors of 28 are 1, 2, 4, 7, 14, 28. The GCD of 16 and 28 is 4.
02

Divide the Numerator and the Denominator by the GCD

Divide both the numerator and the denominator by the GCD found in the previous step. So, divide 16 and 28 by 4. \( \frac{16 \div 4}{28 \div 4} = \frac{4}{7} \).
03

Confirm the Fraction is in Simplest Form

Check if \( \frac{4}{7} \) is in its simplest form. The factors of 4 are 1, 2, 4 and the factors of 7 are 1, 7. Since 4 and 7 do not have any common factors other than 1, \( \frac{4}{7} \) is in its simplest form.

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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 a key concept when simplifying fractions. It is the largest number that can evenly divide both the numerator and the denominator of a fraction. By discovering the GCD, you can easily reduce a fraction to its simplest form. Here's what you need to do:
  • List the factors of the numerator.
  • List the factors of the denominator.
  • Identify the largest factor that appears in both lists.
In the case of simplifying \( \frac{16}{28} \), we see that the GCD of 16 (with factors 1, 2, 4, 8, 16) and 28 (with factors 1, 2, 4, 7, 14, 28) is 4. Finding the GCD helps in breaking down fractions into the simplest terms easily, without any guesswork.
Numerator and Denominator
Understanding numerators and denominators is crucial in fraction simplification. A fraction consists of two parts:
  • The numerator, which is the number above the fraction bar and represents how many parts we have.
  • The denominator, which is the number below the fraction bar and shows the total number of equal parts the whole is divided into.
For example, in the fraction \( \frac{16}{28} \):- The numerator is 16, indicating the number of parts we consider.- The denominator is 28, indicating the total number of equal parts in the whole.By dividing both the numerator and the denominator by their GCD, we effectively simplify the fraction without altering its value. So, dividing 16 and 28 by their GCD (which is 4) gives us the simplified fraction \( \frac{4}{7} \).
Simplest Form
A fraction is in its simplest form when the numerator and the denominator have no common divisors other than 1. Achieving the simplest form is like finding the most reduced version of the fraction. Here's how it works:After dividing the fraction \( \frac{16}{28} \) by its GCD, we get \( \frac{4}{7} \).
  • The factors of 4 are 1, 2, and 4.
  • The factors of 7 are 1 and 7.
Since 4 and 7 share no common factors other than 1, \( \frac{4}{7} \) is indeed in its simplest form. This means no further division to simplify is possible, and you have successfully reduced the fraction to its smallest possible size.

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Most popular questions from this chapter

Find each sum. Write in simplest form. $$\frac{1}{10}+\frac{1}{3}$$

Find each quotient. Write in simplest form. $$\frac{a c}{5} \div \frac{c}{d}$$

Explain how rational numbers are related to other sets of numbers. Illustrate your reasoning with examples of numbers that belong to more than one set and examples of numbers that are only rational.

Write each fraction or mixed number as a decimal. Use a bar to show a repeating decimal. $$-7 \frac{4}{5}$$

Find the least common multiple (LCM) of each pair of numbers or monomials. $$21,28$$
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