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Chapter 4: Problem 28

Write each fraction in simplest form. \(-\frac{21}{49}\)

Short Answer

Expert verified
The simplest form of \(-\frac{21}{49}\) is \(-\frac{3}{7}\).

Step by step solution

01

Identify the Common Factor

Determine the greatest common factor (GCF) of 21 and 49. Both numbers can be divided by 7, which is the largest factor they both share.
02

Divide Both Numerator and Denominator by the GCF

Divide the numerator (-21) and the denominator (49) by their GCF, which is 7. Calculating this gives: \[-\frac{21 \div 7}{49 \div 7} = -\frac{3}{7}\]
03

Confirm the Fraction is in Simplest Form

Verify that -3 and 7 have no common factors other than 1. Since 3 is a prime number and does not divide into 7, the fraction \(-\frac{3}{7}\) 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 Factor
When you're simplifying fractions, identifying the greatest common factor (GCF) of the numerator and the denominator is crucial. The GCF is the largest number that can evenly divide both the numerator and the denominator without leaving a remainder. In our example with the fraction \(-\frac{21}{49}\), we need to find the largest number that divides both 21 and 49. Here's how to do it:
  • List out all the factors of each number. For 21, the factors are 1, 3, 7, and 21. For 49, they are 1, 7, and 49.
  • Identify the largest factor that is common to both lists. In this case, it's 7.
Finding the GCF allows you to simplify fractions effectively. Simplification is achieved by dividing both the numerator and the denominator by this common factor.
Prime Numbers
Understanding prime numbers is essential when working with fractions. A prime number is a number greater than 1 that has no other divisors except 1 and itself. Recognizing these numbers can help you determine whether you've simplified a fraction adequately.In the solution for \(-\frac{21}{49}\), we saw that 3 appeared as part of the simplified fraction \(-\frac{3}{7}\). Confirming whether a number is prime can help you ensure the fraction is in its simplest form:
  • 3 is a prime number since its only divisors are 1 and 3.
  • Checking that prime numbers remain in your fraction means they can't be simplified further with the other fraction's elements.
Seeing prime numbers as part of your fraction in simplest form is a good indication that further reduction is impossible.
Numerator and Denominator Division
Dividing the numerator and denominator by their greatest common factor simplifies the fraction. This division needs to be accurate to ensure you've reduced the fraction properly.Here's how the division process works:
  • In the fraction \(-\frac{21}{49}\), divide both 21 and 49 by the GCF, which we identified as 7.
  • Perform the calculations: 21 divided by 7 equals 3, and 49 divided by 7 equals 7. So, the fraction becomes \(-\frac{3}{7}\).

Now, verify that no number other than 1 can further divide both resulting numbers (3 and 7). Since they have no common divisors other than 1, \(-\frac{3}{7}\) is indeed the simplest form. Mastering this division step is essential in the simplification process, ensuring the fraction is reduced to the smallest possible equivalent.

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Most popular questions from this chapter

Write each fraction in two other equivalent ways by inserting the negative sign in different places. $$-\frac{21}{4}=\quad=$$

Perform the indicated operations. $$ -4 \frac{2}{5} \cdot 2 \frac{3}{10} $$

Perform the indicated operations. $$ -5 \frac{1}{8}-19 \frac{3}{4} $$

Answer true or false for each statement.What operation should be performed first to simplify $$\frac{1}{5} \cdot \frac{5}{2}-\left(\frac{2}{3}+\frac{4}{5}\right)^{2} ?$$ Explain your answer.

Translate each phrase into an algebraic expression. Divide a number by \(-6 \frac{1}{11}\)
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