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Chapter 1: Problem 72

\(-\frac{16}{21} \div \frac{12}{27}\)

Short Answer

Expert verified
The solution to \(-\frac{16}{21} \div \frac{12}{27}\) is \(\frac{12}{7}\)

Step by step solution

01

Convert Divison to Multiplication

First, change the division operation into multiplication by flipping the second fraction. The problem then becomes \(-\frac{16}{21} \times -\frac{27}{12}\)
02

Multiply numerators and denominators

Multiply the numerators together and do the same for the denominators. \(-16 \times -27 = 432\) and \(21 \times 12 = 252\). Therefore, the new fraction is \(\frac{432}{252}\)
03

Simplify the Result

Simplify the resulting fraction to its lowest terms. The Greatest Common Divisor (GCD) of 432 and 252 is 36. Dividing both numerator and denominator by 36 will give \(\frac{432}{36}\) / \(\frac{252}{36}= \frac{12}{7}\)

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Key Concepts

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

Multiplying Fractions
When dividing fractions, the first step is to change the division to multiplication. This involves flipping the second fraction, known as finding its reciprocal. By doing so, a division problem like \(-\frac{16}{21} \div \frac{12}{27}\) becomes a multiplication problem: \(-\frac{16}{21} \times -\frac{27}{12}\).Multiplying fractions is straightforward, and involves:
  • Multiplying the numerators (the top numbers) of the fractions together.
  • Multiplying the denominators (the bottom numbers) together.
For our example, this means calculating \(-16 \times -27 = 432\) for the numerator and \(21 \times 12 = 252\) for the denominator, resulting in the new fraction \(\frac{432}{252}\).Be sure to multiply both numerators and denominators straight across, as this is essential in obtaining the correct product.
Simplifying Fractions
Once you have the product of multiplying fractions, it is important to simplify the fraction to its simplest form. Simplification makes the fraction easier to understand and work with. The fraction \(\frac{432}{252}\) can be simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD).To simplify fractions:
  • Find the GCD of the numerator and the denominator.
  • Divide both numerator and denominator by this GCD.
In our case, the GCD of 432 and 252 is 36. By dividing both by 36, \(\frac{432}{36} = 12\) and \(\frac{252}{36} = 7\), we get the simplest form: \(\frac{12}{7}\). Always check if the fraction can be further reduced to ensure it is fully simplified.
Greatest Common Divisor (GCD)
The Greatest Common Divisor (GCD) is a key concept in simplifying fractions. It is the largest number that can divide both the numerator and the denominator without leaving a remainder. Finding the GCD helps reduce fractions to their simplest form by:
  • Identifying the highest common factor of two numbers.
  • Enabling the division of each by this common factor for simplification.
In the fraction \(\frac{432}{252}\), calculating the GCD involves finding common factors of 432 and 252, which is 36 as determined in this example.There are various methods to find the GCD, such as:
  • Listing out all factors of each number and choosing the largest common one.
  • Using the Euclidean algorithm, which involves repeated division.
By using the GCD, you ensure that the simplified fraction is accurate and in its simplest term.

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Most popular questions from this chapter

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