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Chapter 4: Problem 56

Divide the fractions, and simplify your result. $$\frac{-22}{7} \div \frac{-18}{17}$$

Short Answer

Expert verified
The simplified result is \(\frac{187}{63}\).

Step by step solution

01

Understand the Division of Fractions

To divide one fraction by another, we actually multiply by the reciprocal of the second fraction. This means \[\frac{-22}{7} \div \frac{-18}{17} = \frac{-22}{7} \times \frac{17}{-18}\].
02

Multiply the Fractions

Now multiply the numerators together and the denominators together:\[\frac{-22 \times 17}{7 \times -18}\].
03

Calculate the Products

Calculate the product of the numerators and the denominators:\[-22 \times 17 = -374\] and \[7 \times -18 = -126\]. This gives us the fraction:\[\frac{-374}{-126}\].
04

Simplify the Fraction

A fraction with both a negative numerator and denominator becomes positive. Now find the greatest common divisor (GCD) of 374 and 126 to simplify the fraction:The GCD is 2. Divide both the numerator and denominator by 2:\[\frac{374}{126} = \frac{187}{63}\].
05

Verify Simplification

Check if the simplified fraction can be reduced further by finding GCD of 187 and 63, which is 1. Thus, \(\frac{187}{63}\) is already in its simplest form.

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Key Concepts

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

Mathematics Education
Understanding how to divide fractions is a key concept in mathematics education, as it builds essential problem-solving skills. When dealing with fractions, division can initially seem more complex than multiplication. However, the process becomes intuitive once we remember to multiply by the reciprocal.
  • Start by flipping the second fraction. This changes division into multiplication, which is typically more straightforward for students.
  • Next, multiply the numerators and the denominators separately, maintaining a clear and organized setup of the fractions.
  • This process reinforces multiplication skills as students must manage both positive and negative numbers.
These steps help build a strong foundation in fraction arithmetic, supporting further learning in algebra and more advanced mathematical fields.
Fractions Simplification
Simplifying fractions is a crucial skill that aids in understanding and performing efficient calculations. At the core, simplification makes a fraction as simple as possible while retaining its value.
  • Firstly, recognize when both the numerator and the denominator can be divided by the same number. This common factor helps reduce the fraction to its simplest form.
  • The simplified form of a fraction reveals the most reduced and useful version of the number.
  • In the given exercise, simplifying \(\frac{374}{126}\) to \(\frac{187}{63}\), using their greatest common factor, makes the computation much clearer.
Understanding simplification improves accuracy and lends greater insight into how numbers relate to one another, sharpening numerical reasoning skills.
Greatest Common Divisor
The greatest common divisor (GCD) is a key concept in fractions simplification and mathematics. It determines the largest number that can divide two given numbers evenly.
  • To find the GCD, list all divisors of each number and identify the largest one common to both.
  • The GCD simplifies fractions effectively, aiding in reducing even complex numbers to simpler forms without changing their value.
  • This is vital in the simplification process of \(\frac{374}{126}\), as identifying 2 as the GCD allows the reduction to \(\frac{187}{63}\).
  • Applying the GCD ensures that students can manage sizable calculations more effectively.
Learning about the GCD enhances cognitive skills and is an indispensable tool in problem-solving across various mathematical applications.

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