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Chapter 7: Problem 78

perform the indicated operation and simplify. $$\frac{2}{x} \div \frac{8}{y}$$

Short Answer

Expert verified
\frac{y}{4x}

Step by step solution

01

Understand Division of Fractions

To divide fractions, multiply the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is obtained by swapping the numerator and the denominator.
02

Identify the Reciprocal

The reciprocal of \(\frac{8}{y}\) is \(\frac{y}{8}\).
03

Multiply by the Reciprocal

Replace the division with multiplication by the reciprocal: \(\frac{2}{x} \times \frac{y}{8}\).
04

Multiply the Fractions

Multiply the numerators and the denominators separately: \(\frac{2 \times y}{x \times 8} = \frac{2y}{8x}\).
05

Simplify the Fraction

Simplify \(\frac{2y}{8x}\) by dividing the numerator and the denominator by the common factor, which is 2: \(\frac{2y \div 2}{8x \div 2} = \frac{y}{4x}\).

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Key Concepts

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

Reciprocals
In mathematics, the reciprocal of a fraction is simply flipping the numerator (top part) and the denominator (bottom part) of the fraction. For example, the reciprocal of \(\frac{8}{y}\) is \(\frac{y}{8}\). This concept is essential when dividing fractions, as it allows us to turn the division into a multiplication problem.
To find the reciprocal of a fraction:
  • Swap the numerator and the denominator.
  • Ensure that neither part of the fraction is zero, as zero does not have a reciprocal.
Understanding reciprocals helps in simplifying complex algebraic fractions, making division easier and more manageable.
Fraction Multiplication
Once you have the reciprocal of the second fraction in a division problem, multiply it with the first fraction. This step transforms a division problem into a straightforward multiplication.
To multiply fractions:
  • Multiply the numerators (top parts) of the two fractions together.
  • Multiply the denominators (bottom parts) of the two fractions together.
For instance, in our problem, we turn \(\frac{2}{x} \div \frac{8}{y}\) into \(\frac{2}{x} \times \frac{y}{8}\). Then, we multiply the numerators: 2 \times y = 2y, and denominators: x \times 8 = 8x, resulting in \(\frac{2y}{8x}\). This new fraction now represents our product.
Simplifying Fractions
Simplifying a fraction means making it as simple as possible. This often involves reducing it to its lowest terms using common factors for the numerator and the denominator. For the fraction \(\frac{2y}{8x}\), we look for common factors. Both 2y and 8x are divisible by 2.
To simplify:
  • Divide both the numerator and the denominator by the greatest common divisor (GCD).
  • In our problem, the GCD of 2y and 8x is 2.
We divide 2y and 8x each by 2: \(\frac{2y \div 2}{8x \div 2} = \frac{y}{4x}\). By doing this, we simplify \(\frac{2y}{8x}\) to \(\frac{y}{4x}\). It's crucial to always simplify where possible to make the fraction as simple and easy to understand as possible.

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