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Chapter 1: Problem 19

Calculate and express each result in its simplest form:\(\frac{2}{3} \div \frac{5}{3}\)

Short Answer

Expert verified
The simplest form is \( \frac{2}{5} \).

Step by step solution

01

Understand Division of Fractions

To divide fractions, you multiply the first fraction (dividend) by the reciprocal of the second fraction (divisor). The reciprocal of a fraction is obtained by swapping its numerator and denominator.
02

Find the Reciprocal of Divisor

The divisor in this exercise is \( \frac{5}{3} \). Its reciprocal is \( \frac{3}{5} \) because the numerator and denominator are interchanged.
03

Multiply the Fractions

Multiply the dividend \( \frac{2}{3} \) by the reciprocal of the divisor \( \frac{3}{5} \): \[\frac{2}{3} \times \frac{3}{5} = \frac{2 \times 3}{3 \times 5}\]This yields \( \frac{6}{15} \).
04

Simplify the Result

Simplify the fraction \( \frac{6}{15} \) by finding the greatest common divisor (GCD) of 6 and 15, which is 3. Divide both the numerator and the denominator by the GCD:\[\frac{6 \div 3}{15 \div 3} = \frac{2}{5}\]

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Key Concepts

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

Reciprocal of a Fraction
A fraction can be thought of as a way to represent a division of two numbers. The reciprocal of a fraction is simply a fraction flipped upside down. To find the reciprocal of a fraction:\[\text{If the fraction is } \frac{a}{b}\text{, then its reciprocal is } \frac{b}{a}.\]
Finding the reciprocal is essential when dividing fractions because division by a fraction is the same as multiplying by its reciprocal.
This is why, in fraction division, we swap the numerator and denominator of the divisor. For example, the reciprocal of \(\frac{5}{3}\) is \(\frac{3}{5}\). This concept is rooted in the fundamental property of division that dividing by a number is equivalent to multiplying by its inverse.
Simplifying Fractions
When working with fractions, simplifying them is an important step. A simplified fraction is one where the greatest common divisor (GCD) of its numerator and denominator is 1.
This means that the numerator and denominator have no common factors other than 1.To simplify a fraction:
  • Find the GCD of the numerator and the denominator.
  • Divide both the numerator and the denominator by the GCD.
This process reduces the fraction to its simplest form. For example, to simplify the fraction \(\frac{6}{15}\), find the GCD of 6 and 15 that is 3. Then, divide both numbers by 3, resulting in \(\frac{2}{5}\). Simplification helps to make fractions easier to work with and understand.
Greatest Common Divisor
The greatest common divisor (GCD) is the largest positive integer that divides two or more integers without leaving a remainder.
It is a key tool when simplifying fractions.
To find the GCD:
  • List the factors of both the numerator and the denominator.
  • Identify the common factors shared by both numbers.
  • Select the largest factor from this list of common factors.
For our example, with the fraction \(\frac{6}{15}\):
- Factors of 6 are 1, 2, 3, and 6.
- Factors of 15 are 1, 3, 5, and 15.
The common factors are 1 and 3.
Thus, the GCD is 3. The operation of finding the GCD enables you to simplify fractions to their most reduced form easily, making calculations cleaner and more intuitive.

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Most popular questions from this chapter

Simplify if possible:\(a^{3} / a^{2}\)

The root mean square speed of the particles of an ideal gas at temperature \(T\) is \(c=(3 R T / M)^{1 / 2}\), where \(R=8.31447 \mathrm{~J} \mathrm{~K}^{-1} \mathrm{~mol}^{-1}\) and \(M\) is the molar mass. Confirm that \(c\) has dimensions of velocity.

Express as decimal fractions:\(10^{-2}\)

Express as decimal fractions:\(\frac{1}{25}\)

Calculate and express each result in its simplest form:\(\frac{11}{12}+\frac{3}{16}\)
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