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0.2 Adding fractions  (Page 3/2)

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This module is part of a collection of modules written for students enrolled in a Pre Calculus Course for Pre Engineers (MATH 1508) at the University of Texas at El Paso. The module introduces how fractions can be added through the use of the lowest common denominator method. To illustrate the principle, an introduction to Ohm's Law is presented. In addition, the rules that govern the determination of the equivalent resistance of series and parallel connections of resistors are introduced. Finally, exercises involving the addition of fractions in the context of parallel connections of resistors are included to emphasize the importance of the lowest common denominator method in engineering calculations.

Adding fractions

Introduction

In order to enjoy success as an engineer, it is important to learn how to add fractions. In this module, you will learn to add fractions using the lowest common denominator (LCD) method. Also, you will learn the role that the addition of fractions plays in determining the equivalent resistance of resistors connected in parallel.

Lowest common denominator (lcd) method

In the course of working algebraic problems, one often encounters situations that require the addition of fractions with unequal denominators. For example, let us consider the following

3 10 + 1 12 size 12{ { {3} over {"10"} } + { {1} over {"12"} } } {}

In order to add the two fractions, it is important to rewrite each fraction with the same denominator. In order to accomplish this, we begin by expressing the denominator of the first fraction in terms of a product of its factors

10 = 2 × 5 size 12{"10"=2 times 5} {}

We do the same with the denominator of the second fraction

12 = 2 × 2 × 3 = 2 2 × 5 size 12{"12"=2 times 2 times 3=2 rSup { size 8{2} } times 5} {}

We can express the lowest common denominator as

LCD = 2 2 × 3 × 5 = 60 size 12{ ital "LCD"=2 rSup { size 8{2} } times 3 times 5="60"} {}

We proceed to express the sum of fractions using the lowest common denominator just found

3 10 + 1 12 = 3 × 6 10 × 6 + 1 × 5 12 × 5 = 18 60 + 5 60 = 23 60 size 12{ { {3} over {"10"} } + { {1} over {"12"} } = { {3 times 6} over {"10" times 6} } + { {1 times 5} over {"12" times 5} } = { {"18"} over {"60"} } + { {5} over {"60"} } = { {"23"} over {"60"} } } {}

Thus we obtain the result of the addition as 23/60.

Let us consider another example in which three fractions are to be added

2 3 + 3 2 + 5 7 size 12{ { {2} over {3} } + { {3} over {2} } + { {5} over {7} } } {}

The denominator of each fraction is a prime number, so the lowest common denominator is their product

LCD = 3 × 2 × 7 = 42 size 12{ ital "LCD"=3 times 2 times 7="42"} {}

We must rewrite each fraction as an equivalent fraction with a denominator of 42

2 3 = 2 × 2 × 7 3 × 2 × 7 = 28 42 size 12{ { {2} over {3} } = { {2 times 2 times 7} over {3 times 2 times 7} } = { {"28"} over {"42"} } } {}

size 12{~} {}

3 2 = 3 × 3 × 7 3 × 2 × 7 = 63 42 size 12{ { {3} over {2} } = { {3 times 3 times 7} over {3 times 2 times 7} } = { {"63"} over {"42"} } } {}
5 7 = 5 × 3 × 2 3 × 2 × 7 = 30 42 size 12{ { {5} over {7} } = { {5 times 3 times 2} over {3 times 2 times 7} } = { {"30"} over {"42"} } } {}

Next, we express the sum of fractions using those equivalent fractions just determined

2 3 + 3 2 + 5 7 = 28 42 + 63 42 + 30 42 = 28 + 63 + 30 42 = 121 42 size 12{ { {2} over {3} } + { {3} over {2} } + { {5} over {7} } = { {"28"} over {"42"} } + { {"63"} over {"42"} } + { {"30"} over {"42"} } = { {"28"+"63"+"30"} over {"42"} } = { {"121"} over {"42"} } } {}

So 121/42 is the desired result.

Application: combining resistors in parallel

Figure 1 depicts a physical device known as a resistor. A resistor is often used in an electrical circuit to control the amount of current that flows throughout the circuit. The relationship between voltage, current and resistance in an electric circuit is governed by a fundamental law of Physics known as Ohm’s Law. Stated in words, Ohm’s Law tells us that the potential difference ( V ) measured in Volts across a resistor is directly proportional the current ( I ) measured in Amps that flows through the resistor. Additionally, the constant of proportionality is the value of the resistance ( R ), measured in Ohms. Ohm’s Law can be stated mathematically as

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Source:  OpenStax, Math 1508 (laboratory) engineering applications of precalculus. OpenStax CNX. Aug 24, 2011 Download for free at http://cnx.org/content/col11337/1.3
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