GE142 - Course Introduction

2. Fractions

Dr Robert Batzinger
Instructor Emeritus

8/15/22

1 Chapter 2: Fractions

2 Divisibility

\[\eqalign{\phantom{30)0}15\\30\overline{)451}\\ \phantom{30)}\underline{-30}\phantom{0}\\ 151\\ \underline{-150}\\ 1 }\] \[Ans: 15 \tiny\raise 1ex\frac{1}{30}\]

If there is no remainder, the dividend is divisible by the divisor
Prime numbers can only divided by themselves and 1

3 Finding prime numbers

Prime =   2   3   5   7   11   13   17   19   23   29   31   37   41   43   47   53   59   61   67   71   73   79   83   89   97   101   103   107   109   113   127   131   137   139   149   151   157   163   167   173   179   181   191   193   197   199   211   223   227   229   233   239   241   251   257   263   269   271   277   281   283   293   307   311   313   317   331   337   347   349   353   359   367   373   379   383   389   397   401   409   419   421   431   433   439   443   449   457   461   463   467   479   487   491   499

Walk through a range of numbers
Check possible factors for each number
if factors are only 1 or n: Print out the prime number

3.1 Fractions

\[Fraction = \frac{Numerator}{Denominator}\]

3.2 Equivalent fractions

\[\eqalign{\frac{1}{3} &=& \frac{1\color{yellow}{\times 1}}{3\color{yellow}{\times 1}} &=& \frac{1}{3}\\ &=& \frac{1\color{yellow}{\times 2}}{3\color{yellow}{\times 2}} &=& \frac{2}{6}\\ &=& \frac{1\color{yellow}{\times 3}}{3\color{yellow}{\times 3}} &=& \frac{3}{9}\\ &=& \frac{1\color{yellow}{\times 4}}{3\color{yellow}{\times 4}} &=& \frac{4}{12}\\ &=& \frac{1\color{yellow}{\times 5}}{3\color{yellow}{\times 5}} &=& \frac{5}{15}\\ } \]

3.3 Equivalents

3.4 Writing a fraction in simplest form

Remove integers from the expression and leave the remainder as a faction
Remove common factors from the remaining fraction

\[\frac{20}{6} = 3 \frac{2}{6}=3 \frac{1}{3}\]

3.5 Comparing fractions

Convert the fractions to have the same denominator
Include the equivalent integer amounts
Compare the numerators

\[\frac{32}{3}\ \fbox{?}\ 10\frac{5}{6} \rightarrow \frac{64}{6}\ \fbox{?}\ 10\frac{5}{6} \rightarrow \frac{64}{6}\ \fbox{?}\ \frac{60+5}{6}\]

\[\frac{64}{6} < \frac{65}{6}\]

3.6 Adding or subtracting fractions

Combine the terms into fractions
Convert all the fractions to the same denominator
Carry out the operation
Reduce the answer to its simplest form

\[\eqalign{2\frac{1}{2} + 1\frac{1}{6} +3 \frac{2}{3}&=&\frac{5}{2}+\frac{7}{6} +\frac{11}{3} =\frac{5\color{yellow}{\times 3}}{2\color{yellow}{\times 3}}+\frac{7}{6} +\frac{11\color{yellow}{\times 2}}{3\color{yellow}{\times 2}}=\\ &=&\frac{15+7+22}{6}=\frac{44}{6}= 7\frac{1}{3}\\ }\]

3.7 Multiplying or dividing fractions

Combine the terms into fractions
Multiply across numerators and denominations
Reduce the answer to its simplest form

\[\eqalign{2\frac{1}{2} \times 1\frac{1}{6} \times 3 \frac{2}{3}&=& \frac{5}{2}\times \frac{7}{6} \times \frac{11}{3}=\\ \frac{5\times 7\times 11}{2\times 6 \times 3}&=& \frac{385}{36} = 10 \frac{25}{36}\\}\]

3.8 Cross products

Equivalent fractions also have equivalent cross-products.

\[\eqalign{\frac{\color{yellow}{3}}{\color{red}{4}} &=& \frac{\color{red}{6}}{\color{yellow}{8}}\\ \color{yellow}{3 \times 8} &=& \color{red}{4 \times 6}\\ \color{yellow}{24} &=& \color{red}{24} \\ }\]

4 Example 1

A gym that is open every day of the week offers aerobics classes every third day and Tai Chi classes every fourth day. A student took both classes this morning. In how many days will the gym offer both classes on the same day?

4.1 Example 2

Suppose that a Senate seat and a House of Representatives seat were both filled this year. If the Senate seat is filled every 6 years and the House seat every 2 years, in how many years will both seats be up for election?

5 Exponents / Scientific Notation

\(\eqalign{2,048×1,536×48×24×3,600&=&\\ 2.048 \times 10^3 \times 1.536\times10^3\times4.8\times10\times\\ 2.4\times10x\times3.6\times10^3&=&\\ 2.048\times1.536\times4.8\times2.4\times3.6\\\times10^3\times10^3\times10\times10\times10^3&=&\\ 130.459631616\times10^{(3+3+1+1+3)} &=&\\ 1.3046\times 10^2 \times 10^{11} &=&\\ 1.3046\times 10^{13}&=&\\ 13,045,963,161,600&=&\\ }\)

5.1 Product rule

\[\eqalign{a^b \times a ^c &=& a^{(b+c)}\\ 3^2 \times 3^3 &=& 3^{2+3} = 3^ 5\\ }\]

5.2 Quotient rule

\[\eqalign{a^b / a^c &=& a ^{(b-c)}\\ 3^3 / 3 &=& 3^{(3-1)} = 3^2\\ 1 / 3^3 &=& 3^{-3}\\ }\]