Chapter 1

Mathematical Essentials

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Objectives

▪Perform basic operations with integers▪Perform basic operations with fractions▪Evaluate expressions using order of

operations▪Perform basic operations with decimals▪Use percents▪Use Roman numerals

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How to Calculate: Sign Rules for Adding Two Integers

▪When adding two integers with:• Same sign:

– Sign of answer will have sign as integers added

• Different signs:– Sign of the answer will have sign of larger integer

▪Examples:• −5 + 3 = −2• −2 + (−4) = −6

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How to Calculate: Sign Rules for Subtracting Two Integers

▪The negative sign means the opposite• a − b = a + (−b)• 3 − 3 = 3 + (−3)

▪Examples:• −5 − 4 = −9• 6 − 8 = − 2• −8 − (−5) = −8 + 5 = −3

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How to Calculate: Sign Rules for Multiplying Integers

▪ (+)(+) = +▪ (−)(−) = +▪ (−)(+) = −▪ (+)(−) = −

▪Examples:• (5)(−6)• (−7)( −5) = 35• (−2)(8) = −16

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How to Calculate: Sign Rules for Dividing Two Integers

▪ (+) ÷ (+) = +▪ (−) ÷ (−) = +▪ (−) ÷ (+) = −▪ (+) ÷(−) = −

▪Examples:• = 4 • -12 ÷ 2 = − 6• = −5

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How to Calculate: Multiplying Fractions

▪Common denominators are:• Necessary when adding or subtracting • Not necessary when multiplying or dividing

▪Example:

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How to Calculate: Multiplying Fractions

▪Reducing fractions:• .

▪Reducing fractions with “canceling”:• .

• .

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How to Calculate: Dividing Fractions

▪To divide fractions, invert second fraction and multiply

▪Example:• .

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How to Calculate: Adding and Subtracting Fractions

▪ If denominators are equivalent, add or subtract numerators

▪Examples:• .

• .

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How to Calculate: Adding and Subtracting Fractions

▪ If denominators are not equivalent, find least common denominator (LCD)• Both divide into the LCD evenly

▪Example:• .

• The LCD is 9

How to Calculate: Adding or Subtracting Mixed Numbers

▪Convert each mixed number into an improper fraction

▪Add or subtract ▪Convert answer to mixed number▪Example:

• .

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How to Calculate: Simplifying Complex Fractions

▪Complex fraction: fraction(s) in numerator, denominator or both

▪Example:• .

• .

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How to Calculate: Order of Operations

▪Simplify parentheses and brackets▪Evaluate exponents▪Multiply and divide left to right▪Add and subtract left to right▪Remember order with acronym PEMDAS

• Parentheses, exponents, multiplication, division, addition, and subtraction

How to Calculate: Order of Operations

▪Example:• .• .• .• .

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How to Calculate: Adding and Subtracting Decimals

▪Adding example:• .

▪Subtracting example:• .

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How to Calculate: Multiplying and Dividing Decimals

▪Multiplying example:• .

▪Dividing example:• .• .• .

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How to Calculate:Rounding Decimal Numbers

▪ If number to direct right is:• 5 or larger, round up one number and drop

everything following• 4 or smaller, leave position being rounded and

drop everything following

▪Example:• Round 12.486 to the tenths position• 12.5

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How to Calculate: Converting Fractions to Percents

▪Write fraction as decimal▪Multiply decimal by 100%▪Example:

• .

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How to Calculate: Converting Mixed Numbers to Percents

▪Write mixed number as improper fraction▪Write improper fraction as a decimal▪Multiply decimal by 100%▪Example:

• .

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How to Calculate: Converting Percents

▪Convert percent to decimal: divide by 100%

▪ Example: 40%• .

▪Convert decimal to percent: multiply by 100%

▪Example: 0.246 • Multiplying by 100% yields (0.246)(100%) =

24.6%© 2010 Delmar, Cengage Learning. 21

Roman Numerals

▪ I (or i) = 1▪V (or v) = 5▪X (or x) = 10▪ L (or l) = 50▪C (or c) = 100▪D (or d) = 500▪M (or m) = 1,000

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How to Calculate:Roman Numerals

▪Read from right to left▪ If left numeral is greater than right

numeral, add numerals▪ If left is smaller than right, subtract left

value from right value▪Example:

• XVI = 10 + 5 + 1 = 16

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Additional rules not listed in the book

▪The symbols "I", "X", "C", and "M" can be repeated three times in succession, but no more. "D", "L", and "V" can never be repeated.[5] [6]

▪ "I" can be subtracted from "V" and "X" only. "X" can be subtracted from "L" and "C" only. "C" can be subtracted from "D" and "M" only. "V", "L", and "D" can never be subtracted [7]

▪Only one small-value symbol may be subtracted from any large-value symbol.[8]

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Summary

▪Multiplying two like signed numbers results in a positive• Different signs results in negative• Same rules apply for division

▪When adding or subtracting fractions, you must have common denominators

▪ LCD is the least common multiple

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Summary (cont’d.)

▪Order of operations: PEMDAS▪Convert a decimal to percent: multiply by

100%▪Convert a fraction to a percent: convert

fraction to a decimal; then multiply decimal by 100%

▪Add values of numerals if in descending order; subtract if in ascending order

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