Chapter 1 Mathematical Essentials

© 2010 Delmar, Cengage Learning. 2 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

© 2010 Delmar, Cengage Learning. 3 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

© 2010 Delmar, Cengage Learning. 4 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

© 2010 Delmar, Cengage Learning. 5 How to Calculate: Sign Rules for Multiplying Integers ▪(+)(+) = + ▪(−)(−) = + ▪(−)(+) = − ▪(+)(−) = − ▪Examples: (5)(−6) (−7)( −5) = 35 (−2)(8) = −16

© 2010 Delmar, Cengage Learning. 6 How to Calculate: Sign Rules for Dividing Two Integers ▪(+) ÷ (+) = + ▪(−) ÷ (−) = + ▪(−) ÷ (+) = − ▪(+) ÷(−) = − ▪Examples: = ÷ 2 = − 6 = −5

© 2010 Delmar, Cengage Learning. 7 How to Calculate: Multiplying Fractions ▪Common denominators are: Necessary when adding or subtracting Not necessary when multiplying or dividing ▪Example:

© 2010 Delmar, Cengage Learning. 8 How to Calculate: Multiplying Fractions ▪Reducing fractions:. ▪Reducing fractions with “canceling”:.

© 2010 Delmar, Cengage Learning. 9 How to Calculate: Dividing Fractions ▪To divide fractions, invert second fraction and multiply ▪Example:.

© 2010 Delmar, Cengage Learning. 10 How to Calculate: Adding and Subtracting Fractions ▪If denominators are equivalent, add or subtract numerators ▪Examples:.

© 2010 Delmar, Cengage Learning. 11 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:. © 2010 Delmar, Cengage Learning.

How to Calculate: Simplifying Complex Fractions ▪Complex fraction: fraction(s) in numerator, denominator or both ▪Example:. © 2010 Delmar, Cengage Learning. 13

© 2010 Delmar, Cengage Learning. 14 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:. © 2010 Delmar, Cengage Learning. 15

How to Calculate: Adding and Subtracting Decimals ▪Adding example:. ▪Subtracting example:. © 2010 Delmar, Cengage Learning. 16

How to Calculate: Multiplying and Dividing Decimals ▪Multiplying example:. ▪Dividing example:. © 2010 Delmar, Cengage Learning. 17

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 to the tenths position 12.5 © 2010 Delmar, Cengage Learning. 18

How to Calculate: Converting Fractions to Percents ▪Write fraction as decimal ▪Multiply decimal by 100% ▪Example:. © 2010 Delmar, Cengage Learning. 19

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:. © 2010 Delmar, Cengage Learning. 20

How to Calculate: Converting Percents ▪Convert percent to decimal: divide by 100% ▪Example: 40%. ▪Convert decimal to percent: multiply by 100% ▪Example: 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 © 2010 Delmar, Cengage Learning. 22

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 = = 16 © 2010 Delmar, Cengage Learning. 23

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] [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] [7] ▪Only one small-value symbol may be subtracted from any large-value symbol. [8] [8] 24

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 © 2010 Delmar, Cengage Learning. 25

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 © 2010 Delmar, Cengage Learning. 26