ࡱ> XZWK'` V*bjbjLULU ..?.?"6NNNNNNNb&(&(&(8^(\)dbRp**b2"222Y8Y8Y8pRrRrRrRrRrRrR$aThVxRNY8 7NY8Y8Y8RNN22]RG?G?G?Y8XN2N2pRG?Y8pRG?G?PNNQ2* O&(8QpRR0R,QAW{<tAW(QAWNQY8Y8G?Y8Y8Y8Y8Y8RR>XY8Y8Y8RY8Y8Y8Y8bbb"&(bbb&(bbbNNNNNN Fractions Least Common Multiple (LCM) The smallest multiple that two or more numbers have in common. Number Multiples 4 4, 8, 12, 16, 20, 24, 28, 32, 36 6 6, 12, 18, 24, 30, 36, 42 8 6, 16, 24, 32, 40, 48 LCM = 24 Greatest Common Factor (GCF) The largest factor that two or more numbers have in common. Number Factors 4, 2, 1  EMBED Equation.3  6, 3, 2, 1 8 8, 4, 2, 1 GCF = 2 Simplifying Fractions To simplify a fraction divide the numerator and the denominator by their GCF.  EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3  Or, you can keep dividing the numerator and denominator by a common factor until there are no more factors in common.  EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3  Change Improper Fraction to Mixed Number Divide the numerator by the denominator. Place the remainder over the denominator. Simplify the fraction.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  = 3 EMBED Equation.3  = 3 EMBED Equation.3  Change Mixed Number to Improper Fraction Multiply the denominator by the whole number. Add the product to the numerator. Place the sum over the denominator. 4 EMBED Equation.3  5 x 4 = 20 + 3 = 23 4 EMBED Equation.3  =  EMBED Equation.3  Adding or Subtracting Fractions Fractions must have a common denominator.  EMBED Equation.3  +  EMBED Equation.3  = 1. Find the LCM of 6 and 4. LCM = 12 2. Convert each fraction to an equivalent fraction with denominator of 12.  EMBED Equation.3  x  EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3  x  EMBED Equation.3  =  EMBED Equation.3  3. Rewrite the equation using the new fractions.  EMBED Equation.3  +  EMBED Equation.3  = 4. Add or subtract the numerators only. Denominators do not change.  EMBED Equation.3  +  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  5. Simplify answer if possible. Multiplying Fractions Multiply the numerators. Multiply the denominators.  EMBED Equation.3  x  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Simplify the resulting fraction.  EMBED Equation.3   EMBED Equation.3   EMBED Equation.3  =  EMBED Equation.3  Dividing Fractions Keep the first fraction as it is. Change the operation to multiplication. Invert (flip) the second fraction.  EMBED Equation.3   EMBED Equation.3    EMBED Equation.3  x  EMBED Equation.3  Multiply the fractions.  EMBED Equation.3  x  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Simplify your answer.  EMBED Equation.3  EMBED Equation.3  EMBED Equation.3  =  EMBED Equation.3  Operations with Mixed Numbers Change each fraction to a mixed number. Simplify each fraction. Perform the operation. Simplify your answer. Examples: Addition and Subtraction 2 EMBED Equation.3  + 1 EMBED Equation.3  =  EMBED Equation.3  +  EMBED Equation.3  =  EMBED Equation.3  +  EMBED Equation.3  =  EMBED Equation.3  Simplify  EMBED Equation.3  =  EMBED Equation.3  = 3 EMBED Equation.3  Multiplication and Division 3 EMBED Equation.3  x 2 EMBED Equation.3  =  EMBED Equation.3  x  EMBED Equation.3  =  EMBED Equation.3  Simplify  EMBED Equation.3  = 8 EMBED Equation.3  Comparing Fractions In order to compare fractions, the fractions must have a common denominator. Compare  EMBED Equation.3  and  EMBED Equation.3  LCM = 15  EMBED Equation.3  =  EMBED Equation.3  and  EMBED Equation.3  =  EMBED Equation.3   EMBED Equation.3  >  EMBED Equation.3  Prime Factorization Factoring a number to the point where all of the factors are prime numbers. EMBED Equation.3  120  5 24  3 8 120 = 2 x 3 x 5 4  2 2 Scientific Notation Scientific Notation uses powers of ten to express very large and very small numbers. Scientific notation using positive exponents. Notice the pattern in the powers of 10 10 = 10 x 1 =  EMBED Equation.3  100 = 10 x 10 =  EMBED Equation.3  1,000 = 10 x 10 x 10 =  EMBED Equation.3  10,000 = 10 x 10 x 10 x 10 =  EMBED Equation.3  100,000 = 10 x 10 x 10 x 10 x 10 =  EMBED Equation.3  1,000,000 = 10 x 10 x 10 x 10 x 10 x 10 =  EMBED Equation.3  Example 1: Express 23500 in scientific notation. 23500 = 2350 x  EMBED Equation.3  Move the decimal one place to the left. = 235.0 x  EMBED Equation.3  Move the decimal one place to the left. = 23.5 x  EMBED Equation.3  Move the decimal one place to the left. = 2.35 x  EMBED Equation.3  Now your number is in scientific notation. Example 2: Express 3.65 x  EMBED Equation.3  in standard notation. 3.65 x  EMBED Equation.3  = 3.65 x 10 x 10 x 10 x 10 = 36,500 OR Simply move the decimal point 4 spaces to the right, filling in with zeros as place holders. 3.65 = 36,500 Scientific notation using negative exponents. Scientific notation with negative exponents of ten are used to write very small decimal numbers. Notice the patterns in the table below. Number Decimal Form Exponential Form 100 100  EMBED Equation.3  10 10  EMBED Equation.3  1 1  EMBED Equation.3   EMBED Equation.3  0.1  EMBED Equation.3   EMBED Equation.3  0.01  EMBED Equation.3   EMBED Equation.3  0.001  EMBED Equation.3   EMBED Equation.3  0.0001  EMBED Equation.3  Use this information to write in scientific notation. Example 1: Express 0.0004 in scientific notation. 0.0004 =  EMBED Equation.3  = 4 x  EMBED Equation.3  = 4 x  EMBED Equation.3  Example 2: Express 5.23 x  EMBED Equation.3  in standard notation. 5.23 x  EMBED Equation.3  = 5.23 x  EMBED Equation.3  = 5.23 x 0.001 = 0.00523 OR Simply move the decimal point 3 spaces to the left, filling in zeros as place holders between the decimal point and the first non-zero digit. 5.23 = 0.00523 Fractions to Decimals to Percents Fraction to Decimal Divide the numerator by the denominator. Ex: = 1 4 = 0.25 Decimal to Percent Move the decimal two spaces to the right or Multiply the decimal by 100. Ex: 0.24 = 24% or 0.24 x 100 = 24% Percent to Fraction Drop the percent sign, put the percent over 100 and reduce. or Change the percent to a decimal and the decimal to a fraction. Ex: 35% =  EMBED Equation.3  =  EMBED Equation.3  or 35% = .35 =  EMBED Equation.3  =  EMBED Equation.3  Decimal to Fraction Terminating Decimal Read the decimal using place values and write as a fraction then reduce or Count the number of decimal places in the number. Place the number (without the decimal point) over 10 to the power that equals the number of decimal places. Ex: .456 = 45 thousandths =  EMBED Equation.3  =  EMBED Equation.3  or .456 has 3 decimal places =  EMBED Equation.3  =  EMBED Equation.3  =  EMBED Equation.3  Repeating Decimal Count the number of decimal places in the number. Place the number (without the decimal point) over the same number of 9s that are in the repeat. 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