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Algebra Review: Fractions ALGEBRA REVIEW: FRACTIONS DEFINITION A fraction is a numerical value described as one value divided by another. Ex: 1 ÷ 3 = 1/3 The three parts of a fraction are the numerator, the denominator, and the division symbol. Numerator Denominator TYPES OF FRACTIONS Division symbol Simple. Both numerator and denominator are integers. 1 57 2 212 Complex. There is a fraction or decimal within the fraction. 4.3 1/3 7 8 5 2/5 Compound. A whole number is combined with a simple fraction. 3 12 7 5 13 Improper. A simple fraction with a numerator that is greater than the denominator. 4 5 2 WRITING DECIMALS AS FRACTIONS 17 13 Terminating decimals and complex fractions. Multiply top and bottom by 10 until there is no longer a decimal. (See Multiplying Fractions). .57 1 Ex: 0.57 = .57 1 × 10 10 = 5.7 10 5.7 10 × 10 10 = 57 100 1 Algebra Review: Fractions Infinitely repeating decimals. Follow the steps in the example below. Ex: Write 0.272727 . . . as a fraction. Pick a letter. I’ll use ‘n'. Let n = 0.2727272727 . . . This decimal breaks up to a repeating set of two digits (0. 27 27 27 27 27 . . . ) so I’ll multiply both sides of my equation by a one followed by two zeros, one hundred. 100n = 27.2727272727 . . . Subtracting my original equation, I get 100n = 27.27272727 . . . -100n = 00.27272727 . . . 99n = 27 Divide both sides by 99. n = 27 99 27 99 0.27272727 . . . = Non-terminating, non-repeating decimals. There is no way to write these as exact fractions. CONVERTING FRACTIONS Improper to Compound. Divide the numerator by the denominator. Bring the answer out to the front and leave the remainder in the fraction. Ex: 17 13 is the same as 17 ÷ 13. 17 ÷ 13 = 1 R4 17 13 = 1 4 13 2 Algebra Review: Fractions Compound to Improper. Multiply the outside number by the denominator, and add that number to the numerator. Ex: 2 1 2 = (2 × 2) + 1 2 = 4+1 2 = 5 2 Complex fractions. Rewrite the fraction as a division problem. Follow the steps covered in Dividing Fractions. Ex: 1/3 = 1 ÷ 5 See Dividing Fractions. 5 3 REDUCING SIMPLE FRACTIONS Factor the numerator and the denominator. If they have a common factor, then it can be removed from the fraction. Ex: 27 3×3×3 3 = = 99 3 × 3 × 11 11 This applies to multiplication only. Ex: ADDING AND SUBTRACTING FRACTIONS 3 3+x CANNOT be simplified to 1 x Common denominators. Fractions can only be added or subtracted if they have the same denominator. For example, (1/2 + 1/3) cannot be simplified as is, but (3/6 + 2/6) can. NOTE: This is the same sort of idea as combining like terms. (a + b) is as simplified as it can get, but (a + 2a) can be simplified to (3a). Finding a common denominator. First, find the least common multiple for the denominators. Ex: 3 5 + 6 8 Multiples of 8: 8, 16, 24, 32, 40 Multiples of 6: 6, 12, 18, 24, 30 24 is the smallest number that is divisible by both 8 and 6. 3 Algebra Review: Fractions Second, change both fractions so that the least common multiple is the denominator of both. This is the opposite of reducing fractions. Ex: 5 4 20 × = 6 4 24 (See Multiplying Fractions.) 3 3 9 × = 8 3 24 So 5 + 3 6 8 becomes 20 + 9 = 29 24 24 24 The same steps can be used for subtraction of fractions. The normal rules of addition and subtraction apply. MULTIPLYING FRACTIONS To multiply fractions, multiply straight across the top and bottom, as shown in the example below. Ex: DIVIDING FRACTIONS 5 5 5×5 25 0×0 = = 6 7 6×7 42 Reciprocals. The reciprocal of a fraction is the fraction flipped. Ex: The reciprocal of 2 7 is 7 2 Dividing fractions. Dividing by a fraction is the same as multiplying by the reciprocal of the fraction. Ex: Ex: RATIONAL EXPRESSIONS 5 2 5 7 35 ÷ = × = 6 7 6 2 12 1 1 1 1 ÷ 5 = × = 3 3 5 15 The rules of fractions apply even to more complicated rational expressions like 3x2 + 4xy + y3 + 7 x4 - 5y7 4