Fractions To work well with fractions, it is necessary to understand some basic concepts. SIMPLIFYING FRACTIONS Rule: ᎏ a bc c ᎏ = ᎏ a b ᎏ ■ To simplify fractions, identify the Greatest Common Factor (GCF) of the numerator and denominator and divide both the numerator and denominator by this number. Example Simplify ᎏ 6 7 3 2 ᎏ . The GCF of 63 and 72 is 9 so divide 63 and 72 each by 9 to simplify the fraction: ᎏ 6 7 3 2 ÷ ÷ 9 9 = = 7 8 ᎏ ᎏ 6 7 3 2 ᎏ = ᎏ 7 8 ᎏ ADDING AND SUBTRACTING FRACTIONS Rules: To add or subtract fractions with the same denominator: ᎏ a b ᎏ ± ᎏ b c ᎏ = ᎏ a ± b c ᎏ To add or subtract fractions with different denominators: ᎏ a b ᎏ ± ᎏ d c ᎏ = ᎏ ad b ± d cb ᎏ ■ To add or subtract fractions with like denominators, just add or subtract the numerators and keep the denominator. Examples ᎏ 1 7 ᎏ + ᎏ 5 7 ᎏ = ᎏ 6 7 ᎏ and ᎏ 5 8 ᎏ – ᎏ 2 8 ᎏ = ᎏ 3 8 ᎏ – THEA MATH REVIEW– 100 ■ To add or subtract fractions with unlike denominators, first find the Least Common Denominator or LCD. The LCD is the smallest number divisible by each of the denominators. For example, for the denominators 8 and 12, 24 would be the LCD because 24 is the smallest number that is divisible by both 8 and 12: 8 ϫ 3 = 24, and 12 ϫ 2 = 24. Using the LCD, convert each fraction to its new form by multiplying both the numerator and denominator by the appropriate factor to get the LCD, and then follow the directions for adding/subtracting fractions with like denominators. Example ᎏ 1 3 ᎏ + ᎏ 2 5 ᎏ = ᎏ 1 3 ( ( 5 5 ) ) ᎏ + ᎏ 2 5 ( ( 3 3 ) ) ᎏ = ᎏ 1 5 5 ᎏ + ᎏ 1 6 5 ᎏ = ᎏ 1 1 1 5 ᎏ MULTIPLICATION OF FRACTIONS Rule: ᎏ a b ᎏ ϫ ᎏ d c ᎏ = ᎏ a b ϫ ϫ c d ᎏ ■ Multiplying fractions is one of the easiest operations to perform. To multiply fractions, simply multiply the numerators and the denominators. Example ᎏ 4 5 ᎏ ϫ ᎏ 6 7 ᎏ = ᎏ 2 3 4 5 ᎏ If any numerator and denominator have common factors, these may be simplified before multiplying. Divide the common multiples by a common factor. In the example below, 3 and 6 are both divided by 3 before multi- plying. Example ᎏ 5 3 1 ΋ ᎏ ϫ ᎏ 1 6 2 ΋ ᎏ = ᎏ 1 1 0 ᎏ – THEA MATH REVIEW– 101 DIVIDING FRACTIONS Rule: ᎏ a b ᎏ ÷ ᎏ d c ᎏ = ᎏ a b ᎏ ϫ ᎏ d c ᎏ = ᎏ a b ϫ ϫ d c ᎏ ■ Dividing fractions is equivalent to multiplying the dividend by the reciprocal of the divisor. When divid- ing fractions, simply multiply the dividend by the divisor’s reciprocal to get the answer. Example (dividend) ÷ (divisor) ᎏ 1 4 ᎏ ÷ ᎏ 1 2 ᎏ Determine the reciprocal of the divisor: ᎏ 1 2 ᎏ → ᎏ 2 1 ᎏ Multiply the dividend ( ᎏ 1 4 ᎏ ) by the reciprocal of the divisor ( ᎏ 2 1 ᎏ ) and simplify if necessary. ᎏ 1 4 ᎏ ÷ ᎏ 1 2 ᎏ = ᎏ 1 4 ᎏ ϫ ᎏ 2 1 ᎏ = ᎏ 2 4 ᎏ = ᎏ 1 2 ᎏ COMPARING FRACTIONS Rules: If ᎏ a b ᎏ = ᎏ d c ᎏ , then ad = bc If ᎏ a b ᎏ < ᎏ d c ᎏ , then ad < bc If ᎏ a b ᎏ > ᎏ d c ᎏ , then ad > bc Sometimes it is necessary to compare the size of fractions. This is very simple when the fractions are famil- iar or when they have a common denominator. Examples ᎏ 1 2 ᎏ < ᎏ 3 4 ᎏ and ᎏ 1 1 1 8 ᎏ > ᎏ 1 5 8 ᎏ ■ If the fractions are not familiar and/or do not have a common denominator, there is a simple trick to remember. Multiply the numerator of the first fraction by the denominator of the second fraction. Write this answer under the first fraction. Then multiply the numerator of the second fraction by the denomi- nator of the first one. Write this answer under the second fraction. Compare the two numbers. The larger number represents the larger fraction. – THEA MATH REVIEW– 102 Examples Which is larger: ᎏ 1 7 1 ᎏ or ᎏ 4 9 ᎏ ? Cross-multiply. 7 ϫ 9 = 63 4 ϫ 11 = 44 63 > 44, therefore, ᎏ 1 7 1 ᎏ > ᎏ 4 9 ᎏ Compare ᎏ 1 6 8 ᎏ and ᎏ 2 6 ᎏ . Cross-multiply. 6 ϫ 6 = 36 2 ϫ 18 = 36 36 = 36, therefore, ᎏ 1 6 8 ᎏ = ᎏ 2 6 ᎏ CONVERTING DECIMALS TO FRACTIONS ■ To convert a non-repeating decimal to a fraction, the digits of the decimal become the numerator of the fraction, and the denominator of the fraction is a power of 10 that contains that number of digits as zeros. Example Convert .125 to a fraction. The decimal .125 means 125 thousandths, so it is 125 parts of 1,000. An easy way to do this is to make 125 the numerator, and since there are three digits in the number 125, the denominator is 1 with three zeros, or 1,000. .125 = ᎏ 1 1 ,0 2 0 5 0 ᎏ Then we just need to reduce the fraction. ᎏ 1 1 ,0 2 0 5 0 ᎏ = ᎏ 1 1 ,0 2 0 5 0 ÷ ÷ 1 1 2 2 5 5 ᎏ = ᎏ 1 8 ᎏ ■ When converting a repeating decimal to a fraction, the digits of the repeating pattern of the decimal become the numerator of the fraction, and the denominator of the fraction is the same number of 9s as digits. Example Convert .3 ៮ to a fraction. – THEA MATH REVIEW– 103 You may already recognize .3 ៮ as ᎏ 1 3 ᎏ . The repeating pattern, in this case 3, becomes our numerator. There is one digit in the pattern, so 9 is our denominator. .3 ៮ = ᎏ 3 9 ᎏ = ᎏ 3 9 Ϭ ÷3 3 ᎏ = ᎏ 1 3 ᎏ Example Convert .36 ៮ to a fraction. The repeating pattern, in this case 36, becomes our numerator. There are two digits in the pattern, so 99 is our denominator. .36 ៮ = ᎏ 3 9 6 9 ᎏ = ᎏ 3 9 6 9 ÷ ÷ 9 9 ᎏ = ᎏ 1 4 1 ᎏ CONVERTING FRACTIONS TO DECIMALS ■ To convert a fraction to a decimal, simply treat the fraction as a division problem. Example Convert ᎏ 3 4 ᎏ to a decimal. .75 4ͤ3. ෆ 00 ෆ So, ᎏ 3 4 ᎏ is equal to .75. CONVERTING MIXED NUMBERS TO AND FROM IMPROPER FRACTIONS Rule: a ᎏ b c ᎏ = ᎏ ac c + b ᎏ ■ A mixed number is number greater than 1 which is expressed as a whole number joined to a proper frac- tion. Examples of mixed numbers are 5 ᎏ 3 8 ᎏ ,2 ᎏ 1 3 ᎏ , and –4 ᎏ 5 6 ᎏ . To convert from a mixed number to an improper fraction (a fraction where the numerator is greater than the denominator), multiply the whole number and the denominator and add the numerator. This becomes the new numerator. The new denominator is the same as the original. Note: If the mixed number is negative, temporarily ignore the negative sign while performing the conversion, and just make sure you replace the negative sign when you’re done. Example Convert 5 ᎏ 3 8 ᎏ to an improper fraction. Using the formula above, 5 ᎏ 3 8 ᎏ = ᎏ 5 ϫ 8 8+3 ᎏ = ᎏ 4 8 3 ᎏ . – THEA MATH REVIEW– 104 Example Convert –4 ᎏ 5 6 ᎏ to an improper fraction. Temporarily ignore the negative sign and perform the conversion: 4 ᎏ 5 6 ᎏ = ᎏ 4 ϫ 6 6 +5 ᎏ = ᎏ 2 6 9 ᎏ . The final answer includes the negative sign: – ᎏ 2 6 9 ᎏ . ■ To convert from an improper fraction to a mixed number, simply treat the fraction like a division prob- lem, and express the answer as a fraction rather than a decimal. Example Convert ᎏ 2 7 3 ᎏ to a mixed number. Perform the division: 23 ÷ 7 = 3 ᎏ 2 7 ᎏ . Percents Percents are always “out of 100.” 45% means 45 out of 100. Therefore, to write percents as decimals, move the dec- imal point two places to the left (to the hundredths place). 45% = ᎏ 1 4 0 5 0 ᎏ = 0.45 3% = ᎏ 1 3 00 ᎏ = 0.03 124% = ᎏ 1 1 2 0 4 0 ᎏ = 1.24 0.9% = ᎏ 1 . 0 9 0 ᎏ = ᎏ 1,0 9 00 ᎏ = 0.009 Here are some conversions you should be familiar with: Fraction Decimal Percentage ᎏ 1 2 ᎏ .5 50% ᎏ 1 4 ᎏ .25 25% ᎏ 1 3 ᎏ .333 . . . 33.3 –– % ᎏ 2 3 ᎏ .666 . . . 66.6 –– % ᎏ 1 1 0 ᎏ .1 10% ᎏ 1 8 ᎏ .125 12.5% ᎏ 6 1 ᎏ .1666 . . . 16.6 –– % ᎏ 1 5 ᎏ .2 20% – THEA MATH REVIEW– 105 . be familiar with: Fraction Decimal Percentage ᎏ 1 2 ᎏ .5 50% ᎏ 1 4 ᎏ .25 25% ᎏ 1 3 ᎏ .33 3 . . . 33 .3 –– % ᎏ 2 3 ᎏ .666 . . . 66.6 –– % ᎏ 1 1 0 ᎏ .1 10% ᎏ 1 8 ᎏ .125 12.5% ᎏ 6 1 ᎏ .1666 . . REVIEW– 1 03 You may already recognize .3 ៮ as ᎏ 1 3 ᎏ . The repeating pattern, in this case 3, becomes our numerator. There is one digit in the pattern, so 9 is our denominator. .3 ៮ = ᎏ 3 9 ᎏ = ᎏ 3 9 Ϭ 3 3 ᎏ =. ᎏ 4 9 ᎏ ? Cross-multiply. 7 ϫ 9 = 63 4 ϫ 11 = 44 63 > 44, therefore, ᎏ 1 7 1 ᎏ > ᎏ 4 9 ᎏ Compare ᎏ 1 6 8 ᎏ and ᎏ 2 6 ᎏ . Cross-multiply. 6 ϫ 6 = 36 2 ϫ 18 = 36 36 = 36 , therefore, ᎏ 1 6 8 ᎏ =