ࡱ> TVSy bjbj oL{{vy"""""$FFFPD|F"VV"xxx8":":":":":":"$[$ '^""^"""xx4s"R"x"x8"8": ,x!x-@;! $""0"&!R''x!'"x!^"^"xv"' : Fractions, Decimals, and Percentages Fractions, decimals, and percentages are three ways of expressing what part one this is of another thing. This is often important in chemistry. Chemists may want to find out how much of a sample is made up of a certain element, how much of a solution is made up of a dissolved substance, or how many molecules have broken into ions. Fractions When something is divided into equal parts, the relation of one or more equal parts to the total of equal parts is a number known as a fraction.. If a sample has a total mass of 20 grams, then a 10-gram portion of the sample is 10, or 1, of the sample. 20 2 Similarly, a fraction may express the number of correct answers on a test that is made up of 12 questions. If 8 questions are answered correctly, the fraction of correct answers is 8 or 2. 12 3 The top number of a fraction is called the numerator, and the bottom number the denominator. In order to add or subtract fractions, you must find a common denominator, a number that is divisible by the denominators of the fractions you are adding or subtracting. Example 1 What is the sum of 1 and 1? 2 3 Solution The lowest common denominator for the fractions is 6. Change 1 to 3 by multiplying the numerator and denominator by 3. 2 6 Change 1 to 2 by multiplying the numerator and denominator by 2. 3 6 The answer is 3 + 2 = 5 6 6 6 Example 2 What is 1/3 minus 1/8? Solution The lowest common denominator is 24. Change 1 to 8 and 1 to 3. 3 24 8 24 The answer is 8 3 or 5 24 24 24 In order to multiply fractions, multiply the numerators to obtain the numerator of the answer, and multiply the denominators to get the denominator of the answer. Example 3 What is 1 x 2 ? 2 3 Solution 1 x 2 = 2, or 1 2 3 6 3 When dividing fractions, invert the fraction in the denominator and multiply it by the fraction in the numerator. Example 4 What is (1)/(3/4) Solution 1/3 x 4/3 = 4/9 One way to solve a proportion is to find the lowest common denominator. Example 5 Solve: x/5 = 7/10 Solution 2x/10=7/10; 2x = 7; x = 7/2 Another method for solving proportions is called cross multiplying. Example 6 Solve: x/3 = 14/21 Solution 21x=3(14) x=42/21=2 Decimals A decimal expresses a fractional number with a denominator of some power of 10 (such as 10, 100, and 1000). Only the numerator appears in decimal notation because the denominator is signified by a decimal point. For example. 1/10 = 0.1, 1/100 = 0.01, and 1/1000 = 0.001 Example 7 What is the decimal equivalent of the fraction 24/100? Solution Divide 24 by 1000 and express the answer in decimal form. The answer is 0.024. You may need to change a decimal to a fraction or vice versa Example 8 What is the decimal equivalent of 2/7? Solution Divide 2 by 7; the answer is 0.28 (to two significant figures). Example 9 To what fraction is 0.035 equivalent? Solution Express the decimal as a fraction, 35/1000, and reduce the fraction to lowest terms-that is, 7/200. Percentage The fraction 6/100 can be expressed as 0.06 in decimal form. It can also be expressed in another way known as a percent. The percent symbol (%) represents hundredths, thus 6% means 6 hundredths, and 100% means 100 hundredths, 100/100, or one whole. It is often useful to use percentages in chemistry. For example, a chemist might find that a 30.0gram liquid sample consists of 26.7 grams of oxygen and 3.3 grams of hydrogen. The fraction of oxygen in the sample is 26.7/30.0 = 0.890. The percentage of oxygen in the sample is 0.890/1 x 100/100 = 89.0/100, or 89.0% The percentage composition of a substance can be used to calculate the number of grams of the substance in a sample. Example 10 How many grams of Calcium are in a 12.0-gram sample of a substance that is known to be 40.5% calcium? Solution There are 40.5 grams of calcium in every 100 grams of the substance. Therefore, in 12.0 grams there are 12.0g x 40.5g / 100g = 4.86 g of calcium. Or, simply convert 40.5% directly to a decimal and multiply: .405 x 12.0g = 48.6g calcium The concentrations of solutions are sometimes expressed as a percentage by mass. In a solution of common salt and water there might be 2.5 g of salt dissolved in 100g of water. The fraction by mass of salt in the solution is: 2.5g / (100g + 2.5g) = 2.5g/102.5 = 0.024 The percentage by mass of salt in the solution is 0.024 x 100% = 2.4%. Example 11 How much salt is dissolved in 50 grams of a solution in which the percentage by mass of salt is 3.0%? Solution There is 3.0 g of salt in every 100 g of solution. In 50 g of solution there is half as much salt-that is, 1.5 g. Or simply convert to a decimal and multiply: 0.030 x 50 g = 1.5 g salt Exercises Add 3/7 and 1/8.  31 56Subtract 1/3 from 6/7. 11 21Multiply 4/7 by 3/8. 12 = 3 56 14Divide 5/9 by 3/5. 25 27Solve the proportion: x/3 = 8/12. x = 2Express 3/7 as a decimal (to the hundredths place). 0.43Express 0.0124 as a fraction. 124 = 31 10000 2500Express 4/7 as a percent. 57%How many grams of sodium are there in 20.0 g of sodium chloride? 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