2 MODULE 4. DECIMALS 4a Decimal Arithmetic Adding Decimals Recall the process used for adding decimal numbers. Adding Decimals. To add decimal numbers, proceed as follows: 1. Place the numbers to be added in vertical format, aligning the decimal points. 2. Add the numbers as if they were whole numbers. 3. Place the decimal point in the answer in the same column as the decimal points above it. Alice has $8.63 in her purse and Joanna has $2.29. If they combine their money, what is the total? EXAMPLE 1. Jane has $4.35 in her purse. Jim has $5.62 in his wallet. If they sum their money, what is the total? Solution. Arrange the numbers in vertical format, aligning decimal points, then add. $4.35 +$5.62 $9.97 Answer: $10.91 Together they have $9.97, nine dollars and ninety seven cents. If there are multiple decimal values, we can follow the same procedure. Sometimes it is convenient to add in some additional zeros to make the numbers line up visually. Add: 12.9+4.286 + 33.97 EXAMPLE 2. Find the sum: 12.2 + 8.352 + 22.44. Solution. Arrange the numbers in vertical format, aligning their decimal points in a column. Note that we add trailing zeros to improve the columnar alignment. 12.200 8.352 +22.440 42.992
4A. DECIMAL ARITHMETIC 3 Hence, 12.2+8.352 + 22.44 = 42.992. Answer: 51.156 Subtracting Decimals As with addition, we add trailing zeros to the fractional part of the decimal numbers to help columnar alignment. Subtracting Decimals. To subtract decimal numbers, proceed as follows: 1. Place the numbers to be subtracted in vertical format, aligning the decimal points. 2. Subtract the numbers as if they were whole numbers. 3. Place the decimal point in the answer in the same column as the decimal points above it. EXAMPLE 3. Find the difference: 13.3 8.572. Subtract: 15.2 8.756 Solution. Arrange the numbers in vertical format, aligning their decimal points in a column. Note that we add trailing zeros to the fractional part of 13.3 to improve columnar alignment. 13.300 8.572 4.728 Hence, 13.3 8.572 = 4.728. Answer: 6.444 Adding and Subtracting Signed Decimal Numbers The rules for adding integers with like signs also apply to adding decimals with like signs. Adding Two Decimals with Like Signs. To add two decimals with like signs, proceed as follows: 1. Add the magnitudes of the decimal numbers. 2. Prefix the common sign.
4 MODULE 4. DECIMALS The rules for adding integers with unlike signs also apply to adding decimals with unlike signs. Adding Two Decimals with Unlike Signs. unlike signs, proceed as follows: To add two decimals with 1. Subtract the smaller magnitude from the larger magnitude. 2. Prefix the sign of the decimal number with the larger magnitude. Simplify: 5.7+( 83.85) EXAMPLE 4. Simplify: 3.2+( 18.95). Solution. To add like signs, first add the magnitudes. 3.20 +18.95 22.15 Answer: 89.55 Prefix the common sign. Hence, 3.2 +( 18.95) = 22.15 Remember that subtraction means add the opposite. Simplify: 2.384 ( 15.2) EXAMPLE 5. Simplify: 8.567 ( 12.3). Solution. Subtraction must first be changed to addition by adding the opposite. 8.567 ( 12.3) = 8.567 + 12.3 We now have unlike signs. magnitude. Subtract the smaller magnitude from the larger 12.300 8.567 3.733 Prefix the sign of the decimal number with the larger magnitude. Hence: 8.567 ( 12.3) = 8.567 + 12.3 =3.733 Answer: 12.816
4A. DECIMAL ARITHMETIC 5 Multiplying Decimals When multiplying decimal numbers, use the following method. Multiplying Decimal Numbers. To multiply two decimal numbers, perform the following steps: 1. Ignore the decimal points in the factors and multiply the two factors as if they were whole numbers. 2. Count the number of digits to the right of the decimal point in each factor. Sum these two numbers. 3. Place the decimal point in the product so that the number of digits to the right of the decimal point equals the sum found in step 2. EXAMPLE 6. Simplify: (8.235)(2.3). Multiply: (9.582)(8.6) Solution. Using the convenient vertical format, write 8.235 2.3 24705 16 470 18.9405 The factor 8.235 has three digits to the right of the decimal point; the factor 2.3 has one digit to the right of the decimal point. Therefore, there must be a total of four digits to the right of the decimal point in the product or answer. Answer: 82.4052 EXAMPLE 7. Simplify: (5.68)( 0.012). Multiply: (9.23)( 0.018) Solution. Ignore the signs to do the multiplication, then consider the signs in the final answer.
6 MODULE 4. DECIMALS The first factor has two digits to the right of the decimal point, the second factor has three. Therefore, there must be a total of five digits to the right of the decimal point in the product or answer. This necessitates prepending a zero in front of our product. Unlike signs give a negative product. Hence: (5.68)( 0.012) = 0.06816 Answer: 0.16614 5.68 0.012 1136 568 0.06816 Powers of Ten Consider: 10 1 =10 10 2 =10 10 = 100 10 3 =10 10 10 = 1, 000 10 4 =10 10 10 10 = 10, 000 Note the answer for 10 4, a one followed by four zeros! Do you see the pattern? Powers of Ten. In the expression 10 n,theexponentmatchesthenumberof zeros in the answer. Hence, 10 n will be a 1 followed by n zeros. Multiplying Decimal Numbers by Powers of Ten Let s multiply 1.234567 by 10 3, or equivalently, by 1,000. Ignore the decimal point and multiply the numbers as whole numbers. 1.234567 1000 1234.567000 The sum total of digits to the right of the decimal points in the factors is 6. Therefore, we place the decimal point in the product so that there are six digits to the right of the decimal point. However, the trailing zeros may be removed without changing the value of the product. That is, 1.234567 times 1000 is 1234.567. Note that the decimal
4A. DECIMAL ARITHMETIC 7 point in the product is three places further to the right than in the original factor. This observation leads to the following result. Multiplying a Decimal Number by a Power of Ten. Multiplying a decimal number by 10 n will move the decimal point n places to the right. EXAMPLE 8. Simplify: 1.234567 10 4 Simplify: 1.234567 10 2 Solution. Multiplying by 10 4 (or equivalently, by 10,000) moves the decimal 4 places to the right. Thus, 1.234567 10, 000 = 12345.67. Answer: 123.4567 Dividing Decimals We begin by reviewing the division algorithm for dividing a decimal number by a whole number. Dividing a Decimal by a Whole Number. To divide a decimal number by a whole number, proceed as follows: 1. Set up the long division as you would the division of two whole numbers. 2. Perform the division as if the numbers were both whole numbers, adding zeros to the right of the decimal point in the dividend as necessary to complete the division. 3. Place the decimal point in the quotient immediately above the decimal point in the dividend. EXAMPLE 9. Divide: 155.2 25. Divide: 42.55 23 Solution. Use long division and arrange as if dividing whole numbers. 6.2 25)155.2 150 52 50 2
8 MODULE 4. DECIMALS We still have a nonzero remainder. Adding one zero does no good. 6.20 25)155.20 150 52 50 20 However, if we add tw zeros, the division completes with a zero remainder. 6.208 25)155.200 150 52 50 200 200 0 Answer: 1.85 Thus, 155.2 divided by 25 is 6.208. If you are dividing a decimal number by another decimal number, then there are additional steps that must be included in the division algorithm. Dividing by a Decimal Divisor. If the divisor contains a decimal, proceed as follows: 1. Move the decimal to the end of the divisor. 2. Move the decimal in the dividend an equal number of places. Divide: 0.0113 0.05 EXAMPLE 10. Divide: 0.03 0.024. Solution. First, divide the magnitudes. Move the decimal in the divisor to the end of the divisor. Move the decimal in the dividend an equal number of places (three places) to the right. Note that this requires writing a trailing zero in the dividend. 0.024 )0.030 Our problem then becomes:
4A. DECIMAL ARITHMETIC 9 24)30 We can now follow the algorithm for dividing a decimal number by a whole number. Note that we have to add two trailing zeros in the dividend to complete the division with a zero remainder. 1.25 24)30.00 24 60 48 120 120 0 Finally, because the quotient of unlike signs is negative, 0.03 divided by 0.024 is 1.25. That is, 0.03 0.024 = 1.25. Answer: 0.226 Dividing by Powers of Ten Dividing a Decimal by a Power of Ten. Dividing a decimal number by 10 n will move the decimal point n places to the left. EXAMPLE 11. Simplify: 123456.7 10 4 Simplify: 123456.7 10 2 Solution.Dividing by 10 4 (or equivalently, 10,000) moves the decimal point four places to the left. Thus, 123456.7 10 4 =12.34567. Answer: 1234.567