Section 3.5 Decimal Numbers and Fractions HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Objectives o Change decimal numbers to fractions. o Change fractions to decimal numbers. o Operate with both fractions and decimal numbers. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Changing Decimal Numbers to Fractions To Change from Decimal Numbers to Fractions A terminating decimal number can be written in fraction form by writing a fraction with the following: 1. a numerator that consists of the whole number formed by all the digits of the decimal number, and 2. a denominator that is the power of ten that names the position of the rightmost digit. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Changing Decimal Numbers to Fractions 25 5 51 1 0.25 100 2 5 2 5 4 hundredths HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 2: Changing Decimal Numbers to Fractions 32 48 8 0.32 100 4 25 25 hundredths HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Changing Decimal Numbers to Fractions 131 0.131 1000 thousandths HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 4: Changing Decimal Numbers to Fractions 75 25 3 3 0.075 1000 25 40 40 thousandths HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 5: Changing Decimal Numbers to Fractions 26 2 13 13 2.6 10 2 5 5 tenths or, as a mixed number, 23 6 3 2.6 2 2 2 25 10 5 tenths HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 6: Changing Fractions to Decimal Numbers 3 8 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. 0.3 75 8 3.000 24 60 56 40 40 0 3 0.375 8 Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 7: Changing Fractions to Decimal Numbers 5 4 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. 1. 2 5 4 5.00 4 10 8 20 20 0 5 1.25 4 Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8: Changing Fractions to Decimal Number 7 12 0.5833 12 7.0000 60 1 00 96 40 36 40 36 4 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. The 3 will repeat without end. Continuing to divide will give a remainder of 4 each time. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 8: Changing Fractions to Decimal Number (Cont.) 7 = 0.58333... where the three dots (called We write 12 an ellipsis) mean “and so on” or to continue the pattern without stopping. Or, if we agree to round, we 7 can write = 0.583 (to the nearest thousandth). 12 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 9: Changing Fractions to Decimal Number 1 7 0. 142857 7 1.000000 7 30 28 20 14 60 56 40 35 50 49 1 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. The six digits will repeat in the same pattern without end. We write 1 = 0.142857142857142857... 7 The remainder will repeat in sequence 1,3,2,6,4,5,1, and so on. Therefore, the digits in the quotient will also repeat in sequence. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Operating with Both Fractions and Decimal Numbers Notes In some cases changing fractions to decimal form may involve rounding the decimal form of a number and settling for an approximate answer. To have a more accurate answer, we may need to change the decimals to fraction form and then perform the operations. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 10: Operating with Both Fractions and Decimal Numbers 1 3 Find the sum 10 + 7.32 + 5 in decimal form. 2 5 Solution 1 1 = 0.50 = 10.50 10 2 2 7.32 = 7.32 3 +5 5 = 5.60 23.42 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. 3 = 0.60 5 Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 11: Operating with Both Fractions and Decimal Numbers 3 3 Determine whether 16 is larger than 0.18 by changing 16 to decimal form and then comparing the two numbers. Find the difference. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 11: Operating with Both Fractions and Decimal Numbers (cont.) Solution Divide first. 0.1875 So 3 0.1875. 16 3.0000 16 16 1 40 1 28 120 112 80 80 0 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 11: Operating with Both Fractions and Decimal Numbers (cont.) Now subtract. 0.1875 0.1800 0.0075 difference 3 Thus is larger than 0.18 and their difference is 16 0.0075. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problems Convert the following to fraction form. 1. 0.475 2. 2.9 3. 0.003 4. 1.17 Convert the following to decimal form. Write repeating decimals using bar notation. 1 13 2 18 5. 6. 7. 8. 20 16 7 37 1 4 9. Find the sum 3 + 5.16 + 6 in decimal form. 4 5 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problem Answers 19 1. 40 117 4. 100 29 2. 10 3 3. 1000 5. 0.8125 6. 0.05 7. 0.285714 8. 0.486 9. 15.21 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.