50 MATHCOUNTS LECTURES (6) OPERATIONS WITH DECIMALS

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1 BASIC KNOWLEDGE 1. Decimal representation: A decimal is used to represent a portion of whole. It contains three parts: an integer (which indicates the number of wholes), a decimal point (which separates the integer on the left from the decimal on the right), and a decimal number (which represents a number between 0 and 1). The decimal is read three and fourteen thousand one hundred fifty nine hundred-thousandths. The dot is the decimal point and is read as and Units And Tenths Hundredths Thousandths Tenthousandths Hundredthousandths Whole numbers can also be written with one or more zeros after the decimal point: 3 = 3.0 = 3.00 = The number line representations of decimals: Examples: How would you write six and nine hundredths as a decimal? Answer: How would you write three and nine tenths as a decimal? Answer: 3.9. How would you write five hundredths as a decimal? Answer:

2 2. Changing Decimals to Fractions Move the decimal points to the right such that the numerator becomes an integer. Do the same thing for the denominator

3 3. Decimal addition and subtraction Adding and subtracting decimal numbers is the same as adding whole numbers. The key point is to line up the decimal points. Examples: Compute: (1) = 83

4 (2) = 84

5 (3) =

6 ( ) = = 7. (4) = 86

7 = ( ) = = (5) = 87

8 ( ) = = (6) = 88

9 ( ) = = (7) = 89

10 ( ) = = (8) = 90

11 ( ) = ( ) 0.02 = = (9) = 91

12 = = (10) = ( ) +( ) ( ) = =

13 4. Decimal multiplication and division Multiplication of Two Decimals: (1) Change both decimals to integers by moving the decimal point to the right. (2) Count n, the total number of times the decimal point is moved. (3) Multiply two integers and get the product. (4) Put a decimal point on the right of the units digit of the product. (5) Move the decimal point n times to the left. Note that we don't have to line up the decimal points to multiply. 93

14 Examples: (1) = (we move the decimal point to the right one time) (we move the decimal point to the right one time) One time + one time = two times = (we move the decimal point to the left two times) (2) = 94

15 (we move the decimal point to the right two times) (we move the decimal point to the right one time) Two times + one time = three times = (we move the decimal point to the left three times) Division of Two decimals: (1) Change the divisor from decimal to integer by moving the decimal point to the right. (2) Count n, the total number of times to move the decimal point of the divisor to the end of the number. (3) Move the decimal point of the dividend n times to the right. (4) Perform the division and move the decimal point up to the quotient. Examples: Compute: (1) = 95

16 = (2) = 96

17 ( ) ( ) = (56 7) (165 11) = 8 15 = 120. (3) = 97

18 (4) = = 1.25 ( ) = =

19 19 (3+ 0.1) = = = (5) = 99

20 ( ) ( ) 8 = = (6) = 100

21 ( ) ( ) = = (7) = 101

22 (8) = ( ) ( ) = =

23 (9 0.01) 6 = = = (9) 0.32 ( ) = 103

24 ( ) 0.25 = (32 8) 0.25 = = 1. (10) ( ) (0.5 8) = (12 4) 4 = Decimals and fractions: Decimal Fraction Common fraction or or

25 6. Terminating Decimals and Repeating Decimals Any rational number can be expressed as either a terminating decimal or a repeating decimal. A decimal such as 0.25, which stops, is called a terminating decimal. A rational number a/b in lowest terms results in a repeating decimal if a prime other than 2 or 5 is a factor of the denominator A rational number a/b in lowest terms results in a terminating decimal if the only prime factor of the denominator is 2 or 5 (or both) , 0. 2, Example: How many positive integers less than 100 have reciprocals with terminating decimal representations? 105

26 Solution: 14. The reciprocals of numbers can be a terminating decimal: Repeating block (1). If the denominator has only 2 and 5 as its factors, this fraction can become a terminating decimal. The length of the decimal part equals the greater power of 2 or 5. Example: Length of the decimal part (2). If the denominator has only factors other than 2 and 5, then this fraction can become repeating decimal. The length of the repeating block (period) is the smallest number of nines needed for the number containing 9 s to be divisible by the denominator. 106

27 1 99 Examples: For 1, the repeating block is 2 since 9. ( ) For 1, the repeating block is 3 since 27. ( ) For 1, the repeating block is 4 since 99. ( ) For 1, the repeating block is 6 since ( ) For 1, the repeating block is 6 since ( )

28 (3) If the denominator has 2 or 5 as factors and other prime factors, this fraction can become a mixed repeating decimal. The non-repeating block length is the greater power of 2 or 5. The repeating block length (the repeating period) is the smallest number of nines needed to be divisible by the denominator. (4). Given 1/n, n is a positive integer, the repeating block is r and r n 1. 1 For 1, n = 7 and r n 1 = 6. In fact r (5) (mod p). r is the smallest positive integer. Repeating Block of 1/p (for primes p) p block p block p block p block p block p block

29 6. Operation with Repeating Decimals Examples: (1) Convert the repeating decimal 01. to a fraction. 109

30 Solution: Let x = 01. = (1) Multiply both sides by 10: 10x = (2) (2) (1): 9 x = 1 x = 1/9 (2) Convert the repeating decimal to a fraction. 110

31 Solution: We set x = = and multiply both sides by 100: 100 x = Next we subtract: 100 x = x = x = Next we divide both sides by 99 to get: x = 25/99 5 (3) The fraction of any single digit repeating decimal is the digit over (4) The fraction of any 2-digit repeating decimal is the digits over (5) (6) digits 6 digits 's 2 0's (7) Express as a common fraction. 111

32 Method 1: Method 2: Let x = (1) 100x = (2) 10000x = (3) (3) (2): 9900x = 7173 (4) x (8) Calculate:

33 Solution: Since , so : (9) Calculate: :

34 Solution: Since, and the sum of the first repeating digits ( 4 in and 8 in 2. 83) 5.31 in the addends carries 1, so the last digit of the resulting number needs to be increased by 1: (10) Calculate:

35 Solution: Since , so (11) Calculate:

36 Solution: Since , and the first repeating digit (2 in ) in the top number is smaller than the corresponding digit (3 in ) in the bottom number, so the last digit of the resulting number needs to be decreased by 1: (12) Calculate:

37 Solution: Since the first addend has 2 repeating digits and the second addend has 3 repeating digits, the sum should have 2 3 = 6 repeating digits (the least common multiple of 2 and 3) So (13) Calculate:

38 Solution: and (14) Calculate: , so

39 Solution: Method 1: Method 2: We see that both expressions are the same: (15) Calculate:

40 Solution: (16) Calculate: Solution:

41 EXERCISES Problem 1. Add: Problem 2. Subtract: Problem 3. Find the product: (0.2) (0.5) (4 5) Problem 4. Multiply: Problem 5. Divide: Problem 6. Divide: Problem 7. Simplify: (7 0.25)( ) Problem 8. Simplify: 0.25( ) Problem 9. Simplify: ( ) Problem 10. What is the product of 3.85 and 0.5? Problem 11. Express the product as a decimal number. Problem 12. Find the sum of 18.1, 7.56 and Express your answer as a decimal. Problem 13. Divide 9.62 by 3.7 and express your answer as a decimal. Problem 14. Find the product of 3.9 and 4.7 and express your answer as a decimal. Problem 15. Express the difference as a decimal. Problem 16. Subtract from 3568 and express your answer as a decimal. Problem 17. Express in simplest form: 121

42 10(0.2) 8(0.15) 6(0.5) 8(0.125) Problem 18. Find the sum and express your answer as a decimal. Problem 19. Express as a decimal: Problem 20. Express as a decimal: Problem 21. Express the product as a decimal: (60.5)(0.25)(4.4) Problem 22. Express the quotient as a decimal: Problem 23. Find Express your answer as a decimal. Problem 24. Express as a decimal: Problem 25. Express as a decimal: Problem 26. Express as a decimal: Problem 27. Express as a decimal: Problem 28. Express as a decimal: Problem 29. Express as a decimal rounded to the nearest thousandth: Problem 30. Write as a decimal: Problem 31. Express the product as a decimal: (0.125)(4.8)(0.025)(1.6) Problem 32. Express as a decimal: Problem 33. Express as a decimal to the nearest tenth: Problem 34. Express as a decimal to the nearest thousandth:

43 Problem 35. Express in simplest form: 24( ) 6(0.83 ). Problem 36. Express in simplest form: Problem 37. Express as a decimal: (3.8)(120) Problem 38. Simplify: (0.3)(0.056) Problem 39. Express as a common fraction: Problem 40. Simplify: ( ) + ( ) Problem 41. A sheet of paper is cm thick. How many centimeters high is a stack of 500 sheets? Problem 42. Express the reciprocal of in decimal form. Problem 43. Compute: Problem 44. Compute: Problem 45. Compute: ( ) ( ) Problem 46. Compute: Problem 47. Compute:

44 ANSWER KEYS Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem Problem /10 Problem Problem Problem Problem Problem Problem Problem Problem