Decimals, Real Numbers, and Proportional Reasoning

1. Decimals, Real Numbers, and Proportional Reasoning 7.1 Decimals and Real Numbers 7.2 Computations with Decimals 7.3 Proportional Reasoning 7.4 Percent

2. 7.1 Decimals and Real Numbers

3. PHYSICAL REPRESENTATIONS OF DECIMALS Using Unit Squares, Strips, Small Squares:

4. PHYSICAL REPRESENTATIONS OF DECIMALS Using Base Ten Blocks:

5. PHYSICAL REPRESENTATIONS OF DECIMALS Using Money:

6. Example 7.1 Using Money to Represent Decimals How would you use money to explain the decimal 23.75 to elementary school students? Use 2 ten-dollar bills, 3 dollars, 7 dimes, and 5 pennies. The students will readily see that this collection of bills and coins is worth $23.75, making it easy to explain the expanded notation since a dime is one-tenth of a dollar and a penny is one one-hundredth of a dollar.

7. NEGATIVE NUMBERS AND 0AS EXPONENTS If n is a positive integer, and a ≠ 0 then,

8. Example 7.2 Expanded Form of Decimals Write 234.72 in expanded exponential form: Write 30.0012 in expanded exponential form:

9. MULTIPLYING AND DIVIDING BY POWERS OF 10 If r is a positive integer, the notational effect of multiplying a decimal by 10r is to move the decimal point r places to the right. The notational effect of dividing a decimal by 10r (that is, multiplying by 10‒r) is to move the decimal point r places to the left.

10. Example 7.3 Multiplying and Dividing by Powers of 10 Compute each of the following:

11. DEFINITION: Rational Numbers Represented by Terminating Decimals If a and b are integers with b ≠ 0, if is in simplest form, and if the prime factor other than 2 and/or 5 divides b, then can be represented as a terminating decimal, and conversely.

12. Example 250

13. CHARACTERIZING RATIONAL NUMBERS AS DECIMALS Every rational number can be written as either a terminating or a repeating decimal. Conversely, every terminating or repeating decimal represents a rational number.

14. ORDERING DECIMALS To order two positive decimals, adjoin 0s on the left if necessary so that there are the same number of digits to the left of the decimal point in both numbers, and then determine the first digits from the left that differ. The decimal with the lesser of these two digits is the lesser decimal.

15. Example 7.8: Ordering Decimals Which represents the lesser number, 2.35714 or 2.3570946? The first digits from the left that differ are 1 and 0. Since 0 < 1, it follows that 23570946 , 2.35714

16. DEFINITION: IRRATIONAL AND REAL NUMBERS Numbers represented by nonterminating, nonperiodic decimals are called irrational numbers. The set R consisting of all rational numbers and all irrational numbers is called the set of real numbers.

17. 7.2 Computations with Decimals

18. THE 5-UP RULE FOR ROUNDING DECIMALS To round a decimal to a given place, consider the digit in the next place to the right. • If it is smaller than 5, replace it and all of the digits to its right with 0. • If it is 5 or larger, replace it and all digits to the right by 0 and increase the digit in the given place by one. Replaced digits to the right of the decimal are then dropped to give the rounded decimal.

19. Example 7.11: Rounding Decimals b. Round 3.6147 to the nearest tenth. Consider the digit to the right of 6. • Since 1 < 5, replace it and all of the digits to its right with 0. Replaced digits to the right of the decimal are then dropped to give the rounded decimal.

20. ADDING AND SUBTRACTING DECIMALS Understand the process by relating it to addition or subtraction of fractions.

21. MULTIPLYING DECIMALS Understand the process by relating it to multiplication of fractions.

22. Example Understand the process by relating it to multiplication of fractions.

23. MULTIPLYING DECIMALS To multiply two decimals, do the following: • Multiply as with integers. • Count the number of digits to the right of the decimal point in each factor in the product, add these numbers, and call their sum t. • Finally, place the decimal point in the product obtained so that there are t digits to the right of the decimal point.

24. 7.3 Proportional Reasoning

25. DEFINITION:RATIO If a and b are real numbers with b ≠ 0, the ratio of a to b is the quotient

26. Example Express a ratio of 24 to 16 as a fraction in simplest form.

27. DEFINITION:PROPORTION If are two ratios and this equality is called a proportion.

28. CONDITIONS FOR A PROPORTION The equality is a proportion if, and only if, ad = bc.

29. Example 7.17: Determining Proportions

30. DEFINITION:yisPROPORTIONAL TO x If the variables x and y are related by the equation then yis said to be proportional to x, and k is called the constant of proportionality.

31. 7.4 Percent

32. DEFINITION:PERCENT If r is any nonnegative real number, then r percent, written r %, is the ratio

33. Example 7.24: Expressing Decimals as Percents Write these decimals as percents: a. 0.25 = 25% b. 0.333… = 33.333…% c. 1.255 = 125.5% d. 0.0035 = 0.35%

34. Example 7.25: Expressing Percents as Decimals Express these percents as decimals: a. 40% = 0.40 b. 12% = 0.12 c. 127% = 1.27 d. 0.5% = 0.005

35. Example Write   as a percent.

36. Example 7.29: Calculating A Number of Which a Given Number is a Given Percentage Soo Ling scored 92% on her last test. If she got 23 questions right, how many problems were on the test? Let n = the total # of problems on the test 92% of n = 23

37. CALCULATING COMPOUND INTEREST The value of an investment of P dollars at the end of t years, if interest is paid at the annual rate of r % compounded n times a year, is

38. A COMPOUND INTEREST PROBLEM Many credit card companies charge 18% interest compounded monthly on unpaid balances. Suppose your card was “maxed out” at your credit limit of $10,000 and that you were unable to make payments for two years. Aside from penalties, how much debt would you owe, based on compound interest alone?

39. A COMPOUND INTEREST PROBLEM Principal P = 10000 Rate r = 18 Compounded 12 times/yr t = 12 Number of years n = 2