1. Section 1.6 Division of Real Numbers

2. 1.6 Lecture Guide: Division of Real Numbers Objective: Divide positive and negative real numbers.

3. Notations for the Quotient of x Divided by y for y ≠ 0:

4. Phrases Used To Indicate Division: Key PhraseVerbal ExampleAlgebraic Example Divided by"x divided by y" Quotient"The quotient of 5 and 3" Ratio"The ratio of x to 2"

5. 1. Translate each verbal statement into algebraic form. z divided by two

6. 2. Translate each verbal statement into algebraic form. The quotient of seven and nine

7. 3. Translate each verbal statement into algebraic form. The ratio of three to x

8. Definition of Division: Algebraically Verbally Numerical Example For any real numbers x and y with Dividing two real numbers is the same as multiplying the first number by the multiplicative inverse of the second number.

9. Division of Two Real Numbers: Like signs: Divide the absolute values of the two numbers and use a positive sign for the quotient. Unlike signs: Divide the absolute values of the two numbers and use a negative sign for the quotient. Zero dividend: Zero divisor: for is undefined for every real number x.

10. The Sign of a Product vs. the Sign of a Quotient: 4. Where possible, fill in the correct sign of each product and quotient below. ProductSignQuotientSign (positive)●(positive)=(positive)÷(positive)= (positive)●(negative)=(positive)÷(negative)= (negative)●(positive) =(negative)÷(positive) = (negative)●(negative)=(negative)÷(negative)= (negative or positive)●(0)=(negative or positive)÷(0)= (0)●(negative or positive)=(0)÷(negative or positive)= Although memorization is generally not the best way to learn mathematical concepts, it is very helpful to have the following key facts memorized when performing a division.

11. Mentally evaluate each quotient. 5.

12. Mentally evaluate each quotient. 6.

13. Mentally evaluate each quotient. 7.

14. Mentally evaluate each quotient. 8.

15. Mentally evaluate each quotient. 9.

16. Mentally evaluate each quotient. 10.

17. Algebraically Verbally Numerical Example Dividing Fractions: To divide two fractions, multiply the first fraction by the reciprocal of the second fraction. forand,

18. 11.(a) Why is it important that we require that,, and (b) Can integers like 4 be written as fractions? in the rule for dividing fractions?

19. 12. Divide the following fractions:

20. 13. Divide the following fractions:

21. 14. Divide the following fractions:

22. 15. Divide the following fractions:

23. 16. Divide the following fractions:

24. 17. Divide the following fractions:

25. Properties of radicals will be studied in more detail in Chapter 9. For now, use the fact that for bothandto evaluate each expression. 18. 19.

26. Verbally Numerical Examples Three Signs of a Fraction: For all real numbers a and b with Algebraically Each fraction has three signs associated with it. Any two of these signs can be changed and the value of the fraction will stay the same., and

27. 20. Signs of a fraction: Mentally determine the sign of each expression. Then evaluate each expression on your calculator. (a) (b) Sign: ______ Value: ______ (c) (d) Sign: ______ Value: ______

28. Mean and Range: The mean of a set of numerical scores is an average calculated by dividing the ____________ of scores by the number of scores, and the range of a set of scores is calculated by subtracting the ____________ score minus the ____________ score.

29. On a Beginning Algebra Exam, the following scores were earned: 64, 82, 83, 75, 73, 78, 65, 73, 77, 59, 94, 62, 71, 60. Give the range and the mean for this set of scores. Round the mean to the nearest hundredth. 21. Range = Mean =

30. Percent of error Error of estimate = ____________ value − ____________ value Percent of error = Error of estimate ÷ ____________ value

31. The perimeter of the rectangle in the figure is estimated by using 20 cm for the width and 30 cm for the length. 22. 21.4 cm 32.1 cm (a) Determine the actual perimeter of this rectangle.

32. The perimeter of the rectangle in the figure is estimated by using 20 cm for the width and 30 cm for the length. 22. 21.4 cm 32.1 cm (b) Determine the estimated perimeter of this rectangle.

33. The perimeter of the rectangle in the figure is estimated by using 20 cm for the width and 30 cm for the length. 22. 21.4 cm 32.1 cm (c) Determine the percent of error of this estimate.

34. Objective: Express ratios in lowest terms. Any ratio can be written in fraction form. To express a ratio in lowest terms, simply reduce the fraction. Verbally Numerical Example Ratio: The ratio of a to b is the quotient of a divided by b. Algebraically The ratio a to b can be denoted by either a : b or The ratio 5 to 8 can be denoted by either ____________ or ____________.

35. Express each of the following ratios in lowest terms. 23. Example:

36. Express each of the following ratios in lowest terms. 24.

37. Express each of the following ratios in lowest terms. 25.

38. Use your calculator to reduce the ratios above by entering each on the home screen and then converting the decimal back to a fraction. The calculator will always give the reduced form of a fraction. See Calculator Perspective 1.3.1. 26. Recall that Frac is found by pressing the MATH button. It is very important that you are able to reduce fractions by hand, but the calculator is useful for more challenging problems and as a check on your work.

39. 27. For each pie chart shown, give the ratio of the shaded area to the unshaded area and the ratio of the shaded area to the total area. Express each ratio in lowest terms. Shaded to Unshaded Shaded to Total Example:

40. 27. For each pie chart shown, give the ratio of the shaded area to the unshaded area and the ratio of the shaded area to the total area. Express each ratio in lowest terms. (a) Shaded to Unshaded (b) Shaded to Total

41. 27. For each pie chart shown, give the ratio of the shaded area to the unshaded area and the ratio of the shaded area to the total area. Express each ratio in lowest terms. (c) Shaded to Unshaded (d) Shaded to Total

42. Unit Price Verbally The unit price of an item is the ratio of the price to the number of units. Example Dog food sells for $15.75 for a 50-pound bag. The unit Price = Unit Price=Price  Number of Units

43. 28. Peanut butter is advertised at $2.34 for a 12 oz container and $6.56 for 32 oz container. (a) Find the unit price for each size container. (b) Which size is the better buy?