Write Ratios and Proportions & Cross Products

1. Write Ratios and Proportions&Cross Products Section 3.5 & 3.6

2. Vocabulary • Ratio • Uses division to compare two quantities. These ratios can be expressed in several different ways. They include: • Subtraction Property • Subtracting the same number to each side of an equation produces an equivalent equation • If x+a=b, then x+a-a=b-a, or x=b-a

3. Examples • Wins and Losses • In the 2005-2006 season, the KU Men’s Basketball Team had 25 wins and 8 losses. • What is the ratio of wins to losses • What is the ratio of wins to total games • What is the ratio of losses to total games

4. Vocabulary • Proportion • An equation that states that two ratios are equivalent. The general form is • Subtraction Property • Subtracting the same number to each side of an equation produces an equivalent equation • If x+a=b, then x+a-a=b-a, or x=b-a

5. Vocabulary • Proportion • This can also be written in the form In this format, b & c are called the means and a & d are called the extremes. • Subtraction Property • Subtracting the same number to each side of an equation produces an equivalent equation • If x+a=b, then x+a-a=b-a, or x=b-a

6. Practical Example • Writing and solving proportions • Example 3, page 164 • The elevator that takes passengers from the lobby of the John Hancock Center in Chicago to the observation level travels 150 feet in 5 seconds. The observation level is located on the 94th floor, at 1029 feet above the ground. Find the time it takes the elevator to travel from the lobby to the observation level.

7. Practical Example • Writing and solving proportions • Could we solve this another way?

8. Cross Product • Cross product • Product of the numerator of one ratio and the denominator of the other ratio.

9. Cross products The cross product is really the same as our other format; it just combines two steps. Let’s solve for c

12. Cross Product & Cartography • The use of proportions and cross products becomes very important to us when we start using maps. • All maps have a scale listed on them either on the map or in the legend • We use this scale to determine actual distances based on measured distances.

13. Tennessee Map • What is the scale of this map? • What is the measured distance between Murfreesboro, TN and Shelbyville, TN • What is the approximate actual distance between the two towns?

14. Homework • 3.5 • 1, 2, 31-55 odd • 3.6 • 1, 2, 15-18, 19-41 odd

15. Scale • Scale has many definitions • For our purposes, we’re going to use this one: 3.a graduated line, as on a map, representing proportionate size • Scales on a map are usually shown either as a ratio or as 1”=distance Scale: 1.a succession or progression of steps or degrees; graduated series: the scale of taxation; the social scale. 2.a series of marks laid down at determinate distances, as along a line, for purposes of measurement or computation: the scale of a thermometer. 3.a graduated line, as on a map, representing proportionate size. 4.a table of graduated rates, as of prices or wages: These unions use different scales. 5.a wage that conforms to such rates: How much is scale? 6.Also called union scale.a wage fixed by contract that is the minimum permitted to be paid to or accepted by a particular category of employed persons: All actors and musicians for the performance, including the stars, are working for scale. 7.an instrument with graduated spaces, as for measuring. 8.the proportion that a representation of an object bears to the object itself: a model on a scale of one inch to one foot. 9.the ratio of distances or sometimes of areas on a map to the corresponding values on the earth. 10.a certain relative or proportionate size or extent: They built a residence on a yet more magnificent scale. 11.a standard of measurement or estimation; point of reference by which to gauge or rate: We have no scale by which to judge his achievements. 12.Music. a succession of tones ascending or descending according to fixed intervals, esp. such a series beginning on a particular note: the major scale of C. 13.Education, Psychology. a graded series of tests or tasks for measuring intelligence, achievement, adjustment, etc. 14.Arithmetic. a system of numerical notation: the decimal scale. 15.anything by which one may ascend. 16.Obsolete. a.a ladder. b.a flight of stairs. –verb (used with object) 17.to climb by or as if by a ladder; climb up or over. 18.to make according to scale. 19.to adjust in amount according to a fixed scale or proportion (often fol. by down or up): to scale down wages. 20.to measure by or as if by a scale. 21.Lumbering. a.to measure (logs). b.to estimate the amount of (standing timber). 22.Australian Informal. to ride on (public transportation) without paying the fare. –verb (used without object) 23.to climb; ascend; mount. 24.to progress in a graduated series.

16. Scale • Scale has many definitions • For our purposes, we’re going to use this one: 3.a graduated line, as on a map, representing proportionate size • Scales on a map are usually shown either as a ratio or as 1”=distance Scale: 1.a succession or progression of steps or degrees; graduated series: the scale of taxation; the social scale. 2.a series of marks laid down at determinate distances, as along a line, for purposes of measurement or computation: the scale of a thermometer. 3.a graduated line, as on a map, representing proportionate size. 4.a table of graduated rates, as of prices or wages: These unions use different scales. 5.a wage that conforms to such rates: How much is scale? 6.Also called union scale.a wage fixed by contract that is the minimum permitted to be paid to or accepted by a particular category of employed persons: All actors and musicians for the performance, including the stars, are working for scale. 7.an instrument with graduated spaces, as for measuring. 8.the proportion that a representation of an object bears to the object itself: a model on a scale of one inch to one foot. 9.the ratio of distances or sometimes of areas on a map to the corresponding values on the earth. 10.a certain relative or proportionate size or extent: They built a residence on a yet more magnificent scale. 11.a standard of measurement or estimation; point of reference by which to gauge or rate: We have no scale by which to judge his achievements. 12.Music. a succession of tones ascending or descending according to fixed intervals, esp. such a series beginning on a particular note: the major scale of C. 13.Education, Psychology. a graded series of tests or tasks for measuring intelligence, achievement, adjustment, etc. 14.Arithmetic. a system of numerical notation: the decimal scale. 15.anything by which one may ascend. 16.Obsolete. a.a ladder. b.a flight of stairs. –verb (used with object) 17.to climb by or as if by a ladder; climb up or over. 18.to make according to scale. 19.to adjust in amount according to a fixed scale or proportion (often fol. by down or up): to scale down wages. 20.to measure by or as if by a scale. 21.Lumbering. a.to measure (logs). b.to estimate the amount of (standing timber). 22.Australian Informal. to ride on (public transportation) without paying the fare. –verb (used without object) 23.to climb; ascend; mount. 24.to progress in a graduated series.