Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
RATES, RATIOS and PROPORTIONS RATIOS • A ratio compares the magnitude (size) of two quantities. There are two types of ratios: partpart and part-whole. RATIOS • A ratio compares the magnitude (size) of two quantities. There are two types of ratios: partpart and part-whole. Ex 1: If there are seven boys and twelve girls in a class, then the ratio of boys to girls can be 7 expressed as 7 to 12, 12 , or 7:12 (part-part ratio). RATIOS • A ratio compares the magnitude (size) of two quantities. There are two types of ratios: part-part and part-whole. Ex 1: If there are seven boys and twelve girls in a class, then 7 the ratio of boys to girls can be expressed as 7 to 12, 12 , or 7:12 (part-part ratio). Ex 2: Suppose there is a wall made up of twelve blocks, five white blocks and seven red blocks. The ratio of white blocks to 5 the total number of blocks is , which is a part-whole ratio. 12 5 The ratio of white blocks to red blocks is , which is a part-part 7 ratio. RATES • When the quantities being compared have different dimensions (units), then the ratio is called a rate. RATES • When the quantities being compared have different dimensions (units), then the ratio is called a rate. Ex 1: A phone company charges $0.84 for 7 minutes of long distance. The rate is $ 0.84/7 minutes, which is equal to $0.12 per minute. RATES • When the quantities being compared have different dimensions (units), then the ratio is called a rate. Ex 1: A phone company charges $0.84 for 7 minutes of long distance. The rate is $ 0.84/7 minutes, which is equal to $0.12 per minute. Ex 2: A student reads 10 pages in 8 minutes. This rate is 10 pages/8 minutes, which is equal to 5 pages per 4 minutes. RATES • When the quantities being compared have different dimensions (units), then the ratio is called a rate. Ex 1: A phone company charges $0.84 for 7 minutes of long distance. The rate is $ 0.84/7 minutes, which is equal to $0.12 per minute. Ex 2: A student reads 10 pages in 8 minutes. This rate is 10 pages/8 minutes, which is equal to 5 pages per 4 minutes. Ex 3: If a 12-ounce box of cereal sells for $2.40, and a 16ounce box sells for $2.88, which is the better buy? The unit rate of the first box is $0.20/ounce ($2.40/12 ounces), and the unit rate of the second box is $0.18/ounce ($ 2.88/16 ounces). Therefore, the second box is a better buy. PROPORTIONS • A proportion is a statement of equality between two ratios or rates. PROPORTIONS • A proportion is a statement of equality between two ratios or rates. • In a proportion, if 𝑎 𝑐 = 𝑏 𝑑 then a∙d = b∙c. PROPORTIONS • Ex: A person drives 126 miles in 3 hours. At the same speed (rate), how far (x) would the driver travel in 4 hours? The proportion can be written as 126𝑚𝑖𝑙𝑒𝑠 𝑥 = 3ℎ𝑜𝑢𝑟𝑠 4ℎ𝑜𝑢𝑟𝑠 PROPORTIONS • Multiplying across by the multiplicative inverse yields: (4ℎ𝑜𝑢𝑟𝑠) ∙ 126𝑚𝑖𝑙𝑒𝑠 3ℎ𝑜𝑢𝑟𝑠 = (4ℎ𝑜𝑢𝑟𝑠) ∙ or x = (4)∙(126)/(3) = 168 miles. 𝑥 4ℎ𝑜𝑢𝑟𝑠 In summary… • … rate and proportion, together with ratio, are used for solving many real-world problems that involve comparing different quantities. Let’s take a closer look at rates and ratios • Recall that rates are ratios in which the quantities being compared have different units (and thus, different dimensions). Let’s take a closer look at rates and ratios (cont’d) • Recall that rates are ratios in which the quantities being compared have different units (and thus, different dimensions). • It follows that ratios must compare quantities with the same units (and thus, same dimensions). Let’s take a closer look at rates and ratios (cont’d) • It follows that ratios must compare quantities with the same units (and thus, same dimensions). • Therefore, dimensions on ratios cancel each other out: (𝑙𝑒𝑛𝑔𝑡ℎ) (𝑙𝑒𝑛𝑔𝑡ℎ) → [𝐿] [𝐿] → no dimension Let’s take a closer look at rates and ratios (cont’d) • Therefore, dimensions on ratios cancel each other out: (𝑙𝑒𝑛𝑔𝑡ℎ) (𝑙𝑒𝑛𝑔𝑡ℎ) → [𝐿] [𝐿] → no dimension RATIOS have NO DIMENSIONS. In Physics… … many physical quantities are RATIOS. In Physics… … many physical quantities are RATIOS. RATIOS in Physics are so important, they have (as a class) a name: adimensional quantities. In Physics… … many physical quantities are RATIOS. RATIOS in Physics are so important, they have (as a class) a name: adimensional quantities. This name comes from “a-” + “dimensional” (without) + “dimensional” Adimensional Quantities • The prefix “a-” means “without”, such as in “adiabatic” (without change in pressure); “asymmetric” (without symmetry) and “asexual” (without gender). Adimensional Quantities • The prefix “a-” means “without”, such as in “adiabatic” (without change in pressure); “asymmetric” (without symmetry) and “asexual” (without gender). • The suffix “-less” also means “without”, and can be used in the same way: “dimensionless”, “regardless” and “genderless”. The “Brad Pitt”s and “Angelina Jolie”s of Physics • Famous rates: Density The “Brad Pitt”s and “Angelina Jolie”s of Physics • Famous rates: Density Speed The “Brad Pitt”s and “Angelina Jolie”s of Physics • Famous rates: Density Speed Velocity The “Brad Pitt”s and “Angelina Jolie”s of Physics • Famous rates: Density Speed Velocity Acceleration The “Brad Pitt”s and “Angelina Jolie”s of Physics • Famous rates: Density Speed Velocity Acceleration Field The “Brad Pitt”s and “Angelina Jolie”s of Physics • Famous rates: Density Speed Velocity Acceleration Field Potential The “Brad Pitt”s and “Angelina Jolie”s of Physics • Famous rates: Density Speed Velocity Acceleration Field Potential Coefficients of expansion The “Brad Pitt”s and “Angelina Jolie”s of Physics • Famous adimensional quantities: 1. Trigonometric functions (sin, cos and tan) The “Brad Pitt”s and “Angelina Jolie”s of Physics • Famous adimensional quantities: 1. Trigonometric functions (sin, cos and tan) 2. Magnification (glasses, lenses and mirrors) The “Brad Pitt”s and “Angelina Jolie”s of Physics • Famous adimensional quantities: 1. Trigonometric functions (sin, cos and tan) 2. Magnification (glasses, lenses and mirrors) 3. Mach number (object’s speed relative to sound’s) The “Brad Pitt”s and “Angelina Jolie”s of Physics • Famous adimensional quantities: 1. 2. 3. 4. Trigonometric functions (sin, cos and tan) Magnification (glasses, lenses and mirrors) Mach number (object’s speed relative to sound’s) The number (circumference to diameter ratio for a circle) The “Brad Pitt”s and “Angelina Jolie”s of Physics • Famous adimensional quantities: 1. 2. 3. 4. 5. Trigonometric functions (sin, cos and tan) Magnification (glasses, lenses and mirrors) Mach number (object’s speed relative to sound’s) The number (circumference to diameter ratio for a circle) Friction coefficients The “Brad Pitt”s and “Angelina Jolie”s of Physics • Famous adimensional quantities: 1. 2. 3. 4. 5. 6. Trigonometric functions (sin, cos and tan) Magnification (glasses, lenses and mirrors) Mach number (object’s speed relative to sound’s) The number (circumference to diameter ratio for a circle) Friction coefficients Albedo (percentage of light reflected by celestial bodies) The “Brad Pitt”s and “Angelina Jolie”s of Physics • Famous adimensional quantities: 1. 2. 3. 4. 5. 6. Trigonometric functions (sin, cos and tan) Magnification (glasses, lenses and mirrors) Mach number (object’s speed relative to sound’s) The number (circumference to diameter ratio for a circle) Friction coefficients Albedo (percentage of light reflected by celestial bodies) 7. Oblateness (also called ellipticity and flattening) The “Brad Pitt”s and “Angelina Jolie”s of Physics • Famous adimensional quantities: 1. 2. 3. 4. 5. 6. Trigonometric functions (sin, cos and tan) Magnification (glasses, lenses and mirrors) Mach number (object’s speed relative to sound’s) The number (circumference to diameter ratio for a circle) Friction coefficients Albedo (percentage of light reflected by celestial bodies) 7. Oblateness (also called ellipticity and flattening) 8. Mechanical advantages The “Brad Pitt”s and “Angelina Jolie”s of Physics • Famous adimensional quantities: 1. 2. 3. 4. 5. 6. Trigonometric functions (sin, cos and tan) Magnification (glasses, lenses and mirrors) Mach number (object’s speed relative to sound’s) The number (circumference to diameter ratio for a circle) Friction coefficients Albedo (percentage of light reflected by celestial bodies) 7. Oblateness (also called ellipticity and flattening) 8. Mechanical advantages 9. Efficiency (of simple machines) The “Brad Pitt”s and “Angelina Jolie”s of Physics • Famous adimensional quantities: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. Trigonometric functions (sin, cos and tan) Magnification (glasses, lenses and mirrors) Mach number (object’s speed relative to sound’s) The number (circumference to diameter ratio for a circle) Friction coefficients Albedo (percentage of light reflected by celestial bodies) Oblateness (also called ellipticity and flattening) Mechanical advantages Efficiency (of simple machines) Structural (load) efficiency The “Brad Pitt”s and “Angelina Jolie”s of Physics • Famous adimensional quantities: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. Trigonometric functions (sin, cos and tan) Magnification (glasses, lenses and mirrors) Mach number (object’s speed relative to sound’s) The number (circumference to diameter ratio for a circle) Friction coefficients Albedo (percentage of light reflected by celestial bodies) Oblateness (also called ellipticity and flattening) Mechanical advantages Efficiency (of simple machines) Structural (load) efficiency Opacity/transparency of surfaces The “Brad Pitt”s and “Angelina Jolie”s of Physics • Famous adimensional quantities: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. Trigonometric functions (sin, cos and tan) Magnification (glasses, lenses and mirrors) Mach number (object’s speed relative to sound’s) The number (circumference to diameter ratio for a circle) Friction coefficients Albedo (percentage of light reflected by celestial bodies) Oblateness (also called ellipticity and flattening) Mechanical advantages Efficiency (of simple machines) Structural (load) efficiency Opacity/transparency of surfaces Eccentricity (ratio of large to small axes of an ellipse) The “Brad Pitt”s and “Angelina Jolie”s of Physics • Famous adimensional quantities: 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. Magnification (glasses, lenses and mirrors) Mach number (object’s speed relative to sound’s) The number (circumference to diameter ratio for a circle) Friction coefficients Albedo (percentage of light reflected by celestial bodies) Oblateness (also called ellipticity and flattening) Mechanical advantages Efficiency (of simple machines) Structural (load) efficiency Opacity/transparency of surfaces Eccentricity (ratio of large to small axes of an ellipse) Conversion coefficients The “Brad Pitt”s and “Angelina Jolie”s of Physics • Famous adimensional quantities: 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. Mach number (object’s speed relative to sound’s) The number (circumference to diameter ratio for a circle) Friction coefficients Albedo (percentage of light reflected by celestial bodies) Oblateness (also called ellipticity and flattening) Mechanical advantages Efficiency (of simple machines) Structural (load) efficiency Opacity/transparency of surfaces Eccentricity (ratio of large to small axes of an ellipse) Conversion coefficients Percentage, percent relative and relative errors The “Brad Pitt”s and “Angelina Jolie”s of Physics • Famous adimensional quantities: 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. The number (circumference to diameter ratio for a circle) Friction coefficients Albedo (percentage of light reflected by celestial bodies) Oblateness (also called ellipticity and flattening) Mechanical advantages Efficiency (of simple machines) Structural (load) efficiency Opacity/transparency of surfaces Eccentricity (ratio of large to small axes of an ellipse) Conversion coefficients Percentage, percent relative and relative errors Indexes of refraction, reflection and absorption The “Brad Pitt”s and “Angelina Jolie”s of Physics • Famous adimensional quantities: 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. Friction coefficients Albedo (percentage of light reflected by celestial bodies) Oblateness (also called ellipticity and flattening) Mechanical advantages Efficiency (of simple machines) Structural (load) efficiency Opacity/transparency of surfaces Eccentricity (ratio of large to small axes of an ellipse) Conversion coefficients Percentage, percent relative and relative errors Indexes of refraction, reflection and absorption pH The “Brad Pitt”s and “Angelina Jolie”s of Physics • Famous adimensional quantities: 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. Albedo (percentage of light reflected by celestial bodies) Oblateness (also called ellipticity and flattening) Mechanical advantages Efficiency (of simple machines) Structural (load) efficiency Opacity/transparency of surfaces Eccentricity (ratio of large to small axes of an ellipse) Conversion coefficients Percentage, percent relative and relative errors Indexes of refraction, reflection and absorption pH Decibel ADIMENSIONAL QUANTITIES HAVE NO UNITS!!! NONE – ZERO – ZILCH – NADA THE END Lilian Wehner ©