Download RATES, RATIOS and PROPORTIONS

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 ©