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- Ratios, - Rates, - Proportions, - Units of measurement
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CONTENTS
RATIOS
the ratio of the number of
teaching assistants to the number
of students in any course …
the ratio of the number of defective
chips to the total number of chips …
What is the ratio of the surface
area of a cube to the surface area
of a rectangular solid
SETTING UP A RATIO
The ratio of x to y
: orx
x yy
The ratio of 24 apples
to 12 oranges is
24 2
12 1
apples
oranges
EQUIVALENT RATIOS
The ratio of 24 apples to 12 oranges is
24 2 4 6
12 1 2 3
apples
oranges
Apples Oranges Total
2 1 3
4 2 6
6 3 9
8 4 12
24 12 36
EQUIVALENT RATIOS
SOLVING A PROPORTION
Cross multiply
3
5 4
x
3.75 3
5 4
4 3 5
15
4
3.75
x
x
x
PART TO PART RATIOS
In a classroom, the ratio of
males to females is 1 to 2, then:
1
2
males
females
3
1 : 2
:
students
m f f
2
1
females
males
PART TO WHOLE RATIOS
In a classroom, the ratio of males
to females is 1 to 2, then:
1
3
males
students
2
3
females
students
3
1 : 2
:
students
m f f
In a classroom, the ratio of
males to females is 1 to 2, then:
3
1 : 2
:
students
m f f
The total number of students in the
classroom must be a multiple of 3.
Females Males Total
2 1 3
4 2 6 Times 2
6 3 9 Times 3
8 4 12 Times 4
10 5 15 Times 5
EQUIVALENT RATIOS
RATIOS WITH MORE THAN TWO TERMS
The sum of the parts
does equal the whole
: :a b c 2:3:5
10
2:3:5
The ratio of is .
a b c Total
2 3 5 10
4 6 10 20 Times 2
6 9 15 30 Times 3
8 12 20 40 Times 4
10 15 25 50 Times 5
EQUIVALENT RATIOS
2 3 2,
3 5 5
a b aIf and then
b c c
COMBINED RATIO
The ratio of a:b is 2:3 and the ratio of
b:c is 3:5. What is the ratio of a:c?
: : c
2 :3:5
a b
COMBINED RATIO
The ratio of a:b is 7:3 and the ratio of
b:c is 2:5. What is the ratio of a:c?
7 2, , ?
3 5
a b a
b c c
2 3 6
5 3 15
b
c
7 2, , ?
3 5
a b a
b c c
14
15
athen
c
7 2 14
3 2 6
a
b
RATIO/FRACTION PROBLEMS
A fraction is a ratio
between two numbers.
2 4 6 1 30
3 6 9 1.5 45
2
3
x
x
RATIOS
RATES
Rate
is a fraction that compares two
quantities that are measured in different
units.
Last year the rate of inflation was 1.2
percent, but for the current year …
A train travels at the
rate of 10 miles/hr …
The rate of boys to girls in a
classroom is ….
RATE
Keep the units straight.
20 students 40 students
1 class 2 classes=
Don’t just average the rates.
Total Aaverage A per B
Total B
AVERAGE RATE
Edwin Lapuerta, Feb 2014
distance
speed time
AVERAGE SPEED
Total Distanceaverage speed
Total Time
Total Distanceaverage speed
Total Time
AVERAGE SPEED
distance
0.010.020:030:040:050:06 sec
= 18m
time
18m m3sec6sec
MOTION PROBLEMS
Jane traveled for 2 hours at a rate of 70 kilometers per hour and
for 5 hours at a rate of 60 kilometers per hour. What was her
average speed for the 7-hour period?
Time = 2 h
Speed = 70 km/h
Time = 5 h
Speed = 60 km/h
Jane traveled for 2 hours at a rate of 70 kilometers per
hour and for 5 hours at a rate of 60 kilometers per hour.
What was her average speed for the 7-hour period?
Distance = Rate . Time
A 70 2
B 60 5
Total 7
140
300
440 440 / 7
= 62.8
RATES
PROPORTIONS
DIRECT PROPORTION
Both quantities
increase simultaneously or
decrease simultaneously.
DIRECT PROPORTION
If Biff can shape 3 surfboards in 50 minutes, how many
surfboards can he shape in 5 hours?
Surfboard Time Rate
3 50
6 100
9 150
12 200
3
50
6 3
100 50
9 3
150 50
12 3
200 50
Constant rate
3
50 300
3 300
50
18
x
x
x
INVERSE PROPORTION
The first quantity
increases while the
second quantity
decreases or viceversa.
INVERSE PROPORTION
If 7 workers can assemble a car in 8 hours, how long
would it take 12 workers to assemble the same car?
Workers Time Product
7 8
14 4
28 2
56 1
56
56
56
56
Constant product
7 8 12
7 8
12
4.66
h
h
h
PROPORTIONS
UNITS OF MEASUREMENT
UNITS OF TIME
1 minute = 60 seconds
1 hour = 60 minutes
1 day = 24 hours
1 week = 7 days
1 year = 52 week
1 year = 365 days
1 leap year = 366 days
UNITS OF MASS
1 pound = 16 ounces
UNITS OF CAPACITY
1 pint = 2 cups
1 quart = 2 pints
1 gallon = 4 quarts
UNITS OF LENGHT
1 foot = 12 inches
1 yard = 3 feet
1 yard = 36 inches
1 ft = 12 in 1 ft2 = 144 in2
1 yd = 3 ft 1 yd2 = 9 ft2
UNITS OF AREA
UNITS OF VOLUME
1 ft = 12 in 1 ft3 = 1,728 in3
1 yd = 3 ft 1 yd3 = 27 ft3
UNITS OF
MEASUREMENT
SUMMARY
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