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| Ratios and Proportions
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Ratio
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Proportion
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Ratios
and proportion properties
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Direct proportion
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Cross product, means and extremes
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Proportionality
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Graphical representation of
proportionality
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Proportion practice problems, use
of the rule of three
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Inverse proportion
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Graphical representation of inverse
proportionality
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Coefficient of inverse
proportionality
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Inverse proportion practice
problems |
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| Ratio |
| A ratio is a comparison of two similar quantities obtained by dividing
one quantity by the other. |
| A ratio can be written as, |
| a
:
b
or as a fraction a/b
or by the phrase ' a to
b '. |
| A ratio is a way in which
quantities can be divided or shared. It shows how much bigger one
thing is than |
| another. |
| Make the numbers in ratios smaller so that
they are easier to compare. Do this by dividing each side of the |
| ratio
by the same number, the highest common factor. |
|
| Proportion |
| A proportion is
a statement of the equality of two ratios. If the fractions both reduce
to the same value, the |
| proportion is true. |
| Example: |
|
9
: 6 = 3/2
and, 21
: 14 = 3/2,
follows,
9 :
6 = 21 : 14. |
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| Ratios and proportion properties
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| A ratio's value is not altered if both the first and second term are
multiplied or divided by the same number |
| different from zero. |
|
a
:
b =
(a · c)
:
(b ·
c) |
 |
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| |
 |
 |
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| Two quantities are in
direct proportion when they increase or decrease in the same ratio. |
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| When two ratios are equal, then the
cross products of the ratios are equal. |
|
That is, |
a
:
b = c
:
d |
or |
 |
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|
| the
product of the means is equal to the product of the extremes.
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| The
first and last term in a proportion are called the extremes, the
second and third terms are called the
|
| means.
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| If any three terms in
a proportion are given, the fourth may be found. |
| |
 |
|
 |
|
| |
the
extremes |
|
the
means |
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| Examples: |
a)
2
:
x =
6 :
15 |
|
b)
8
:
(x
-
1)
= 2 :
7 |
|
c)
(7
-
x)
:
3 =
4x :
9 |
| |
 |
|
 |
|
9 · (7
-
x) =
3 ·
4x |
| |
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63
-
9x
= 12x |
| |
x = 5 |
|
x = 29 |
|
21x = 63
| ¸
21 |
| |
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|
x = 3 |
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| The first and third terms, and the second and the fourth terms in a
proportion are the corresponding terms. |
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| Proportionality
|
| Two related variable quantities
x
and y
whose ratio is constant are called proportional or directly
proportional, |
| written
y
:
x =
m. |
| That is,
y =
m
· x ,
where m
(m > 0) is called the constant of proportionality, |
| related quantities
x and
y are proportional if their corresponding
values are a constant multiple of each other. |
| Or, by whatever ratio
one quantity changes, the other changes in the same ratio. |
| It is also
called linear relationship. |
|
| Graphical representation of
proportionality
|
| If
x and
y
are proportional quantities, i.e.,
y
:
x =
m
then, |
| their
corresponding values
x1,
y1 and
x2,
y2 are also proportional
alternately |
|
x1
:
x2 =
y1
:
y2,
i.e., x1
is to
x2
as is
y1
to
y2 . |
| Given four
related variable quantities are Cartesian-coordinates of the points
P1
and
P2
of a straight line |
| shown below: Thus, |
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|
| Proportion practice problems,
use of the rule of three
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| Linear relationship of two variable quantities whose ratio is constant
show the proportions |
| x1
:
y1 =
x2
:
y2
or
x1
:
x2 =
y1
:
y2 |
| that we use
to solve practice problems. |
| Two quantities are in direct proportion,
or in linear relationship, when they increase or decrease in the
same |
| ratio. Or, by whatever ratio one quantity changes, the other
changes in the same ratio. |
| If any three terms in a proportion are
given, the fourth may be found using the principles of solving equations. |
|
| Example:
A lift reaches the third floor in seven seconds. When will reach
the 18th floor? |
| Given quantities make the proportion |
| |
x1
:
y1 =
x2
:
y2. |
That
is, |
3
:
7 =
18 :
x |
=> |
 |
seconds. |
|
|
| We could obtain the same result by setting up the corresponding
values of the proportional quantities in a |
| table. |
| The values of the
same quantity put in the first colon and the corresponding values
of another quantity in the |
| second
colon. The arrows show the sequence of the terms of the
proportion. |
 |
x1 |
|
y1 |
|
|
| x2 |
y2 |
or |
x1
:
x2
=
y1
:
y2 |
|
| Thus, for the values from the above example follows: |
| |
|
3rd
floor |
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|
7
sec. |
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| |
18th
floor |
|
x
sec. |
or |
3
:
18 = 7 :
x |
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| |
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|
| => |
|
seconds. |
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|
| Example:
To
the central angle a = 30°, of a circle of radius
r, corresponds
the arc length equal
p/6
r. |
| What arc length corresponds to the
central angle a = 320°? |
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| Inverse proportion
|
| The two related variable quantities
x and
y whose product is
constant i.e., |
| |
x
· y = k
or |
 |
are
inversely proportional. |
|
|
| That is, related quantities are inversely proportional
when an increase by a certain multiple in one, brings |
| about or is
associated with, a decrease by the same factor in the other. |
| By whatever ratio one quantity changes, the other will change in the
inverse ratio. |
| This means that if one of the quantities triples, then
the other will become one third as large. |
|
| Graphical representation of inverse
proportionality
|
| If
x and
y are inversely proportional quantities, i.e.,
x
· y = k then,
the same relationship holds between their |
| corresponding values x1,
y1 and
x2,
y2. |
| Therefore,
x1
· y1 = k
and
x2
· y2 = k
that is,
x1
· y1 =
x2
· y2. |
| Dividing by
x2
· y1 gives the proportion |
| |
 |
or |
x1
:
x2 =
y2
:
y1 |
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| Inverse proportion practice
problems |
| Example:
If 12 workers can do a job in 15 hours, then how many workers
would be needed to do the job in 9
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|
hours? |
| It’s obvious that more workers will do a job in shorter time. That
means, the time it takes to do the job is |
| inversely proportional to the
number of workers. |
|
Therefore, use the proportion or set up the
table where the corresponding values make the colons and the
|
| arrows show the sequence of the terms of the proportion.
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|
x1 |
|
 |
y1 |
|
|
| x2 |
y2 |
or |
x1
:
x2
=
y2
:
y1 |
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| |
|
12
workers |
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|
15
hours |
|
|
| Thus, |
x
workers |
|
9
hours |
or |
12
:
x =
9 :
15 |
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| |
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|
| => |
 |
workers. |
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| Example:
A distance
d
from place A to place B a car covers in 75 minutes
traveling at the rate of 120 km/h. |
| At what rate it should drive so that
it will travel the same distance in 60 minutes. |
| By driving two times
faster, the same distance it will cover in half less time. |
| Since, rate
r and time
t are inversely proportional quantities use the proportion
x1
:
x2 =
y2
:
y1. |
| As the quantities we are comparing must always
be measured in the same units, we’ll convert minutes to |
| hours, |
|
so 75 min =
75
min
· 1hour
/ 60
min =
1.25
h. |
| In the given example the
distance is constant called coefficient of inverse
proportionality. |
| |
|
1.25
h |
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|
120
km/h |
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| Thus, |
1
h |
|
x
km/h |
or |
1.25
:
1
= x :
120 |
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| |
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|
| => |
 |
km/h. |
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| Copyright
© 2004 - 2012, Nabla Ltd. All rights reserved. |
