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Ratio & Proportion
By:
TestBag Faculty
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RATIO
PROPORTION
1.1
Compound Ratio
2.1
Forth Proportional
1.2
Inverse Ratio
2.2 Third Proportional
1.3 Comparison of Ratio
2.3 Mean Proportional
2.4 Type of Proportional
A.) Direct Proportion
B.) Inverse Proportion
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A ratio is an expression that compares quantities
relative to each other.
Can be written as x to y, x : y, or x/y.
The most common examples involve two quantities,
but any number of quantities can be compared.0
Ratios are represented mathematically by separating
each quantity with a colon -
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The ratio of a to b is written as a : b or a/b = a÷b.
Since a : b is a fraction, b can never be zero.
The two quantities must be of the same kind.
In a : b, a is antecedent and b is consequent.
Example:
The ratio 5 : 9 represents 5/9, here
antecedent = 5,
Consequent=9.
The fraction a/b is usually different from the fraction b/a,
So the order of the terms in a ratio is important.
a
b
— ≠ —
a
b
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
If the ratio between the first and the second quantitie
is a : b and the ratio between the second and third
quantities is c : d, then the ratio among first, second
and third quantities is given by ac : bc : bd.
Example:
The sum of three numbers is 98. If the ratio between the first
and second be 2 : 3 and that between the second and third be 5 :
8, then find the second number.
Solution :
The ratio among the three numbers is
2:3
5:8
10 : 15 : 24
98
∴ The second number = ——————
×15 = 30
10 + 15 + 24
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
If the ratio between then first and the second
quantities is a : b the ratio between the second and the
third quantities is c : d and the ratio between the third and
the fourth quantities is e : f then the ratio among the first,
second, third and fourth quantities is given by
1st
:
2nd
= a
2nd : 3rd =
3rd : 4th =
b
c
d
e
1st : 2nd : 3rd : 4th = ace : bce : bde : bdf
f
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Example :
Solution:
If A : B = 3 : 4, B : C = 8 : 10 and C : D = 15 : 17
then find A : B : C : D.
A:B
=
B:C
3:4
=
C:D
A:B:C:D
8 : 10
=
15 : 17
= 3×8×15 : 4×8×15 : 4×10×15 : 4×10×17
= 9 : 12 : 15 : 17
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1.1 COMPOUND RATIO
Ratios are compounded by multiplying together the
antecedents for a new antecedent, and the consequents
for a new consequent.
Example :
The ratio compounded of the four ratios i.e. 2 : 3, 5 :
11, 18 : 7 and 21 : 4 is
Solution :
2 × 5 × 18 × 21 = ——
45 = 45 : 11
= ———————
3 × 11 × 7 × 4
11
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A ratio, in the simplest form, is when both terms are
integers, and when these integers are prime to one another.
We may multiply or divide both terms of a ratio by the same
number without affecting the value of the ratio. However,
addition or subtraction of same numbers will affect the value
of the ratio.
To express the ratio of two quantities, they must be
expressed in the same units.
Example :
To find the ratio of 8 inches and 6 feet . First, we
change 6 feet into inches, and then will find the ratio.
1 foot = 12 inches, so 6 feet = 12 x 6 = 72 inches.
8
1
The ratio —— = —
9 Or, 1 : 9
72
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1.2
INVERSE RATIO
If a : b be the given ratio, then 1/a : 1/b or b : a
is called its inverse or reciprocal ratio.
Example :
A, B, C and D are four quantities of the same kind such
That A : B = 3 : 4, B : C = 8 : 9, C : D = 15 : 16. Find the
Ratio A : D.
3
B = —,
8
C = 15
Solution : A
— = —,
—
—
—
B
4
C
9
∴
A
— =
D
A B C
—×—×— =
B C D
∴
A : D = 5 : 8
D
16
3 8 15
5
—×—×— = —
4 9 16
8
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1.3 COMPARISON OF RATIOS
a
c
We say that (a : b) > (c : d)  — > —
b
d
Some other Ratios are :-

Duplicate Ratio of (a : b) is (a2 : b2)

Sub-duplicate Ratio of (a : b) is (√a : √b).

Triplicate Ratio of (a : b) is (a3 : b3).

Sub-triplicate Ratio of (a : b) is (a1/3 : b1/3).

If — = —, them ——— = ——— (Componendo and Dividendo)
a
b
c
d
a+b
a-b
c+d
c-d
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A proportion is a statement where two ratios are equal.
If, m/n = p/q, it means m/n is in proportion with p/q and
can be written as m : n :: p : q, where m and q extreme left
and right parts are known as extremes and the middle parts
n and p are known as means or interim.
If four quantities be in proportion, the product of the
extremes is equal to the product of the means or interims.
m
p
or, — = — or m : n : : p : q
n
q
∴ m×q = n×p
Or m × q = Product of extremes And n × p = Product of interims (means)
Product of extremes = Product of interims (means)
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PROPORTION
Means the equality of 2 ratios.
In symbols, it is a/b = c/d
or
a:b = c:d
1st term
2nd term 3rd term
4th term
Note : b and d must not be zero
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Example:
Find x,
x : 5 = 15 : 25
25x = 5(15)
25x = 75
x = 3
The product
of the means equals
the product of the
extremes.
OR
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Identify the MEANS and the EXTREMES and look
for the missing term (x).
1. 3 : x = 9 : 21
9x = 63
x=7
2. (x + 2) : 8 = (3x – 7) : 16
8(3x-7) = 16(x+2)
24x-56 = 16x+32
24x-16x = 32+56
3.
x
= 2
8
90 - x
2(90-x) = 8x
180–2x = 8x
-2x- 8x = -180
-10x = -180
x = 18
8x = 88
x = 11
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2.1 FOURTH PROPORTIONAL
If a : b = c : d, then d is called the fourth proportional
to a, b, c.
Example :
Find the fourth proportional to the numbers 6, 8 and 15.
Solution :
If x be the fourth proportional, then 6 : 8 = 15 : x
∴
8 × 15 = 20
x = ———
6
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2.2
THIRD PROPORTIONAL
If a : b = c : d, then c is called the third proportional
To a and b.
Example :
Find the third proportional to 15 and 20.
Solution :
Here, we have to find a fourth proportional to 15, 20 and 20
If x be the fourth proportional, we have 15 : 20 = 20 : x
∴
2
× 20 = 80
x = 20
———
— = 26—
3
15
3
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2.3
MEAN PROPORTIONAL
Mean proportional between a and b is
—
√ab .
Example :
Find the mean proportional between 3 and 75.
Solution :
If x be the required mean proportional, we have
3 : x : : x : 75
———
∴ x = √3 x 75 = 15
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2.4 TYPES OF PROPORTIONS
A.) Direct Proportions
When any two quantities are
related to each and if one increase or decreases, the other
also increases or decreases to same extent, then they are
said to be directly proportional to each other. Two variables
a and b are directly proportional if they satisfy a relationship
of the form a = kb, where k is a number..
Example :
Solution :
If 5 balls cost Rs 8, what do 15 balls cost ?
5 balls : 15 balls : : Rs 8 : required cost
15 x 8
∴ The required cost = Rs ——— = Rs 24
5
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B.) Inverse Proportion
When the two quantities are
related to each other in a such way that if one increase, the
other decreases to the some extent and vice-versa, then they
are said to be inversely proportional to each other. Two
variables a and b are indirectly proportional if they satisfy
a relationship of the form k = ab, where k is a number.
Example :
If 15 men can reap a field in 28 days, in how many
Days will 10 men reap it?
Solution :
1 :—
1 : : 28
∴ —
15 10
Or, 10 : 15 : : 28
─ The required number of days
─ The required number of days
15 x 28
∴ The required number of day = ———— = 42
10
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Thanking you
and
Good bye…
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