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Ratio & Proportion

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RATIO1.1 Compound Ratio

1.2 Inverse Ratio

1.3 Comparison of Ratio

PROPORTION2.1 Forth Proportional

2.2 Third Proportional

2.3 Mean Proportional

2.4 Type of Proportional A.) Direct Proportion B.) Inverse Proportion

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The most common examples involve two quantities, but any number of quantities can be compared.0

A ratio is an expression that compares quantities

relative to each other. Can be written as x to y, x : y, or x/y.

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 fraction a/b is usually different from the fraction b/a, So the order of the terms in a ratio is important.

— ≠ —baab

The ratio 5 : 9 represents 5/9, here antecedent = 5,Consequent=9.

TestBagTestBag If the ratio between the first and the second

quantitiesis 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.

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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.

The ratio among the three numbers is Solution :2 : 3

5 : 8

10 : 15 : 24

∴ The second number = 30 = —————— ×1510 + 15 + 24

98

TestBagTestBag 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

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1st : 2nd =

2nd : 3rd =

3rd : 4th = e

c

a

d

b

f1st : 2nd : 3rd : 4th = ace : bce : bde : bdf

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Example : If A : B = 3 : 4, B : C = 8 : 10 and C : D = 15 : 17then find A : B : C : D.

Solution: A : B = 3 : 4

B : C = 8 : 10

C : D = 15 : 17

= 3×8×15 : 4×8×15 : 4×10×15 : 4×10×17

A : B : C : D

= 9 : 12 : 15 : 17

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Ratios are compounded by multiplying together the antecedents for a new antecedent, and the consequents for a new consequent.

1.1 COMPOUND RATIO

The ratio compounded of the four ratios i.e. 2 : 3, 5 : 11, 18 : 7 and 21 : 4 is

= 45 : 11 = ——————— = ——3 × 11 × 7 × 4 112 × 5 × 18 × 21 45

Example :

Solution :

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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.Or, 1 : 9 —— = —

72 8

91

The ratio

Example :

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.

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1.2 INVERSE RATIOIf a : b be the given ratio, then 1/a : 1/b or b : ais 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.

Solution : — = —, — = —, — = — B 4 C 9 D 16A 3 B 8 C 15

— = — × — × — = — × — × — = — D B C D 4 9 16 8A A B C 3 8 15 5∴

∴ A : D = 5 : 8

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1.3 COMPARISON OF RATIOS

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).

We say that (a : b) > (c : d) — > —a cb d

If — = —, them ——— = ——— (Componendo and Dividendo)

a cb d

a + ba - b

c + dc - 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 leftand right parts are known as extremes and the middle partsn 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.

or, — = — or m : n : : p : qm pn q

∴ m × q = n × pOr m × q = Product of extremes And n × p = Product of interims (means)

Product of extremes = Product of interims (means)

TestBagTestBagPROPORTION

Means the equality of 2 ratios.

In symbols, it is a/b = c/dor

a : b = c : d1st term 2nd

term

4th term3rd term

Note : b and d must not be zero

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TestBagTestBagExample:

Find x,

x : 5 = 15 : 25

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The productof the means equalsthe product of the extremes.

OR

25x = 5(15)

25x = 75

x = 3

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Identify the MEANS and the EXTREMES and look for the missing term (x).

1. 3 : x = 9 : 21

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2. (x + 2) : 8 = (3x – 7) : 16

3. x90 - x

28

=9x = 63

x = 7

8(3x-7) = 16(x+2)

24x-56 = 16x+32

24x-16x = 32+56

8x = 88

x = 11

2(90-x) = 8x

180–2x = 8x

-2x- 8x = -180

-10x = -180

x = 18

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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

∴ x = ——— = 208 × 156

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2.2 THIRD PROPORTIONAL

If a : b = c : d, then c is called the third proportionalTo 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 20If x be the fourth proportional, we have 15 : 20 = 20 : x

∴ x = ——— = — = 26— 20 × 2015 3

23

80

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2.3 MEAN PROPORTIONAL

Example :

Find the mean proportional between 3 and 75.

Solution :If x be the required mean proportional, we have

Mean proportional between a and b is √ab . —

∴ x = √3 x 75 = 15———

3 : x : : x : 75

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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 relationshipof the form a = kb, where k is a number..

2.4 TYPES OF PROPORTIONS A.) Direct Proportions

Example :If 5 balls cost Rs 8, what do 15 balls cost ?

Solution :5 balls : 15 balls : : Rs 8 : required cost

∴ The required cost = Rs ——— 15 x 8

5= Rs 24

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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 theyare said to be inversely proportional to each other. Twovariables a and b are indirectly proportional if they satisfya relationship of the form k = ab, where k is a number.

B.) Inverse Proportion

Example :If 15 men can reap a field in 28 days, in how manyDays will 10 men reap it?

Solution :

∴ The required number of day = ———— 15 x 28

10

∴ — : — : : 28 ─ The required number of days 1 1 15 10

Or, 10 : 15 : : 28 ─ The required number of days

= 42

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Thanking you and

Good bye…