Ratio And Proportion Concept

Ratio – It is a way of comparing two numbers or quantities and showing the relationship between them.

It is denoted by → ‘:’

Ex: In a class, there are 60 boys and 40 girls. What is the ratio of boys and girls?

Sol.: boys=60 and girls=40

Ratio= boys/girls =60/40=3/2=3:2

3:2 means हर three boys पर 2 girls है | या मैं ऐसा भी बोल सकता हूँ कि 5(i.e 3+2=5) students में 3 boys है or 2 girls है|

→ If boys : girls = 3 : 2 क्या मैं ऐसा भी बोल सकता हूँ कि boys=3 or girls=2 तो मेरा Ans. है नहीं. क्योंकि मैंने common factor को cancel out कर दिया है उसके बाद हमे 3:2 प्राप्त हुआ है |

यदि हमे boys and girls का individual value निकलना हो तो common factor से multiply करना होगा |

that means boys=3×20=60 and girls=2×20=40.here common factor=20 यहाँ पर मुझे common factor मालूम था तो मैंने individual value निकाल लिया |

यदि मुझे common factor मालूम नहीं होता तो मैं boys=3x and girls=2x लिखता|

Ex: In a class, the ratio of boys and girls is 3:2. If there are total 100 students, then how many boys and girls in the class.

Solution:

Let boys=3x and girls=2x

Now,

5x=100

x=20

so,boys=3x=3×20=60and girls=2x=2×20=40

Another way:

In 5 (i.e 3+2=5) students there are 3 boys

so, 1 student, there are 3/5 boys

∴ In  100 student boys are=(3/5)×100=60 boys

In 5 (i.e 3+2=5) students there are 2 girls

so, 1 student, there are 2/5 girls

∴ In  100 student girls are=(2/5)×100=40 girls

OR

girls=100-60=40 girls

Shortcut:

boys=(3/5)×100=60 boys

Girls=(2/5)×100=40 girls

Important Point:

⇒ For a ratio, the two quantities must be in the same unit.

Ex: Ratio of Rs 5 to Rs 30 here unit=Rs. In this example both quantities unit are same.

So, Ratio=5/30=1:6

Ex-2 Ratio of Rs 5 to 30 paise.

Solution: we can’t express in the form of a ratio. because a unit of both quantities is not same.

if u want to express in the form of a ratio, first of all, make the unit of both quantities are same ie. Rs 5 to Rs 0.30 or 500 paise to 30 paise.

Rule:

The multiplication or division of each term of a ratio by the same non-zero number does not effect the ratio.

Ex: 4:5

PROPORTION:

If the ratio of the first and second quantities is equal to the ratio of the third and fourth quantities then it is called proportion.

It is represented by → ‘::’

i.e if a:b=c:d, we write a:b::c:d and we say that a, b, c, d are in proportional

Here a and d are called extremes while b and c are called mean terms.

Ex: check 6, 10, 48, 80 are in proportional

Solution: 6/10=3/5=3:5

and 48/80=3/5=3:5

so, 6, 10, 48, 80 are in proportional.

⇒ Fourth Proportional: If a:b=c:d, then d is called the fourth proportional to a, b, c.

⇒Fourth Proportional(d) =(b×c)/a

⇒ Third Proportional: If a:b=b:c, then c is called the third proportional to a and b.

⇒Third Proportional(c)=b²/a

⇒Mean Proportional between a and b= √ab.

⇒Duplicate ratio of a:b= a²:b² .

⇒Sub-duplicate ratio of a:b= √a:√b .

⇒Triplicate ratio of a:b = a³:b³

A common type of Ratio And Proportion problem that is frequently asked in the various competitive exam.

(Q1)If A:B=2:3 B:C=4:3 Find A:B:C

Solution:

[Concept: A:B and B:C दोनों में B common है so दोनों में जो B का ratio दिया हुआ है उसे equal करेंगें|]

∴ A:B:C=8:12:9 Ans.

2nd Method:

∴ A:B:C=8:12:9 Ans.

(Q2) A:B=2:1 and A:C=1:3 then A:B:C is

Solution:

[Concept: A:B and A:C दोनों में A common है so दोनों में जो A का ratio दिया हुआ है उसे equal करेंगें|]

∴ A:B:C=2:1:6 Ans.

(Q3)If A:B=5:2 B:C=2:3 C:D=5:3 find ratio of A:B:C:D=?

Solution:

[Concept: A:B:C and C:D दोनों में C common है so दोनों में जो C का ratio दिया हुआ है उसे equal करने के लिए 5 से multiply किया गया है|]

(Q4)If A:B=4:9 and A:C=2:3 then (A+B):(A+C) is.

Solution:

[Concept: A:B and A:C दोनों में A common है so दोनों में जो A का ratio दिया हुआ है उसे equal करेंगें|]

∴ A:B:C=4:9:6

Now,

(A+B):(A+C)=(4+9):(4+6)=13:10 Ans.

(Q5)If 2A=3B=4C, then A:B:C is.

Solution:

L.C.M of 2,3,4=12

Now,

2A/12 =3B/12 = 4C/12

A/6 = B/4 = C/3

A:B:C=6:4:3 Ans.

(Q6)If A=(1/4)B and B=(½)C then A:B:C is

Solution:

A/B=1/4 and B/C=1/2

A:B=1:4 and B:C=1:2

[Concept: A:B and B:C दोनों में B common है so दोनों में जो B का ratio दिया हुआ है उसे equal करेंगें|]

So, A:B:C=1:4:8 Ans.

(Q7)If A:B=2:3 and B:C=3:7 then (A+B):(B+C):(C+A) is.

Solution:

[Concept: A:B=2:3 B:C=3:7 दोनों में B common है so दोनों में जो B का ratio दिया हुआ है उसे equal करेंगें| but in this question ratio of B in both cases are equal.]

∴ A:B:C=2:3:7

So, (A+B):(B+C):(C+A)=5:10:9 Ans.

(Q8)If x:y=4:5 then (3x+4y):(5x+3y)=?

Solution:

x:y=4:5

x/y=4/5

Put the value of x/y then

(3x+4y):(5x+3y)=17:35 Ans.

(Q9)If a:b:c=2:3:4 and 2a-3b+4c=33, then the value of c is.

Solution:

a:b:c=2:3:4

∴ a/2 = b/3 = c/4 = K(let)

⇒ a=2k, b=3k, and c=4k

Given that 2a-3b+4c=33

=>2×2k-3×3k+4×4k=33

so, k=3

∴ c=4k=4×3=12 Ans.

(Q10)The fourth proportional to 4, 9, 12 is.

Solution:

Let the fourth proportional to 4, 9, 12 be x.

then 4:9::12:x

⇒4×x=9×12   (i.e a×d=b×c)

∴ x=27 Ans.

(Q11)The third proportional to 16 and 36 is.

Solution:

Let the third proportional to 16 and 36 be x

then 16:36:36:x

⇒ 16×x=36×36

∴ x=81 Ans.

(Q12)The mean proportional between 0.08 and 0.18 is.

Solution:

The mean proportional between 0.08 and 0.18=