an amount of money is distributed among a, b and c in the ratio 6:19:7. if c gives rs.200 from his...
TRANSCRIPT
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mIntroduction:
The Ratio could be perceived as a special case of percentage. All the
concept related to approaching the of the Ratio questions could be
easily correlated with the approach what we had already developed in
the percentage Digi-notes (Specially the application of the Unitary
Method). Like percentage; ratio is extensively used in the Data Analysis.
So you are advised to go through the chapter thoroughly keeping in
mind to learn how we are approaching the question.
What is Ratio?:
RATIO & PROPORTION
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Mathematically: Let a and b be the two quantities.
Then ratio of a and b is defined as the fractionb
a .
This fraction is simply denoted as a : b when we are talkingabout the ratio.
And is is read as “a is to b”.
Practically: Since with the help of the fractions we are making
comparison between the quantities, therefore ratio is thecomparison between two quantities.
The fundamental Property of Ratio: As the multiplication or division by the same number to the
numerator and denominator doesn’t affect a faction; in the sameway the multiplication or division of each term of ratio by thesame non-zero number doesn’t affect the ratio.
An Important Result:
It is to be noted that the individual ratios to be combined:
Should have one quantity in common.
And the common quantity of the Ist ratio should be at thesecond place and that of the IInd ratio at the first place.
st
nd
I Ratio A : B a : b
II Ratio B : C c : d
Then combined Ratio A : B : C ac : bc : bd
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mQue: If A, B, C are three quantities such that A:B = 5:4 and B:C = 3:2,
find:
1) B : C 2) A : C 3) (A + B) : (A - C)
Sol: We have 𝐴 ∶ 𝐵 ∶ 𝐶 = 5 × 3 ∶ 3 × 4 ∶ 4 × 2 = 15 ∶ 12 ∶ 8
Hence
1) B ∶ C = 12 ∶ 8 = 3 ∶ 2
2) A ∶ C = 15 ∶ 8
3) (A + B): (A − C) = (15 + 12): (15 − 8) = 27 ∶ 7
Que: The sum of the money with A and B is Rs.500. If the ratio of the
money they had is 3:2, find the share of A.
Sol:
𝐴 =𝑅𝑠.500
(3+2)× 3 = 𝑅𝑠. 300
IMPORTANT QUESTIONS
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Que: The ratio of the money with A and B is 7:3. If the difference of the
money they had is Rs.160, find the share of B.
Sol:
𝐵 =𝑅𝑠. 160
(7 − 3)× 3 = 𝑅𝑠. 120
Que: A sum of money is divided between two persons in the ratio of
3:5. If the share of one person is Rs.20 less than that of the other, find
the sum.
Sol:
𝐴 =𝑅𝑠. 20
(5 − 3)× (5 + 3) = 𝑅𝑠. 80
Que: The ratio of the shares of three persons A, B and C is 7:5:11. If the
difference of the shares of A and C is Rs.16000; find the total amount of
money with B and C.
Sol:
𝐴 =𝑅𝑠. 16000
(11 − 7)× (7 + 11) = 𝑅𝑠. 72000
Que: The ratio of the shares of three persons A, B and C is 12:9:11. The
total amount of the money with A and C is Rs.5600 more than the
money with B; find the difference of the amount of the money with B
and C.
Sol:
𝐴 =𝑅𝑠.5600
(12+11−9)× (11 − 9) = 𝑅𝑠. 800
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Que: The ratio of the incomes of four persons A, B, C and D is 5:3:9:4.
The sum of the incomes of A and C is 84000; find the difference of the
incomes of B and D?
Sol:
𝐴 =𝑅𝑠. 84000
(5 + 9)× (4 − 3) = 𝑅𝑠. 6000
Que: The ratio of the angles of a triangle is 10:11:15. Find the difference
of the second largest and the smallest angle of the triangle.
Sol:
Required Differenec =1800
(10 + 11 + 15)× (11 − 10) = 50
Que: The ratio of any two angles of a triangle is 5:9. If the third angle is
measured to be 110 degree, then find the difference of the other two
angles.
Sol:
Required Differenec =1800−1100
(9+5)× (9 − 5) = 200
Que: The ratio of the four angles of a quadrilateral is 9:10:14:3. Find the
sum of the second smallest and second largest angle of the
quadrilateral.
Sol:
Required Differenec =3600
(9+10+14+3)× (10 − 9) = 100
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Ques: P, Q and R are three numbers. The ratio of P and Q is 2:3 and that
of Q and R is 2:1. If the sum of P and R is 980; find Q.
Sol:
We have, 𝑃: 𝑄: 𝑅 = 4 ∶ 6 ∶ 3
Hence; 𝑄 =980
(4+3)× 6 = 840
Que: The salary of Amit is 3/4th of the salary of Raju; while the salary of
Raju is 25% less than the salary of Sohan. The difference of the salaries
of Amit and Sohan is Rs.9800; find 142
7% of the sum of the salaries of
Raju and Sohan.
Sol:
A : R = 3 : 4
R : S = 75% : 100% = 3 : 4
Hence A : R : S = 9 : 12 : 16
Required Sum = {9800
16−9× (12 + 16)} ×
1
7= 𝑅𝑠. 5600
Que: Rs.5600 is divided among three persons A, B and C in such a way
that; A received 2/5th of the total share of B and C together. Find:
1) Share of A
2) Share of B
3) Share of C
4) Total share of A and B
5) Total share of B and C
6) Total share of C and A
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Sol:
: 2 :5
:
56001 2 .1600
7
2
3
4
56005 5 .4000
7
6
A B C
Hence
A Rs
CND
CND
CND
B C Rs
CND
Que: Rs.15600 is divided among three persons A, B and C in such a way
that; A received 2/3rd of the total share of B and C together while B
received 3/7th of the total share of A and C together. Find the share of
C.
Sol:
: 2:3
: 3:7
:
156002 .6240
515600
3 .468010
.15600 .6240 .4680 .4680
A B C
and B A C
Hence
A Rs
and B Rs
Hence
C Rs Rs Rs Rs
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Que: Divide Rs.1540 among A, B, C such that A shall receive 2/9 as
much as B and C together and B shall receive 3/11 as much as A and C
together.
Sol:
: 2 : 9
: 3 :11
:
15402 .280
11
15600 3 .330
14
.1540 .280 .330 .930
A B C
and B A C
Hence
A Rs
and B Rs
Hence
C Rs Rs Rs Rs
Que: Rs.2575 is divided among three persons A, B and C in such a way
that if each of their shares be reduced by Rs.25; the ratio of their
remainders would be 2 : 3 : 5. Find the sum of the shares of B and C.
Sol:
2575 25 25 258 25 25 .2050
10B C Rs
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Que: The ratio between two numbers 3:4. If each number be increased
by 6, the ratio becomes 4:5. Find the two numbers.
Sol:
1 2
1
2
:
3 : 4
4 : 5
1 6
3 3 6 18
4 4 6 24
N N
N
N
Que: The ratio between two numbers 5:1. If each number be decreased
by 6, the ratio becomes 7:1. Find the difference between the numbers.
Sol:
1 2
1 2
5 1 30 6
7 1 28 4
2 6
1 3
24 24 3 72
N : N
: :
: :
N ~ N
Que: The ratio between two numbers 3:4. If each number be increased
by 2, the ratio becomes 7:9. Find the sum of the two numbers.
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Sol:
1 2
1 2
3 4 6 8
7 9 7 9
1 2
14 14 2 28
N : N
: :
: :
N N
Que: The incomes of A and B are in the ratio 3:2 and their expenditures
are in the ratio 5:3. If each saves Rs.200, what is the difference of their
incomes?
Sol:
3 2 6 4
5 3 5 3
1 200
2 200 2 400
A : B
I : :
E : :
Rs.
A ~ B Rs.
Que: The incomes of Ram and Shyam are in the ratio 8:11 and their
expenditures are in the ratio 7:10. If each of them saves Rs.500, then
find the sum of their Income.
Sol:
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8 11
7 10
1 500
19 19 500 .9500
R : S
I :
E :
Rs.
R S Rs
Que: The students in the three classes are in the ratio 2:3:5. If 20
students are increased in each class the ratio changes to 4:5:7. What
was the total number of students in the three classes before the
increase?
Sol:
:
2 : 3 : 5
4 : 5 : 7
2 20
2010 10
2100
st nd rd
st nd rd
I : II III
I II III
Que: An amount of money is distributed among A, B and C in the ratio
1:2:9. If C gives Rs.500 from his share to B, the new ratio becomes
1:3:8. Find the total amount of money.
Sol:
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1 2 9
1 3 8
1 500
12 500 12
6000
A : B : C
: :
: :
Rs.
A B C
Rs.
Que: An amount of money is distributed among A, B and C in the ratio
6:19:7. If C gives Rs.200 from his share to B, the new ratio becomes
3:10:3. Find the total amount of money.
Sol:
:
6 : 19 : 7 6:19:7
3 : 10 : 3 6:20:6
1 .200
32 200 32
.6400
A : B C
Rs
A B C
Rs
Que: In 40 litres mixture of milk and water the ratio of milk and water is
3:1. Find the quantity of water to be added to make this ratio 2:1.
Sol:
3 1 6:2
2 1 6:3
40 1 5
8
M : W
:
:
litersliters
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Que: In 60 litres mixture of milk and water the ratio of milk to water is
7:3. Find the quantity of water to be added to make the ratio of milk to
water 3:7.
Sol:
7 3 21:9
3 7 21:49
60 40 80
30
M : W
:
:
lietrsliters
Que: A mixture contains alcohol and water in the ratio 8:3. On adding 3
litres of water, the ratio of alcohol to water becomes 2:1. Find the
difference of quantities of alcohol and water in the original mixture.
Sol:
8 3 8:3
2 1 8:4
3 5 15
1
A : W
:
:
litersliters
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Que: A mixture contains wine and water in the ratio of 3:2.If 4 litres of
water is added to the mixture, wine and water in the mixture become
equal. Find the sum of the quantities of wine and water in the original
mixture.
Sol:
3 2 3:2
1 1 3:3
4 5 20
1
Wn : W
:
:
litersliters
Que: The ratio of number of boys and girls in a school is 2:3. If 200 boys
are additionally admitted in the school the ratio changes to 5:6. Find
the difference of the number of boys and girls in the school originally.
Sol:
2 3 4:6
5 6 5:6
2002 400
1
B : G
:
:
Boys
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Que: The ratio of A’s and B’s income last year was 3:4. The ratio of their
own incomes of last year and this year is 4:5 and 2:3 respectively. If the
total sum of their present incomes is Rs.4160, then find the present
income of A.
Sol:
1 year
[------------------------]
Past Present
3 4
3 34 : 5
4 44 4
2 : 32
A: B :
A
B
2
3 45 3 5 8
4 24160
5 160013
P P
P
A : B : :
A Rs.
Que: One year ago, the ratio between Laxman’s and Gopal’s salaries
was 3:5. The ratio of their individual salaries of last year and present
year are 2:3 and 4:5 respectively. If their total salaries for the present
year are Rs.4300, find the difference of the salaries of Laxman and
Gopal.
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Sol:
1 year
[--------------------------]
Pr
3 5
3 32 : 3
2 25
4 : 4
Past esent
L :G :
L
G
5
54
3 53 5 18 25
2 44300
7 70043
P P
P
A : B : :
A Rs.
Que: A vessel contains liquids A and B in the ratio 5:3. If 16 litres of
mixture are removed and the same quantity of liquid B is added, the
ratio becomes 3:5. What quantity does the vessel hold?
Sol:
3 5 165
8 8 8
210
8
8
1 102
40
liters
liters
Qantity of
Mixture in
Vessel
liters
Question: A bucket contains a mixture of two liquids A and B in the
ratio 7:5. If 9 litres of the mixture is replaced by 9 litres of liquid B, then
the ratio of the two liquids become 7:9. How much of the liquid A was
there in the mixture?
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Sol:
7 7 97
16 12 12
7 21
48 4
21 48
14 7
36
367 21
12
liters
liters
Qantity of
Mixture in
Vessel
liters
A liters
Proportion:
Let a , b, c and d are the four quantities.
If we calculated the ratios a : b and c : d in that order and find that
dcba :: Then we say that a, b, c and d are in proportion in that order.
Hence equality of ratios is called proportion. Here
a is called the first proportional.
b is called the second proportional.
c is called the third proportional.
d is called the fourth proportional.
Since we have
Means
of
Extremes
of
bcad
d
c
b
a
c:da:b
ProductProduct
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Continued Proportion:
Let a , b and c are the three quantities.
If we calculated a : b and b : c in that order and find that
cbba ::
Then we say that a, b and c are in continued proportion in that
order. That’s why it is called continued proportion because b is in
continuation in the equality of the ratios. Here
a is called the first proportional.
b is called the mean/second proportional.
c is called the third proportional.
Since we have
Mean
Square of
Extremes
of
bad
c
b
b
a
b:ca:b
Product
2
Que: Given three quantities in the same unit as 5, 10 and 25. Find the
following in proportion to these quantities:
1) First Proportional
210525
25 ,10 ,5 ,
xx
proportioninarex
2) Second Proportional
5.1225510
25 ,10 , ,5
xx
proportioninarex
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3) Third Proportional
51225510
25105
.xx
oportion are in pr, x, ,
4) Fourth Proportional
5025105
,25 ,10 ,5
xx
proportioninarex
Question: Given two quantities in the same unit as 5 and 25. Find the
following in proportion tom these quantities:
5) First Proportional
1525 2 xx
6) Mean Proportional
55255 2 xx
7) Third Proportional
125255 2 xx
Question: Divide Rs.1350 into three shares proportional to the numbers
2, 3, and 4.
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Sol:
600.41503
450.31502
300.21501
1509
13501
1350.9
Rs
Rs
Rs
Rs
rd
nd
st
Que: Divide Rs.391 into three parts proportional to the fraction ½, 2/3
and ¾.
Sol:
153.9173
136.8172
102.6171
1723
3911
391.23
9:8:64
3:
3
2:
2
1
Rs
Rs
Rs
Rs
rd
nd
st
Question: Divide 581 into three parts such that 4 times of the first may
be equal to 5 times the second and 7 times the third.
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Sol:
1407203
1967282
2457351
783
5811
58183
20:35:283:1:2
4:73:1
5:41:2
rd
nd
st
rdstnd
rdst
stnd
Que: Divide Rs. 2430 among three persons A, B, C such that if their
shares be diminished by Rs.5, Rs.10 and Rs.15 respectively, the
remainders shall be in the ratio 3:4:5.
Sol:
1015.152005
810.102004
605.52003
200.12
24001
2400.12
2400.)15105(2430Re
RsC
RsB
RsA
Rs
Rs
Rsmainder
Que: Rs.425 is divided among 4 men, 5 women and 6 boys such that the
share of a man, a woman and a boy may be in the ratio of 9:8:4. What
is the share of a woman?
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Sol:
34.5
1701
170.1025
425105
25
4251
425.25
6:10:9
64:58:496:5:4
RsW
RsW
Rs
BWM
Que: Divide Rs.1320 among 7 men, 11 women and 5 boys such that
each womanmay have 3 times as much as a boy, and a man as much as
a woman and a boy together. Find how much each person receive?
Sol:
: : 1: 3 : (1 3) 1: 3 : 4
5 :11 : 7 1 5 : 3 11: 4 7
5 : 33: 28
66 .1320
13201 20
66
7 28 20 28 .560
11 33 20 33 .660
5 5 20 5 .100
B W M
B W M
Rs
M Rs
W Rs
B Rs
Que: A certain sum of money is divided A, B and C such that for each
rupee A has, B has 65 paisa and C has 40 paisa. If C’s share is Rs.8, find
the sum of money.
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Sol:
41.418
841
8
81
8.8
8:13:2040:65:100::
RsMoneyofSum
Rs
CBA
Ques: 465 coins consist of rupee, 50 paisa and 25 paisa coins. Their
values are in the ratio 5:3:1. Find the number of each coin.
Sol:
5 : 3:1
100 100 100. 5 : 3 :1
100 50 25
5 : 6 : 4
15 465
4651 31
15
. 1 5 31 5 155
50 6 31 6 186
25 4 31 4
Values
No of Coins
Rs Coins
Paise Coins
Paise Coins
124
Que: A sum of Rs.11.70 consists of rupees, 50 paisa and 5 paisa coins in
the ratio 3:5:7. Find the number of each kind of coins.
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Sol:
. 5 : 3 :1
100 50 53 : 5 : 7
100 100 100
60 : 50 : 7
117 1170
11701 10
117
60 10 60 600
. 1
600. 1 6
100
50
No of Coins
Values
Paise
Values ofPaise
Rs Coins
Rs Coins
Values of
Pa
50 10 50 500
50050 10
50
7 10 7 70
5
705 14
5
Paiseise Coins
Paise Coins
Values ofPaise
Paise Coins
Paise Coins
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NOTES