Chapter 5.2

Ratio & Proportion

of the mixture are removed and the same quantity of liquid A is added, the ratio becomes 4:5 . What quantity does the vessel hold? a) 72 litres b) 90 litres c) 64 litres d) 80 litres

3. A vessel contains liquids A and B in ratio 3 :2. I f 15 litres of the mixture are removed and the same quantity of liquid B is added, the ratio becomes 2:3 . What quantity does the vessel hold? a) 30 litres b) 35 litres c) 40 litres d) 45 litres

4. A vessel contains liquids A and B in ratio 3 : 1. I f 8 litres of the mixture are removed and the same quantity of liquid B is added, the ratio becomes 1:3. What quantity does the vessel hold? a) 12 litres b) 14 litres c) 16 litres d) 10 litres

5. A vessel contains liquids A and B in ratio 7:6. I f 26 litres of the mixture are removed and the same quantity of liquid B is added, the ratio becomes 6:7. What quantity does the vessel hold? a) 142 litres b) 172 litres c) 156 litres d) 182 litres

Answers 1. c 2. a; Hint: Seethe 'note'.

The removed quantity contains

A and 18 -10 = 8 litres of B. Now,

18 5 + 4

x5 = 10litres of

(5 + 4) 2 Total quantity = 2 . x quantity of B in the removed

mixture = 72 litres. 3.d 4. a 5.d

Rule 38 Ex.: An employer reduces the number of his employees in

the ratio 9 : 8 and increases their wages in the ratio 14 : 15. State whether his bil l of total wages increases or decreases, and in what ratio.

Soln: 9: 8 14:15 We know that the total bill = wage per person * no. of total employees. Therefore, the ratio of change in bill

= 9x 14:8x 15 = 126:120=21:20 The ratio shows that there is a decrease in the bill.

Note: For a detailed method let the no. of employees in two cases = 9x & 8x. Wages in two cases be 14y & 15 y Initial wage = 9x * 14y = I26xy Changed wage = 8x * 15y = 120xy This shows the decrease in bill and ratio is 126xy : 120xy=21:20.

Exercise 1. An employer reduces the number of his employees in

the ratio 7 : 6 and increases their wages in i 14. State whether his bill of total wages decreases, and in what ratio? a) Increase, 13:12 b) Decrease, 13 :12 c) Increase, 14: 13 d) Decrease, 13:14

2. An employer reduces the number of his employees in the ratio 8 : 7 and increases their wages in the ratio " 1 State whether his bill of total wages increases or de-creases, and in what ratio? a) Decrease, 4 : 1 b) Increase, 1 :4 c) Decrease, 5:2 d) Increase, 2 : 5

3. An employer reduces the number of his employees in the ratio 7: 5 and increases their wages in the ratio 10 : 9. State whether his bill of total wages increases or de-creases, and in what ratio? a) Decrease, 13:9 b) Decrease 14:9 c) Increase 9:14 d) Increase 9:13

4. An employer reduces the number of his employees in the ratio 8 :3 and increases their wages in the ratio 3 : 8. State whether his bill of total wages increases or de-creases, and in what ratio? a) Remains unchanged, 1:1 b) Decrease, 3 : 1 c) Decrease, 2 : 1 d) Can't be determined

5. An employer reduces the number of his employees in the ratio 9 :4 and increases their wages in the ratio 2 :5 . State whether his bill of total wages increases or de-creases, and in what ratio? a) Decrease, 10:9 b) Increase, 10:9 c) Decrease 9:11 d) Increase, 9:10

Answers l .b 2. a 3.b 4. a 5.d

Rule 39 Theorem: Two candles of the same height are lighted at the

same time. The first is consumed in Tx hours and the sec-

ond in T2 hours. Assuming that each candle burns at a

constant rate, the time after which the ratio of first candle

to second candle becomes x :y is given by

hours.

X

X T,-T2

Illustrative Example Ex.: Two candles of the same height are lighted at the

same time. The first is consumed in 4 hours and the second in 3 hours. Assuming that each candle burns at a constant rate, in how many hours after being lighted, was the first candle twice the height of the second?

118 PRACTICE BOOK ON QUICKER MATHS

Soln: Detail Method: Let the height of candles be h and candle be A and B.

The candle A burns of its height in 1 hour while

1 candle B burns of its height in one hour.

As per question, the height of candle A after x hours be double of height of candle B.

xh Ah xh Height of candle A after x hours = h =

4 4

, xh 3h- xh Height of candle B after x hours = - = -

As per given question,

Ah-xh _ i}h-xh) 4 3 .

or, 12-3* =24-8x or, 5x= 12

12 or, x - hours or 2 hours 24 minutes.

.'. The required answer is 2 hours 24 minutes. Quicker Method: Applying the above theorem, we have,

x2

4x3 (2

the required answer = 12

- x 4 - 3 hours

1

= 2 hours 24 minutes. Note: Here, x : y = 2 :1

Exercise 1. Two candles of the same height are lighted at the same

time. The first is consumed in 8 hours and the second in 6 hours. Assuming that each candle burns at a constant rate, in how many hours after being lighted, the ratio between the first and second candles becomes 2 : 1 . a) 2 hours 24 minutes b) 4 hours c) 1 hour 12 minutes d) 4 hours 48 minutes

2. Two candles of the same height are lighted at the same time. The first is consumed in 5 hours and the second in 4 hours. Assuming that each candle burns at a constant rate, in how many hours after being lighted, the ratio between the first and second candles becomes 3 :2 . a) 3 hours b) 3 hours 45 minutes

20 c) hours d) Can't be determined

3. Two candles of the same height are lighted at the same time. The first is consumed in 7 hours and the second in

6 hours. Assuming that each candle burns at a constant rate, in how many hours after being lighted, the ratio between the first and second candles becomes 3 : 1 . a) 5 hours 36 minutes b) 5 hours c) 5 hours 60 minutes d) 6 hours

4. Two candles of the same height are lighted at the same time. The first is consumed in 3 hours and the second in 1 hour. Assuming that each candle burns at a constant

rate, m flow many hours after being lighted, the ratio between the first and second candles become 2 : 1 . a) 48 minutes b) 1 hour 36 min c) 3 b minutes d) 60 minutes

5. Two candles of the same height are lighted at the same time. The first is consumed in 7 hours and the second in 3 hours. Assuming that each candle burns at a constant rate, in how many hours after being lighted, the ratio between the first and second candles becomes 3 : 1 . a) 2 hours b) 2 hours 20 minutes c) 3 hours 20 minutes d) 3 hours

Answers l . d 2.c 3. a 4.c 5.b

Miscellaneous 1. Find the sum of three numbers in the ratio of 3 : 2 : 5,

such that the sum of their squares is equal to 1862. a) 70 b)75 c)69 d)60

2. A, B, C, D are four quantities of the same kind such that A : B = 3 : 4 , B : C = 8 :9 ,C :D = 15:16 (i) Find the ratio A : D

a)5:8 b)8:5 c)4:5 d)5:4 (ii) Compare A, B,C,D.

9 24 a ) 4 : 3 : - : - b ) 3 : 4 : -

24 5

= )4 :3 : 24 24 9

d) 3 :4 : : -+ J 5 2

4.

Divide 94 into two parts in such a way that fifth part of the first and eighth part of the second are in the ratio 3 : 4. a) 30,64 b)20,44 c)35,69 d)30,65 Divide 1162 into three parts such that 4 times the first may be equal to 5 times the second and 7 times the third. Find the value of smallest part, a) 490 b)492 c)390 d)280

5. Divide Rs. 680 among A, B and C such that A gets - of

what B gets and B gets th of what C gets. What is C's

share? a)Rs280 b)Rs380 c)Rs480 d)Rsl20

Ratio & Proportion 119

6. I f 10% of m is the same as 20% of n, then m: n is equal to: a) 1:2 b ) 2 : l c ) 5 : l d) 10:1

(C.B.I. Exam. 1990) 7. If A : B = 2 : 3 and B : C = 4 : 5, then C : A is equal to:

a)15:8 b) 12:10 c)8:5 d)8:15 (Railway Recruitment Board, 1991)

8. Rs 600 has been divided among A, B and C in such a way that Rs. 40 more than (2/5) of A's share, Rs. 20 more than (2/7) of B's share, Rs. 10 more than (9/17) of C's share, are all equal. A's share is: a)Rs280 b)Rsl70 c)Rsl50 d)Rs200

(Railway Recruitment Board, 1991) 9. 729 ml of a mixture contains milk and water in the ratio 7

: 2. How much more water is to be added to get a new mixture containing milk and water in the ratio 7 :3? a) 600 ml b) 710 ml c) 520 ml d) None of these

(Railway Recruitment Board, 1991) 10. Gold is 19 times as heavy as water and copper 9 times as

heavy as water. The ratio in which these two metals be mixed so that the mixture is 15 times as heavy as water, is: (Delhi Police & CBI1990) a ) l : 2 b)2:3 c)3:2 d) 19:135

1 1 11. I f A is - ofBandBis of C, then A : B : C is

(Police Inspector Exam. 1988) a) 1:3:6 b ) 2 : 3 : 6 c )3 :2 :6 d) 3 :1 :2

12. A certain amount was divided between Kavita and Reena in the ratio 4:3. I f Reena's share was Rs. 2400, the amount is: (SBIPO Exam 1988) a)Rs5600 b)Rs3200 c)Rs9600 d) None of these

13. I f a carton containing a dozen mirrors is dropped, which of the following cannot be the ratio of broken mirrors to unbroken mirrors? a ) 2 : l b ) 3 : l c )3 :2 d)7:5

(SBIPO Exam 1987) 14. A man spends Rs 8100 in buying tables at Rs 1200 each

and chairs at Rs 300 each. The ratio of chairs to tables when the maximum number of tables is purchased, is

(SBIPO Exam 1988) a) 1:4 b)5 :7 c ) l : 2 d ) 2 : l

15. A sum of Rs 86700 is to be divided among A, B and C in such a manner that for every rupee that A gets, B gets 90 paise and for every rupee that B gets, C gets 110 paise. B's share is (LIC AAO Exam 1988) a) Rs 26010 b)Rs 27000 c) Rs 30000 d) None of these

16. Rs 5625 is to be divided among A, B and C, so that A may receive (1/2) as much as B and C together receive and B receives (1/4) of what A and C together receive. The share of A is more than that of B by

(Excise and I. Tax Exam 1988) a)Rs750 b)Rs775 c)Rsl500 d)Rsl600

17. I f the weight of a 13 metres long iron rod be 23.4 kg_ the weight of 6 metres long of such rod will be

(Bank PO Exam 1986) a) 7.2kg b) 12.4kg c) 10.8 kg d)18kg

18. The ratio of the money with Ram and Gopal is 7: 17 and that with Gopal and Krishan is 7 : 17. I f Ram has Rs 490. Krishan has a)Rs2890 b)Rs2330 c)Rsl l90 d)Rs2680

19. The students in three classes are in the ratio 2:3 : 5. If 20 students are increased in each class, the ratio changes to 4 : 5 : 7. The total number of students in the three classes before the increase were (LIC AAO Exam 1988) a) 10 b)90 c)100 d) None of these

20. One year ago the ratio between Laxman's and Gopal's salary was 3:4 . The individual ratios between their last year's and this year's salaries are 4 : 5 and 2 : 3 respec-tively. At present the total of their salary is Rs 4160. The salary of Laxman now, is (SBI Bank PO Exam 1987) a)Rsl600 b)Rs2560 c)Rsl040 d)Rs3120

21. The sum of the squares of three numbers is 532 and the ratio of the first to the second as also of the second to the third is 3 : 2. What is the second number? a) 12 b) 14 c)10 d)8

a b 22. I f b c : a c : a b = l :2 :3 , f ind :

be ca a ) 2 : l b ) 3 : l c ) 4 : l d) 1:4

P_ 23. The sum of two numbers is 'c' and their quotient is .

Find the numbers.

pc qc qp q 3 ) p + q' p + q b ) p + q' (p + q)c

qp qc c) , ' d) None of these

' p+q p+q ' 24. A bag contains rupees, fifty paise, twenty five paise and

ten paise coins in the proportion 1 : 3 : 5 : 7. I f the total amount is Rs 22.25, find the number of twenty-five paise coins. a) 25 b)5 c)15 d)35

25. In a school the number of boys and that of the girls are in the ratio of 2 : 3. I f the number of boys is increased by 20% and that of girls is increased by 10%. What will be the new ratio of the number of boys to that of girls?

(SBI BankPO 2001) a)4:5 b)5:8 c) 8:11 d) Data inadequate

26. An amount of money is to be distributed among P, Q and R in the ratio of 6:19:7 respectively. I f R gives Rs 200 of his share to Q the ratio among P, Q and R becomes 3:10 : 3 respectively. What was the total amount?


(SBI BankPO 2000) a)Rs6400 b)Rs 12800 c) Rs 3200 d) Data inadequate

27. When 3 5 per cent of a number is added to another num-ber, the second number increases by its 20 per cent. What is the ratio between the second number and the first number? (BSRB Mumbai PO 1998) a)4:7 b )7 :4 c) 8:5 d) Data inadequate

28. There is a ratio of 5 : 4 between two numbers. I f 40 per cent of the first number is 12 then what would be the 50 per cent of the second number?

(Bank of Baroda PO 1999) a) 12 b)24 c)18 d) Data inadequate

29. An amount of money is to be distributed among P, Q and R in the ratio of 5 : 8:12 respectively. I f the total share of Q and R is four times that of P, what is definitely P's share? (Bank of Baroda PO 1999) a)Rs3000 b)Rs5000 c) Rs 8000 d) Data inadequate

30. When 30 per cent of a number is added to another num-ber the second number increases to its 140 per cent. What is the ratio between the first and the second num-ber? (Bank of Baroda PO 1999) a)3:4 b)4:3 c) 3 :2 d) Data inadequate

31. I f 25% of a number is subtracted from a second number the second number reduces to its five-sixths. What is the ratio between the first number and the second num-ber? (SBI Associates PO 1999) a)2:3 b )3 :2 c) 1:3 d) Data inadequate

32. When 50% of one number is added to a second number, the second number increases to its four-thirds. What is the ratio between the first number and the second num-ber? (Guwahati PO 1999) a)3:2 b )3 :4 c) 2 :3 d) Data inadequate

33. An amount of money is to be divided among P, Q and R in the ratio of 4:9:16. I f R gets 4 times more than P, what is Q's share in it? (BSRB Calcutta PO 1999) a)Rsl800 b)Rs2700 c) Rs 3600 d) Data inadequate

34. I fa :b = 2:5.Find(3a + 4b):(4a + 5b). a)26:33 b) 14:31 c)25:32 d)33:26

35. A bag contains rupee, 50-paise and 25-paise coins in the ratio 5 : 6 : 7. I f the total amount is Rs 390, find the number of 25-paise coins. a) 280 b)200 c)240 d)260

Answers 1. a; Let the numbers be 3x, 2x, 5x.

we have 9 x 2 +4X2 + 25x2 = 1862

38x 2 = 1862 x2 =49 = 7 2 .-. x = 7. Hence, the required numbers are 21,24 and 35 .-. sum of the numbers = 21+24+ 35 = 70.

2. (i)a; A_3 B__ 8 C__ 15 B~ 4' C~ 9' D~ 16

A B C x x B C D

3 8 15 5 x x = A , D = 5 - 8 4 9 16 8

(i i)b; A : B = 3:4

9 9 B : C = 8 :9= l : - = 4 : -

8 2

, 1 6 9 24 C : D = 1 5 : 1 6 = l : - = - : -

A : B : C : D = 3 :4 : 9 .24 2 ' 5

We put down the first ratio in its original form and change the terms of the other ratios so as to make each antecedent equal to the preceding consequent. (Also see Rule 16).

3. a; Let these parts be x and y. Then,

x y . , - : - = 3:4 0 r ,Sx:5y = 3\4

5y 5x3 15

E or 5 y

4x8 32 '

Thus,x:y=15:32. Now, sum of ratios = 15 + 32 = 47.

94x15 -.-. First part = = 30 >

94x32 second part :

47 = 64

4. d; 4 x (1 st part) = 5 x (2nd part) = 7 * (3rd part) = x (say)

X X X Then, 1st part = , 2nd part = ; 3rd part = .

:. Ratio of divisions : X X X

4 ' I ' l

= - : - : - = 35:28:20 4 5 7

Sum of ratios = 35 + 28 + 20 = 83.

1162x35 .-. I st part =

2nd part

83

1162x28 83

= 490

= 392

Ratio 8 5 Proportion . 2 .

7. a,

i.c;

1162x20 . . . 3rd part = = 280

o3

5. c; Suppose C gets Re 1. Then, B gets Re and A gets

Rs (f*/{] or R s f

Ratios of A, B and C's shares = : : 1 = 2:3:12. 6 4

680x2 So, A's share = Rs = Rs 80.

680x3

B's share = Rs = Rs 120.

680x12 C's share = Rs =Rs480.

6.b; 10%ofm = 20%ofn

m_( 20 \00^ 1 ' n

10 20 m = n

100 100

^100 10 ) 1

A 2 Z? _ 4 A B__2 4 B 7 3 ' C ~ 5 ^ B X C ~ 3 X 5

C 15 C 15

-A + 40 = -B + 20 = --C + \0 = x 5 7 17

. A = -(x~40) = -x-\00; 2 V 7 2

5 = - ( ; t - 2 0 ) = - x - 7 0 2 V y 2

17/ i n x 17 170 and C = (x-\0) = x-.

5 7 17 170 -x-\00 + -x-10 + x- = 600 2 2 9 9

71 7100 or, x = =>x = 100.

A's share: - x l 0 0 - 1 0 0 | = R s l 5 0 .

9.d; Quantity of milk = [ J 2 9 * - J ml = 567 ml.

Quantity of water = (729 - 567) ml = 162 ml

Now, 567

5 6 7 x 3 = 7(162 + x)

10. b;

162 + x 3 => 1701 = 1134 + 7x => x = 81. Let x gm of water be taken Then, gold = 19x gm and copper = 9x gm Let 1 gm of gold be mixed with y gm of copper.

2 Then, 19x + 9 x y = 1 5 x ( l + y ) = > y~~z.

11. a; 1 1 B = i-C,A = -B=-2 3 3

1 -C\ -C-2 J 6

.', A:B:C = -C:-C:C = - : - A or 1 :3 :6 . 6 2 6 2

12. a; Let the amount be Rs x.

13. c;

14. c;

15. b;

Then, Reena's share = Rs I x x ^

3x (^2400x7^ . ' . y = 2400 or x = ^ j = R S 5600.

Sum of the ratios must divide 12. Since 3 + 2 = 5 does not divide 12, so it can not be 3 : 2. Maximum number of tables purchased at Rs 1200 per table spending within Rs 8100 is clearly 6. Remaining amount = Rs (8100 -1200 * 6) = Rs 900 Number of chairs for Rs 900 = (9001300) = 3 -.: Number of chairs: Number of tables = 3 :6 or 1 :2. I f A gets Re 1, B gets 90 paise. Now, i f B gets Re 1, C gets 110 paise.

' U 0 x 9 0 l = 99 100

I f B gets 90 paise, C gets

.-. A : B : C = 1 0 0 : 9 0 : 9 9 .

( 90 1 So, B's share = Rs I 8 6 7 0 0 x ^

J paise.

= Rs 27000.

16. a;

17. c; Let

A + B + C = 5625andB = - (A + C)i.e. A + C = 4B.

.-. 4B + B = 5625 o rB= 1125. AIso,A + C = 4B = 4 x 1125=4500.

, 1

Also, /* = - ( B + C)orB + C = 2AorB = 2A-C.

.-. 2A-C=1125. Now, solving A + C = 4500 and 2A - C = 1125, we get A = 1875 and C = 2625. .-. A - B = (1875- 1125)=Rs750.

13 23.4 ^ 5 i v ^ .Then, x = 23.4x6 13

= 10.8 kg


18. a; Let Ram, Gopal and Krishan have rupees x, y and z respectively.

Then, and i / z 1_

17

x y x y

1 7 x 17 17

49 289

Thus, i f Ram has Rs 49, Krishan has Rs 289.

'289 49

= Rs2890.

I f Ram has Rs 490, Krishan has Rs x490

19. c; Let the number of students in the class be 2x, 3x and 5x respectively. Then, (2x + 20): (3x + 20): (5x + 20):: 4:5 :7.

2s + 20 3s + 20 5.X + 20 4 5 7

Solving these equations, we get x = 10. .-. total number of students in the class

= (2x+3x+5x)=10x=100. 20. a; Let the salaries of Laxman and Gopal one year before

be JC, , y, and now it be x2, y2 respectively. Then,

3 x. 4 s.

1 A s'y2

and x2 +y2 =4160

Solving these equations, we get x2 = 1600

21.a; First number _ 3 3 _ 9

Second number 2 3 6

First number 3 2 6 and ~ . , : _ x x ^ r = : ~ r

Third number 2 2 4 .-. First: Second: Third = 9 : 6 : 4 As per the question .-. (9s)2 +{6xf + (4s) 2 =532

or, 133s2 =532 or,x = 2 Second Number is 6x = 12

22. c; bc=l ac 2

a b

or, a: b = 2:1

a ca _ a be ca be b b2

4 1

23. a; Let the numbers be x, y. .-. x+y = c(given)

and (given)

x+y p+q

* = P c p + q x =

PC p + q

and y = qc_

p + q Since =

y Q. 24. a; Let the number of coins be x, 3x, 5x, 7x respectively as

rupees, fifty paise, twenty five paise and ten paise. Since Number of coin * Value of coin in rupee = Amount in rupees

Now, value of 50-paise coin in rupee :

Value of 25-paise coin in rupee :

1

Value of 10-paise coin in rupee = 10

or,

(xxl)+

89s 20 "

' f 3xx 2 5x x I H

7xx-10

= 22.25

22.5 or, x = 5

:. number of rupee coins = 5 x 1=5 Number of 50-paise coins = 3x = 15 Number of 25-paise coins = 5x = 25 Number of 10-paise coins = 7x = 35

25. c; Let the number of boys be 2x and the number of girls be3x.

120 12* No. of boys is increased by 20% = 2x x =

, 110 33*

No. of girls is increased by 10% = 3x x - =

The new ratio of the number of boys to that of girls 12s 5

33s 10

8:11

26. a; Let the sum of P, Q, and R be 6x, 19x and 7x. .-. total sum = 6x + 19x + 7x=32x From the question 6x:19x+200:7x-200 = 3:10:3 ie6x = 7x-200 .-. x = 200 .-. total sum = 32 x 200 = Rs 6400

27. b; 35percentofx + y = 120 100 y

35s + 100>- 120 ^ or, = y => 35s = 20y

100 100

Ratio & Proportion

^ = 7:4

a b

20

5 , 4 2 -- b = -a Given, (40% of a = ) - a = 12. 4 5 5

123

33. d; Here, neither the total amount nor the individual amount is given. So the share of Q cannot be deter-mined.

a 2 34. a; Given - ^

o J

.-. a = 5 x6and 6 = ^ x 5 x 6 = 24

, 24 , ;. 50% of b = = 12

2 29.d; P :Q:R = 5:8:12

Total share of Q and R 8 + 12 20 = 4

Share of P 5 5 So, we see that not new information has been given in question and P's share can't be determined.

30. b; Let the first and the second numbers be x and y re-spectively then y + 30% of x = 140% of y or, y + 0.3x = 1.4y or, 0.3x = 0.4y .-. x : y = 0.4:0.3 = 4:3

31. a; Let the first and second number be x and y respec-tively

y-xx 25 100

yx-

1 x : y = 2:3

Let the numbers be y and x respectively.

4x 4x y x + 50%of ^ = or, -

Now, 3a + 4b 4a + 5b A a .

4 | - | + 5

[Dividing numerator and

denominator by b.]

3 x - + 4 5

4 x ~ + 5 5

26

33

.-. (3a + 4b): (4a + 5b) = 26:33

35. a; Ratio of values of the coins : 5 6 7 1 2 4

= 20:12:7

Value of one-rupee coins = Rs

Value of 50-paise coins = Rs 3 9 0 X T T = R S 120

390 , 3 v 3 9

12

Rs 200

39

Value of 25-paise coins = Rs ^ 3 9 0 x ^ j = Rs 70

.-. Number of one-rupee coins = 200 Number of 50-paise coins = (120 x 2) = 240 Number of 25-paise coins = (70 M ) = 280.

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Chapter 5.2