Quicker Maths Aligations

ALLIGATION

Allegation: It is the rule that enables us to find the ratio in which two or more ingredients at the given price must be mixed to produce a mixture of a desired price.

1     In what proportion must rice at Rs 3.10 per kg be mixed with rice at Rs 3.60 per kg, so that the mixture be worth Rs 3.25 a kg?

a)    3 : 7                      b) 7 : 3                  c) 5 : 2                  d) 2: 5

3.60 – 3.25 / 3.25 – 3.10

35/ 15

7 : 3

Or

If required ratio is a : b

3.10 x a + 3.60 x b = 3.25 x (a + b)

310 a + 360 b = 325 a + 325 b

(360 – 325) b = (325 - 310) a

35 b = 15 a

a = 35

b = 15

a: b

35 : 15

7 : 3

2     How many kg of salt at 42 paisa per kg must a man mix with 25 kg of salt at 24 paisa per kg. so that he may, on selling the mixture at 40 per kg, gain 25% on the out lay?

a)    25 kg                    b) 15 kg                c) 20 kg                d) 17.5 kg

[100 x 40 – 24 x (100 + 25) / 42 (100 + 25) – (100 x 40)] x 25

20 kg

Or

40 ----- 125% (100% + 25%)

? ------- 100%

40 x 100% = ? x 125

? = 32 paisa

a x 42 + 25 x 24 = (25 + a) 32

42 a + 600 = 800 – 32a

a = 20

Or

40 ----- 125% (100% + 25%)

? ------- 100%

40 x 100% = ? x 125

? = 32 paisa

(24 – 32) / (32 - 42)

8 : 10

4 : 5

5 ------ 25

4 ------ ?

 

       x5

5 ----------- 25

4 ----------- ?

4 x 5 = 20

3     A mixture of a certain quantity of milk with 16 liters of water is worth 90 paisa per liter. If pure milk be worth 108 paisa per liter how much milk is there in the mixture?

a)    70 liters                b) 75 liters            c) 85 liters            d) 80 liters

16 x (90/ 108 – 90)

80 liters

Or

a x 108 + 16 x 0 = (a + 16) 90  (water cost is zero)

a = 80 liters

or

108 – 90/ 90 – 0

1 : 5

1 ------ 16

5 ------ ?

 

       x16

1 ----------- 16

5 ----------- ?

5 x 16 = 80

4     In what proportion must water be mixed with spirit to gain 16 (2/3)% by selling it at cost price?

a)    1:3                        b) 2:5                              c) 1:6                    d) 3:7

16 2/3%

(50/3)%

50/ 300

1/6

1:6

5     There are 65 students in a class, 39 rupees are distributed among them so that each boy gets 80 paisa and each girl gets 30 paisa. Find the number of boys and girls in that class?

a)    26 and 39     b) 39 and 26       c) 30 and 35         d) 35 and 30

Rs 39

3900 paisa

3900 - 65 x 30 / (80 - 30)

39 boys

65 - 39

26 girls

65 x 80 – 3900 / (80 - 30)

26 girls

65 - 29

39 boys

Or

3900/ 65

60 paisa

If A is number of girls

A x 30 + (65 - A) x 80 = 65 x 60

A = 26 (girls)

65 - 26

39 (boys)

Or

(30 – 60)/ (60 - 80)   (Rs 39; 3900/65 = 60P)

3:2

5 ------ 65

3 ------ ?

 

       x13

5 ----------- 65

3 ----------- ?

3 x 13 = 39 (boys)

 

       x13

5 ----------- 65

2 ----------- ?

2 x 13 = 26 (girls)

6     A person has a chemical of Rs 25 per liter. In what ratio should water be mixed in that chemical so that after selling the mixture at Rs 20 per liter he may get a profit of 25%?

a)    9:16                      b) 16:9                           c) 3:4                    d) 4:3

20/ (25 - 20) + 25% x 25

16:9

Or

20 ------ 125% (100% + 25%)

? -------- 100%

20 x 100% = ? x 125

? = 16

25 x a + 0 x b = (a + b)16

25a = 16a + 16b

9a = 16b

a = 16

b = 9

Or

(0 – 16)/ (16 - 25)

16 : 9

7     A person travels 285 km in 6 hours in two stages. In the first part of the journey, he travels by bus at the speed of 40 km per hr. In the second part of the journey, he travels by train at the speed of 55 km per hr. how much distance did he travel by train?

a)    160km                  b) 180km              c) 195km              d) 165 km

285 – 40 x 6/ (55 - 40)

165 km

Or

40 x a + 55 (6 - a) = 285

40 a + 330 – 55 a = 95 x 3

a = 3

3 x 55

165 km

Or

(55 – 285/6) / (285/6 - 40)

1: 1

55 x 3

165km

8     A trader has 50 kg of pulses, part of which he sells at 8% profit and the rest at 18% profit. He gains 14% on the whole. What is the quantity sold at 18% profit?

a)    40 kg                    b) 35 kg                c) 30 kg                d) 25 kg

(14 – 8/ 18 – 8) x 50

30 kg

Or

18 x a + 8 (50 - a) = 14 x 50

18a + 400 – 8a = 700

a = 30

Or

18 – 14/ 14 - 8

2: 3

5 ------ 50

3 ------ ?

 

       x10

5 ----------- 50

3 ----------- ?

3 x 10 = 30

9     A trader has 50 kg of rice, a part of which he sells at 10% profit and the rest at 5% loss. He gains 7% on the whole. What is the quantity sold at 10% gain and 5% loss?

a)    10 kg, 40 kg         b) 40 kg, 10 kg   c) 30 kg, 20 kg    d) 20kg, 30 kg

(7 + 5/ 10 + 5) x 50

40 kg (10% profit)

(10 – 7/ 10 + 5) x 50

10 kg (5% loss)

Or

If a is the quantity of sold at 10% profit

-5% -------(50 - a)

a x 10% - (50 - a) 5% = 50 x 7%

10 a – 250 + 5a = 350

15 a = 600

a = 40 (10% of profit)

50 - 40

10 (5% of loss)

Or

-5 – 7/ 7 - 10

12 : 3

4 : 1

5 ------- 50

4 -------- ?

 

       x10

5 ----------- 50

4 ----------- ?

4 x 10 = 40 (10% of profit)

50 - 40

10 (5% of loss)

10  A trader has 50 kg of rice, a part of which he sells at 14% profit and the rest at 6% loss. On the whole his loss is 4%. What is the quantity sold at 14% gain and 6% loss?

a)    45 kg, 5 kg            b) 5 kg, 45 kg   c) 30 kg, 20 kg    d) 20kg, 30 kg

(6 - 4/ 14 + 6) x 50

5 kg (14% profit)

(14 + 4/ 14 + 6) x 50

45 kg (6% loss)

Or

14% ----------- a

-6% ------- (50 - a)

a x 14 – (50 - a) x 6 = 50 x (-4)

14a – 300 + 6a = -200

20a = 100

a = 5 kg

50 – 5

45 kg

11  Mira’s expenditure and saving are in the ratio 3:2. Her income increases by 10%. Her expenditure also increases by 12%. By how many % does her saving increase?

a)    3%                        b) 5%                               c) 7%                    d) 9%

((3/2) +1 ) x 10 – (3/2) x 12

25 – 18

7%

12  A vessel of 80 liters is filled with milk and water. 70% of milk and 30% of water is taken out of vessel. It is found that the vessel is vacated by 55%. Find the initial quantity of milk water?

a)    30 liters, 50 liters b) 50 liters, 30 liters c) 35 liters, 45 liters d) 45 liters, 35 liters

a, 80 – a (milk and water)

70% of a + 30% of (80 - a) = 55% x 80

70 a + 2400 – 30 a = 4400

40 a = 2000

a = 50 (milk)

80 – 50

30 (water)

13  A jar contains a mixture of two liquids A and B in the ratio 4:1. When 10 liters of the mixture is taken out and 10 liters of liquid B is poured into the jar, the ratio becomes 2:3. How many liters of liquid A was contained in the jar?

a)    15 liters                b) 12 liters            c) 18 liters            d) 16 liters

4 : 1

4a : 1a

4a + 1a =5a

5a -------- 10

4a -------- ?

8 liters

10 – 8 = 2 liters

(4a – 8) : (1a – 2 + 10) :: 2 : 3

4a – 8 : 1a + 8 :: 2 : 3

12a – 24 = 2a + 16

10a = 40

a = 4

4 a

4 x 4

16 liters milk

14  729 liters of a mixture contains milk and water in the ratio 7 : 2. How much water is to be added to get a new mixture contained milk and water in the ratio 7 : 3?

a)    50 liters                b) 70 liters            c) 80 liters            d) 81 liters

7a + 2a = 729

a = 81

7 x 81 : 2 x 81

7 x 81 : (2 x 81 + Z) :: 7 : 3

3 x 7 x 81 = 7 (2 x 81 + Z)

Z = 81

15  In three vessels each of 10 liters capacity, mixture of milk and water is filled. The ratios of milk and water are 2 : 1, 3 : 1 and 3 : 2 in the three respective vessels. If all the three vessels are emptied into a single large vessel, find the proportion of milk and water in the mixture?

a)    59 : 121                b) 121 : 59            c) 11 : 9                d) 9 : 11

Milk in liters                  Water in liters

20/3                              10/3

15/2                              5/2

6                                    4

121/6                            59/6

121 : 59