**Mixture and Alligation Problems for CAT Set-2:**

Ratio proportion mixtures and alligations is one of the important topic for CAT. Mixtures and alligation is the application of the ratio and proportion. We have provided some important questions on this topic with detailed explanations. Learn mixtures and alligation concepts and formulas for CAT, before trying these problems.

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**Question 1: **In a test the average marks obtained by boys and girls in a class are 72 and 56 respectively. The average marks obtained by the class in the same test is 60. If in an other test the average marks obtained by the boys and girls are 64 and 80 respectively, what is the average marks obtained by the class in the test?

a) 76

b) 74

c) 72

d) 70

**Question 2: **Two varieties of rice R1 and R2 cost Rs. 20 and Rs. 50 per kg. In what ratio must R1 and R2 be mixed such that the cost price of the resultant rice is Rs. 45 per kg?

a) 1:4

b) 3:8

c) 2:7

d) 1:3

e) 1:5

**Question 3: **In what ratio must acid solutions of concentrations 10%, 15% and 50% be mixed to get an acid solution of 40% concentration?

a) 2:2:11

b) 1:1:6

c) 2:3:15

d) 5:4:9

**Question 4: **Two solutions have milk:water ratio of 2:3 and 4:5. In what ratio must they be mixed such that the resultant solution has milk:water ratio of 3:4?

a) 8:3

b) 3:8

c) 5:9

d) 9:5

**Question 5:** In what ratio must sugar costing Rs. 60, Rs. 90 and Rs. 100 per kg be mixed to get a mixture costing Rs. 70 per kg?

a) 1:1:5

b) 5:1:1

c) 7:2:1

d) More than one of the above

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**Question 6: **In what ratio must milk costing Rs. 30, Rs. 40 and Rs. 45 per litre be mixed to form a mixture costing Rs. 35 per litre?

a) 1:1:1

b) 2:3:4

c) 1:2:3

d) 5:1:2

**Question 7: **In what ratio must 3 varieties of rice whose price is 20, 25 and 40 rupees per kg have to be mixed such that the resulting mixture has a price of 30 rupees per kg?

a) 1:1:1

b) 1:2:1

c) 1:2:2

d) 1:1:2

**Question 8:** A rice trader trades two qualities of rice: standard rice which costs him Rs 40/kg and premium rice which costs Rs 90/kg. He mixes some quantity of the standard rice with the premium rice and sells it a Rs 100/kg thereby making a profit of 30%. What is the ratio in which the standard and premium rice have been mixed?

a) 2:3

b) 3:2

c) 48:17

d) 17 : 48

**Question 9:** Four containers (equal in size) have mixtures of milk and water in the ratio 4:3, 5:4 , 1:2 and 2:5 respectively. Mixtures from all the four containers are poured into a big vessel which is 4.2 times of the size of each container and the remaining part of big vessel is filled with water. Find the ratio of milk to water in the big vessel ?

a) 1100:1546

b) 40:120

c) 12:14

d) 168:504

**Question 10:** Two milk containers, A and B, contain milk and water in the ratio 7:1 and 1:3 respectively. In what ratio should they be mixed to get a final solution containing milk and water in the ratio 1:1?

a) 1:2

b) 2:3

c) 3:4

d) None of these

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**Solutions for Mixture and Alligation Problems for CAT Set-2:**

**Solutions:**

**1) Answer (A)**

Ratio of boys to girls in the class can be obtained using the method of alligation.

Ratio of boys to girls is

(60-56) : (72-60)

= 4 : 12

= 1 : 3

In the second test, the average marks obtained by boys and girls are 64 and 80 respectively.

Average marks obtained by the class = [(64*1)+(80*3)]/4 = 304/4 = 76

Hence Option A.

**2) Answer (E)**

Using Alligation we get,

20 50

\ /

45

/ \

5 25

Thus the ratio is 5:25 or 1:5.

**3) Answer (A)**

Let the ratio of acids whose concentrations are 10%, 15% and 50% be a:b:c.

Acids of concentration 10% and 50% can be mixed to get acid of concentration 40%.

This can be done using the alligation formula,

10 50

\ /

40

/ \

10 30

Thus the ratio of a:c = 1:3.

Also acids of concentration 15% and 50% can be mixed to get acid of concentration 40%.

This can be done using the alligation formula,

15 50

\ /

40

/ \

10 25

Thus the ratio of b:c = 2:5.

If we add up both the ratios a:c and b:c in the ratio 2:1 we get a:b:c = 2:2:11.

**4) Answer (C)**

The amount of milk with respect to the whole in solutions 1,2 and the resultant mixture is 2/5, 4/9 and 3/7.

Now we can perform alligation as shown,

2/5 4/9

\ /

3/7

/ \

1/63 1/35

Thus the ratio is (1/63):(1/35) = 5:9

**5) Answer (D)**

Let the ratio of sugar costing 60, 90 and 100 rupees per kg be a:b:c.

Sugar costing 60 and 90 rupees can be mixed to get rice costing 70 rupees per kg.

This can be done using the alligation formula,

60 90

\ /

70

/ \

20 10

Thus the ratio of a:b = 2:1.

Also sugar costing 60 and 100 rupees can be mixed to get sugar costing 70 rupees per kg.

This can be done using the alligation formula,

60 100

\ /

70

/ \

30 10

Thus the ratio of a:c = 3:1.

If we add up both the ratios a:b and a:c we get a:b:c = 5:1:1.

But more than one of the above is also an option.

So let’s check for other possibilities.

If we add up the ratios a:b and a:c in the ratio 1:2 we get a:b:c = 7:2:1.

Thus both 5:1:1 and 7:2:1 are right and the answer is option D.

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**6) Answer (D)**

Let the ratio of milk costing 30, 40 and 45 rupees per litre be a:b:c.

Milk costing 30 and 40 rupees can be mixed to get rice costing 35 rupees per kg.

This can be done using the alligation formula,

30 40

\ /

35

/ \

5 5

Thus the ratio of a:b = 1:1.

Also milk costing 30 and 45 rupees can be mixed to get milk costing 35 rupees per litre.

This can be done using the alligation formula,

30 45

\ /

35

/ \

10 5

Thus the ratio of a:c = 2:1.

If we add up both the ratios a:c and b:c in the ratio 1:2 we get a:b:c = 5:1:2

**7) Answer (C)**

Let the ratio of rice costing 20, 25 and 40 rupees per kg be a:b:c.

Rice costing 20 and 40 rupees can be mixed to get rice costing 30 rupees per kg.

This can be done using the alligation formula,

20 40

\ /

30

/ \

10 10

Thus the ratio of a:c = 1:1.

Also rice costing 25 and 40 rupees can be mixed to get rice costing 30 rupees per kg.

This can be done using the alligation formula,

25 40

\ /

30

/ \

10 5

Thus the ratio of b:c = 2:1.

If we add up both the ratios a:c and b:c we get A:B:C = 1:2:2.

**8) Answer (D)**

Hence, Cost price of rice = 100/1.3. By alligation formula, the ratio for mixing the two rices = (90-100/1.3)/(100/1.3-40)= (17/1.3)/(48/1.3) = 17:48.

**9) Answer (A)**

Let the size of each container be 63 parts

Required ratio = (36+35+21+18) : (27 +28+ 42+ 45 +(63)0.2)=110:154.6

**10) Answer (B)**

Fraction of A containing milk is 7/8

Fraction of B containing milk is 1/4

Final solution contains 1/2 milk. Using alligation.

$$\frac{QuantityA}{QuantityB} = \frac{1/2-1/4}{7/8-1/2} = \frac{1/4}{3/8} = \frac{2}{3}$$

This CAT problems on mixtures and alligations will be useful for CAT preparation.