Concepts Examples

How To Solve Allegations

Suppose the average weight of a group of objects O₁ is A₁ and average weight of group of objects O₂ is A₂ and A₁ < A₂, then we can write the data in the manner as shown in the figure. (A_w−A₂) : (A_w−A₁) gives the ratio of O₁ : O₂.

Allegations does not give the actual volume to be mixed but only the ratio in which volumes are to be mixed.

Question 1 A shopkeeper sells two types of rice A and B. If he sold 30% of type A rice at a profit of 50% and sold 90% of the rice of type B at a profit of 10%. What is the averge profit percent of the shopkeeper, if he sells only these two types of rice.

35%	45%
40%	55%

C

Since the profit on rice of type B = 10% ∴ A₁ = 10

Since the profit on rice of type A = 50% ∴ A₂ = 50

Let the average profit be Rs x ∴ A_w = x

$\frac{50-x}{x-10}=\frac{3}{1}\Rightarrow50-x=3x-30$50−xx−10 =31 ⇒50−x=3x−30

⇒ x = 20

∴ Average profit of the shopkeeper is 40%

Question 2 How much water should be mixed with 6L pure milk to make the concentration of milk 60%.

4L	2L
3L	5L

A

Since the concentration of milk in water = 0 ∴ A₁ = 0

Since the concentration of pure milk = 100 ∴ A₂ = 100

Since the concentration of milk in milk-water solution = 60 ∴ A_w = 60

∴ Ratio in which water and milk should be added = 2:3

∴ Volume of water to be mixed = 4L

Question 3 A dealer buys 11 kg of wheat at Rs 275 and mixes it with another quantity of wheat in the ratio of 4:5. The price of the resulting mixture is Rs 30 per kg. Finf the price of this other quantity of wheat.

Rs 37 per kg	Rs 31 per kg
Rs 24 per kg	Rs 34 per kg

D

Price per kg of wheat = $\frac{275}{11}=25$27511 =25 Rs

Since the of wheat = Rs 25 ∴ A₁ = 25

Let the price of rice of other type be Rs x ∴ A₂ = x

The average price of wheat = Rs 30 ∴ A_w = 30

∴$\frac{5}{x-30}=\frac{5}{4}\Rightarrow20=5x-150$5x−30 =54 ⇒20=5x−150

⇒ x = 34

∴ Price of wheat of other type = Rs 34 per kg

Question 4 If the average weight of a class of 60 students is 50 and the average weight of a class of 40 students is 45 , then what will be the average weight of both the classes combined.

24	26
45	48

D

Since the average weight of class of 40 students = 45 ∴ A₁ = 45

Since the average weight of class of 60 students = 50 ∴ A₂ = 50

Let the average weight of both the classes combined be x kg.

O₁:O₂ = 40:60 = 2:3

$\frac{50-x}{x-45}=\frac{2}{3}$50−xx−45 =23 

$150-3x=2x-90$150−3x=2x−90

$x=48$x=48

∴ Average weight of both the classes combined = 48 kg

Question 5 A person travels at a speed of 25 km/hr for 30min and then travels at 40km/hr for next 20 min to reach his destination. Find his average speed in the whole journey.

21 km/hr	31 km/hr
33 km/hr	None of these

B

Since the person travels for 30 min at speed of 25 km/hr ∴ A₁ = 25

Since the person travels for 20 min at speed of 40 km/hr ∴ A₂ = 40

Let the average speed of the person be x km/hr ∴ A_w = 500

$\frac{40-x}{x-25}=\frac{3}{2}\Rightarrow80-2x=3x-75$40−xx−25 =32 ⇒80−2x=3x−75

⇒ x = 31

∴ Average speed of the person in the whole journey was 31 Km/hr

Question 6 In what ratio should a 65% milk solution and 85% milk solution should be mixed so that the new solution is a 80% milk solution.

1:2	1:3
1:4	2:3

B

Since the concentration of milk in water = 65 ∴ A₁ = 65

Since the concentration of pure milk = 85 ∴ A₂ = 85

Since the concentration of milk in milk-water solution = 80 ∴ A_w = 80

∴ Ratio in which the two solutions should be mixed = 1:3

Concepts Examples

When we have to calculate the average of some groups of different things, having different number of elements is being calculated, then it is called the weighted average.

If the number of elements in n different groups be N₁,N₂,N₃ .. N_n and their respective averages be A₁,A₂,A₃...A_n ,

then Weighted average = $\frac{\mathbb{N}_1\cdot A_1+\mathbb{N}_2\cdot A_2+\mathbb{N}_3\cdot A_3...\mathbb{N}_n\cdot A_n}{\mathbb{N}_1+\mathbb{N}_2+\mathbb{N}_3...\mathbb{N}_n}$N₁·A₁+N₂·A₂+N₃·A₃...N_n·A_nN₁+N₂+N₃...N_n 

Question 1 Average cost of buying 5 quality A crackers is 60 and cost of buying 10 quality B crackers is 80. WHhat will be average cost of buying all the crackers.

71.33	72.33
73.33	74.33

C

Total number of employees = 5 + 10 = 15

Average cost of all the crackers = $\frac{5\cdot60+10\cdot80}{5+10}=\frac{300+800}{15}=\frac{1100}{15}=73.33$5·60+10·805+10 =300+80015 =110015 =73.33

Question 2 Average weight of a class of 40 students is 30 and average weight of a class of 20 students is 15. Find the average weight of both the classes combined.

21	22
24	25

D

Average weight = $\frac{40\times30+20\times15}{40+20}=25$40×30+20×1540+20 =25

Question 3 There are two sections A and B of a class, consisting of 36 and 44 students' respectively. If the average weight of section A is 40kg and that of section B is 35kg, find the average weight of the whole class.

37.25	74.5
36.25	36

A

Total weight of (36+44) Students = (36×40 + 44×35) kg = 2980 kg

Therefore average weight of the whole class = $\frac{2980}{80}$298080  kg

Therefore average weight = 37.25 kg

Question 4 Average salary of 12 employees in company A is 15000 pm and 15 employees is company B is 16000 pm. What will be the average salary of all the employees.

17000	16888
15000	14888

B

Total number of employees = 15 + 12 = 27

Average salary = $\frac{12\cdot18000+15\cdot16000}{12+15}$12·18000+15·1600012+15 

= $\frac{456000}{27}=16888.88$45600027 =16888.88