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# Arithmetic

Averages-1

__Averages__

__Introduction__

Suppose you have to tell your friend that when your expenditure of Goa was Rs.3500 first day for travelling, accommodation and food. Second day’s expenditure was Rs.3200 for accommodation, and food, third day’s expenditure was Rs.3810 for accommodation and food and on fourth day’s expenditure was Rs.5000 on food, shopping and return tickets, how would you tell him? In this case you can see that the expenditure is not same on all days. Average expenditure is the sum value of per day in such a way that will give you same total expenditure of all the three days. Mean is a synonym of Average.

** Example 2** In the first 20 overs of gully cricket, the run rate was only 3.2. What should be the run rate in the remaining 40 overs to reach the target of 284?

** Solution: **20 * 3.2 + x * 40 = 284

64 + 40 x = 284

40x = 220; x = 5.5

__Change in average__

**When an entity is being removed**

Suppose we have a set of numbers 1, 2, 3, 4, 5

Average of this is 3, this means all entities are 3, 3, 3, 3 and 3

1) If 3 is being removed from the group, Does the average change?

The answer is no, Why? Since 3 is taking 0 value to remain 3.

2) When 1 is being removed. Does the average change now?

The answer is YES, But why? The reason is 1 was taking +2 to become 3 when it is removed the average increases by 2 over 4 (number of elements left) = 2/4= 0.5

3) When 5 is being removed. Does the average change now?

Yes it does, why? 5 was giving 2 to become 3 so the average decreases by 2/4 i.e. 0.5

So the average becomes 0.5

**When an entity is being added**

4) Suppose 6 is being added to the group

It gives 3 more than the existing average of 3, so the increase in the total average will be of 3/6 =0.5. So, the overall average increases by 0.5

New average will be 3.5

** Example 3** Ages of the persons in the group are 25, 26, 28 and 41. A person of age 40 years is added to the group, what is the increase in the average age?

When age 40 is being added to the group the age of the group increases by 10/5 = 2

New average age will be 32

** Example 4** The average weight of 8 person's increases by 2.5 kg when a new person comes in

Place of one of them weighing 65 kg. What might be the weight of the new person?

Solution: if after the new person has come the average weight increases by 2.5 kg, means the total weight increases by 2.5*8 = 20kgs

So the weight of new person is 20kgs more than the one being replaced

Answer is 85

Weighted average

Weighted average is also an average/ Mean calculated by giving values, when you have two or more groups whose individual averages are known.

** Example 5**: In a class there are two sections A and B, Section A has 30 students and Section B has 35 students. Section A scored an average marks of 60 and Section B scored an average marks of 65. Find the average marks of the whole class?

Solution: Average marks of the Class = = 23.61

** Example 6:** The average monthly income of P and Q is Rs. 5000. The average monthly income of Q and R is Rs. 6000 and the average monthly income of P and R is Rs. 4000. The monthly income of P is?

** Solution:** total monthly income of P and Q = P + Q = 5000 * 2 = 10000

Total monthly income of Q and R = Q + R = 6000 * 2 = 12000

Total monthly income of P and R = P + R = 4000 * 2 = 8000

Solving above equations

P = 3000

Q = 7000

R= 5000

** Example 7:** The average age of three brothers Ram, Shyam and Sundar 3 years ago was 27 years and that of Shyam and Sundar 5 years ago was 20 years. The present age of Shyam is?

** Solution:** present age of husband, child and wife are h, c and w respectively

3 years ago

Ram: R-3

Shyam: Sh-3

Sundar: Su-3

R + Sh + Su =90 -------------------- (1)

5 years ago

Shyam: Sh – 5

Sundar: Su -5

Sh + Su =50 -------------------- (2)

Solving (1) and (2) we get Rams age =40 years

** Example 8:** with an average speed of 40 kmph, a train reaches its destination in time. If it goes with an average speed of 35 kmph, it is late by 15 minutes. The length of the total journey is?

** Solution:** the train needs to travel 15mins extra @35kmph, hence it is behind 8.75 kms. The rate of losing distance 5 kmph. Hence the train must have travelled 8.75/5 = 1 hours 45 minutes @ 40kmph

Answer= 70kms

** Example 9:** 19 persons went to a hotel for a combined dinner party. 13 of them spent Rs.79 each on their dinner and the rest spent Rs.4 more than the average expenditure of all the 19. What was the total money spent by them

** Solution:** let the average be x

13 * 79 +6 (4 + A) = 19A

A= 79

Answer: 19A = 1501

** Example 10:** There are 5 boxes in a cargo hold. The weight of the first box is 200kg and weight of the second box is 20% higher than the weight of the third box, whose weight is 25% higher than the first box. The fourth box is at 350kg is 30% lighter than the fifth box. Find the difference in average weight of the four heaviest boxes and the four lightest boxes.

** Solution:** Weights of boxes

1^{st} box = 200 kg

2^{nd} box = 300 kg

3^{rd} box = 250 kg

4^{th} box = 350 kg

5^{th} box = 500 kg

Difference between heaviest boxes and the four lightest boxes = 300

Difference in their averages = 75