Average - MathsTips.com

The average o two or more similar quantities is the quantity obtained by dividing the sum of the quantities by the number of quantities.

$Average= \dfrac{Sum \: of \: the \: quantities}{Number \: of \: the \: quantities}$
Also, $Sum \: of \: the \: quantities= Average \times Number \: of \: quantities$

The average value of a quantity need not be the usual value of the quantity. It need not even be the value that occurs most often. For example, if three girls are 10, 14 and 16 years old, their average age is $\dfrac{10+14+16}{3}=10$ . We cannot say that most of the girls are 10 years old.

Note: If an object covers equal distances in equal intervals of time then we can say that it has uniform speed. In reality, an object does not usually have uniform speed. In such instances we usually take the average speed of the object.
$Average \: speed=\dfrac{Total \: distance \: covered}{Total \: time \: taken}$

Example 1: The monthly savings of 8 friends are Rs 2326, Rs 2676, Rs 2751, Rs 2401, Rs 2351, Rs 2451, Rs 2521 and Rs 2651. Find their average savings.

Solution:

Sum of the savings= Rs (2326+2676+2751+2401+2351+2451+2521+2651) = Rs 20,128.
Number of persons=8

$\therefore$ $Average \: Savings= \dfrac{Sum \: of \: the \: savings}{Number \: of \: persons}$

$=Rs. \dfrac{20,128}{8}= Rs. 2516$

Example 2: The average weight of 30 students of a class is 40Kg. If the weight of the teacher is added, the average weight goes up by 1 Kg. Find the weight of the teacher.

Solution:

The average weight of 30 students= 40 Kg.
Therefore, the total weight of 30 students $= 40 Kg \times 30= 1200 Kg$ .
Given, the average weight of 30 students and 1 teacher= (40+1) Kg=41 Kg.
Therefore, the total weight of 30 students and 1 teacher $= 41 Kg \times 31= 1271 Kg$ .
$\therefore$ the weight of the teacher= the total weight of 30 students and 1 teacher- the total weight of 30 students
$=1271 Kg- 1200 Kg =71 Kg$ .

Example 3: The average age of 5 men is 46 years and the average age of 4 of them is 43. What is the age of the fifth man?

Solution:

The average age of 5 men is 46 years.
Therefore, the sum of their ages $= 46 years \times 5=230 years$ .
The average age of 4 of the men is 43 years.
Therefore, the sum of the ages of the 4 men $=43 years \times 4=172 years$ .
Therefore, the age of the fifth man= [Sum of the ages of 5 men] – [Sum of the ages of 4 men]
=230 years- 172 years
=58 years.

Example 4: A train covers a distance of 200 Km at the speed of 50 Km/hr and the next 200 Km at the speed of 75 Km/hr. Find its average speed.

Solution:

When the speed is 50 Km/hr, the time taken to cover 200 Km
$=\dfrac{Distance}{Speed}=\dfrac{200 Km}{50 Km/hr}=4 hr$
When the speed is 75 Km/hr, the time taken to cover 200 Km
$= \dfrac{Distance}{Speed}=\dfrac{200 Km}{ 75 Km/hr}=\dfrac{8}{3} hr$
Total distance covered=200 Km+200 Km= 400 Km.
Total time taken $= \dfrac{4+8}{3} hr= \dfrac{20}{3} hr$
$\therefore$ Average speed during the whole journey

$=\dfrac{Total \: distance \: covered}{Total \: time \: taken}$

$= \dfrac{400 Km}{\frac{20}{3}}$
$=400 \times \dfrac{3}{20} Km/hr=60 Km/hr$

Example 5: Ayesha covers the first 200 Km of her journey at 40 Km/hr, the next 150 Km at 45 Km/hr and the last 400 Km at 60 Km/hr. Find her average speed.

Solution:

Time taken to cover 200 Km $= \dfrac{Distance}{Speed} = \dfrac{200 Km}{40 Km/hr} = \dfrac{200}{40} hr= 5 hr$
Time taken to cover next 150 Km $= \dfrac{Distance}{Speed} = \dfrac{150 Km}{45 Km/hr} = \dfrac{150}{45} hr= \dfrac{10}{3} hr$
Time taken to cover last 400 Km $= \dfrac{Distance}{Speed} = \dfrac{400 Km}{60 Km/hr} = \dfrac{400}{60} hr= \dfrac{20}{3} hr$ .
$\therefore$ the total time taken $= (5+\dfrac{10}{3}+\dfrac{20}{3}) hr = 15 hr$
Total distance covered $= (200+150+400) Km = 750 Km $
$\therefore$ average speed $= \dfrac{750 Km}{15 hr}= 50 Km/hr$

Exercise:

1) (i) Rahul scored 80, 137, 88, 54 and 96 runs in 5 innings. What was his average score?
(ii) The average weight of 4 men is 75 Kg. Find their total weight.

2) (i) The average height of 4 boys is 145 cm. if the heights of 3 of them are 150 cm, 142 cm and 146 cm, find the height of the fourth boy.
(ii) A class has 26 boys and 14 girls. The average weight of boys of the class is 40 Kg and the average weight of the girls of the class is 35 Kg. Find the average weight of the students of the class.

3) The average monthly income of three members of a family is Rs 8000. Another member of the family gets a job, and his monthly salary is Rs 10,000. What is the average monthly income of the four earning members of the family?

4) A boat travelled at 15 Km/hr for 4 hours and then at 12 Km/hr for 2 hours. Calculate the average speed of the boat.

5) A car travelled at 60 Km/hr for 2(1/2) hours. During the next two hours it travelled at 56Km/hr. For the last 1(1/2) hours of the journey it travelled at 50 Km/hr. Calculate the average speed of the car during the whole journey.

6) A man drives from one place to another at 60 Km/hr and returns at 40 Km/hr. Find his average speed.

7) The average runs scored by a batsman in the first four one-day matches of a five- match series is 39. How much must he score in the last match to raise his average in the series by 10?

8) Ram covers a certain distance at the speed of 40 Km/hr and comes back at the speed of 60Km/hr. Find his average speed during the entire journey.

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