Problems on Averages

 

1. Basic Problems: Finding the Average of Given Numbers

The average (arithmetic mean) of a set of numbers is calculated using the formula:

$$\text{Average}=\frac{\text{Sum of all values}}{\text{Total number of values}}$$

Example:

Find the average of 12, 18, 24, and 30.

Solution:
Sum of numbers = 12 + 18 + 24 + 30 = 84
Total numbers = 4

$$\text{Average}=\frac{84}{4}=21$$

So, the average is 21.

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2. Missing Number Problems: Finding the Missing Value When Average is Given

When the average of a group is known, and one number is missing, we use the formula:

$$\text{Total Sum}=\text{Average}\times \text{Total numbers}$$

Example:

The average of 4 numbers is 20. Three of the numbers are 18, 22, and 19. Find the missing number.

Solution:
Total sum = Average × Total numbers

$$\text{Total Sum}=20\times 4=80$$

Sum of given numbers = 18 + 22 + 19 = 59
Missing number = 80 - 59 = 21
So, the missing number is 21.

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3. Changing Average Problems: Impact on the Average When a Number Is Added/Removed

If a number is added or removed, the new average is recalculated using the updated total sum and count.

Example 1 (Adding a Number):

The average of 5 numbers is 30. A new number, 40, is added. Find the new average.

Solution:
Old total sum = 30 × 5 = 150
New total sum = 150 + 40 = 190
New number of values = 6

$$\text{New Average}=\frac{190}{6}=31.67$$

So, the new average is 31.67.

Example 2 (Removing a Number):

The average of 6 numbers is 25. If one number, 30, is removed, find the new average.

Solution:
Old total sum = 25 × 6 = 150
New total sum = 150 - 30 = 120
New number of values = 5

$$\text{New Average}=\frac{120}{5}=24$$

So, the new average is 24.

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4. Average Speed Problems

Average speed is calculated using the formula:

$$\text{Average Speed}=\frac{\text{Total Distance}}{\text{Total Time}}$$

Example:

A car travels 60 km in 2 hours and then 90 km in 3 hours. Find the average speed.

Solution:
Total distance = 60 + 90 = 150 km
Total time = 2 + 3 = 5 hours

$$\text{Average Speed}=\frac{150}{5}=30\text{ km/h}$$

So, the average speed is 30 km/h.

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5. Average Marks Problems: Calculating Class Performance

The class average is calculated by summing all student scores and dividing by the total number of students.

Example:

A class has 5 students with marks 50, 60, 70, 80, and 90. Find the average marks.

Solution:
Total marks = 50 + 60 + 70 + 80 + 90 = 350
Total students = 5

$$\text{Average Marks}=\frac{350}{5}=70$$

So, the average marks are 70.

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6. Daily Life Applications of Average

Averages are used in everyday situations to analyze and compare values.

1. Temperature:

• The average temperature of a city over a week helps understand weather patterns.

• Example: If the temperatures for a week are 28°C, 30°C, 29°C, 27°C, 31°C, 32°C, and 30°C, the average temperature is:

$$\text{Average}=\frac{28+30+29+27+31+32+30}{7}=\frac{207}{7}=29.57\mathrm{°}C$$

So, the average temperature is 29.57°C.

2. Money:

• The average monthly expenses help in budgeting.

• Example: If a person spends ₹5000 in January, ₹6000 in February, ₹5500 in March, and ₹6200 in April, the average expense is:

$$\text{Average}=\frac{5000+6000+5500+6200}{4}=\frac{22700}{4}=\text{₹}5675$$

So, the average monthly expense is ₹5675.

3. Age:

• The average age of a family or team helps in understanding group demographics.

• Example: If a family has members aged 5, 10, 35, and 40, the average age is:

$$\text{Average}=\frac{5+10+35+40}{4}=\frac{90}{4}=22.5$$

So, the average age is 22.5 years.

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Multiple Choice Questions (MCQs)

1. Basic Average Problem

The average of 10, 15, and 25 is:
a) 20
b) 15
c) 10
d) 25

Answer: a) 20

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2. Missing Number Problem

The average of 4 numbers is 50. Three numbers are 45, 55, and 60. What is the missing number?
a) 40
b) 50
c) 45
d) 60

Answer: b) 40

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3. Changing Average Problem

The average of 6 numbers is 24. If a new number 30 is added, what is the new average?
a) 24
b) 25
c) 26
d) 27

Answer: b) 25

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4. Average Speed Problem

A person travels 120 km in 2 hours and 180 km in 3 hours. What is the average speed?
a) 40 km/h
b) 50 km/h
c) 60 km/h
d) 70 km/h

Answer: c) 60 km/h

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5. Daily Life Applications

If a person spends ₹10,000 in 5 months, what is the average monthly expense?
a) ₹2000
b) ₹2500
c) ₹3000
d) ₹4000

Answer: a) ₹2000