Average and Ages

🔹 Average

1. What is Average?

The average is a way to find the central or typical value in a set of numbers. It helps summarize the entire data with a single representative number.

2. How to Calculate Average?

Add all the numbers together, then divide by the total count of numbers.

$$\text{Average}=\frac{\text{Sum of all numbers}}{\text{Total count of numbers}}$$

3. Types of Averages

• Mean (most common average)

• Median (middle value)

• Mode (most frequent value)

📌 Examples of Average:

Suppose the marks scored by 5 students in a test are: 70, 75, 80, 85, and 90.

• Mean:

$$\frac{70+75+80+85+90}{5}=\frac{400}{5}=80$$

• Median:
  Arrange in order: 70, 75, 80, 85, 90
  Middle value (3rd) = 80

• Mode:
  All values occur once, so no mode here.

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🔹 Ages

1. What Are Age Problems?

Age problems involve finding the age of a person or thing using clues, such as ratios or differences given in a problem.

2. How to Solve?

Use algebraic equations, ratios, or proportions based on the information given.

3. Why Important?

They are common in real life — for example, figuring out someone's current age, or the difference between ages.

📌 Example of Age Problem:

Problem:
A father is 36 years old, and his son is 12 years old. After how many years will the father’s age be twice the son’s age?

Solution:
Let the number of years be $x$.

After $x$ years:
Father’s age = $36 + x$
Son’s age = $12 + x$

Set up equation:

$$36+x=2(12+x)$$$$36+x=24+2x$$$$36-24=2x-x$$$$12=x$$

Answer: After 12 years.

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🔹 Mean, Median, and Mode: Measures of Central Tendency

Mean

• Definition: The sum of all values divided by the total number of values.

• Formula:

$$\text{Mean}=\frac{\text{Sum of all values}}{\text{Number of values}}$$

Median

• Definition: The middle value when all values are arranged in order.

• How to find:

  • If odd number of values → middle value

  • If even number → average of two middle values

Mode

• Definition: The value that appears most frequently in the dataset.

• How to find: Look for the value with the highest frequency.

📌 More Examples

• Dataset: 4, 8, 6, 8, 10

  • Mean:

  $$\frac{4+8+6+8+10}{5}=\frac{36}{5}=7.2$$

• Median: Arrange → 4, 6, 8, 8, 10 → middle = 8

• Mode: 8 (appears twice)

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Here are 5 MCQs based on the topic Average, Ages, Mean, Median, and Mode, complete with answers and explanations:

MCQ 1: Average Calculation

Q. The average of 5 numbers is 18. If four of the numbers are 15, 20, 18, and 22, what is the fifth number?

A) 15
B) 17
C) 18
D) 19

Answer: ✅ D) 19

Explanation:
Total sum = Average × Number of values = 18 × 5 = 90
Sum of 4 numbers = 15 + 20 + 18 + 22 = 75
Fifth number = 90 – 75 = 15

Oops! Correction here, based on options — the fifth number is:

90 – 75 = 15 → Answer: A) 15

Corrected Answer: ✅ A) 15

MCQ 2: Age Problem

Q. A person is 24 years older than his son. After 6 years, the father will be twice as old as the son. What is the son's current age?

A) 12 years
B) 18 years
C) 20 years
D) 24 years

Answer: ✅ B) 18 years

Explanation:
Let son's age = x. Then father's age = x + 24
After 6 years:
Father’s age = x + 24 + 6 = x + 30
Son’s age = x + 6
Now, x + 30 = 2(x + 6)
x + 30 = 2x + 12 → 30 - 12 = 2x - x → x = 18

MCQ 3: Mean of Data Set

Q. What is the mean of the numbers: 10, 15, 20, 25, 30?

A) 20
B) 22
C) 18
D) 25

Answer: ✅ A) 20

Explanation:
Sum = 10 + 15 + 20 + 25 + 30 = 100
Mean = 100 ÷ 5 = 20

MCQ 4: Median

Q. What is the median of the dataset: 9, 5, 12, 7, 3?

A) 7
B) 5
C) 9
D) 12

Answer: ✅ A) 7

Explanation:
Arrange in order: 3, 5, 7, 9, 12
Median = middle value = 7

MCQ 5: Mode

Q. Find the mode in the following list: 4, 2, 7, 4, 9, 2, 4, 1

A) 2
B) 4
C) 7
D) 1

Answer: ✅ B) 4

Explanation:
Frequency of each number:

• 4 → 3 times

• 2 → 2 times

• Others → once
  So, mode = 4