# Averages

**Averages**

In the Quantitative Ability section of the TCS NQT (National Qualifier Test), questions related to averages are common. Averages, also known as arithmetic means, are used to represent the central tendency of a set of numbers. Here's what you may encounter in this section:

__Arithmetic Mean:__

In this we find the average of a given set of numbers.

The arithmetic mean of a set of 'n' numbers is the sum of all the numbers divided by 'n'.

Example: Find the average of the numbers 10, 15, 20, 25, and 30.

Example 1: Find the arithmetic mean of the following set of numbers: 10, 15, 20, 25, and 30.

Solution: Arithmetic Mean = (Sum of numbers) / (Number of numbers) Arithmetic Mean = (10 + 15 + 20 + 25 + 30) / 5 Arithmetic Mean = 100 / 5 Arithmetic Mean = 20

The arithmetic mean of the given set of numbers is 20.

Example 2: The test scores of five students in a math exam are as follows: 85, 90, 92, 88, and 95. Calculate the arithmetic mean.

Solution: Arithmetic Mean = (Sum of scores) / (Number of students) Arithmetic Mean = (85 + 90 + 92 + 88 + 95) / 5 Arithmetic Mean = 450 / 5 Arithmetic Mean = 90

The arithmetic mean of the test scores is 90.

Example 3: The monthly salaries (in dollars) of seven employees in a company are as follows: 2500, 3000, 2800, 3200, 2700, 2900, and 3100. Calculate the arithmetic mean of their salaries.

Solution: Arithmetic Mean = (Sum of salaries) / (Number of employees) Arithmetic Mean = (2500 + 3000 + 2800 + 3200 + 2700 + 2900 + 3100) / 7 Arithmetic Mean = 20200 / 7 Arithmetic Mean ≈ 2885.71 (rounded to two decimal places)

The arithmetic mean of the monthly salaries is approximately $2885.71.

These are some examples of finding the arithmetic mean of a set of numbers. Remember that the arithmetic mean represents the central tendency of a given set of values and is commonly used in various statistical calculations and real-life scenarios.

__Weighted Average:__

Calculating the average of a set of numbers with different weights assigned to each number.

The weighted average is the sum of the products of each number and its corresponding weight, divided by the sum of the weights.

Example: The marks obtained by a student in three subjects are 80, 90, and 70, with respective weights of 2, 3, and 4. Calculate the weighted average.

Example 1: A student's final grade in a course is calculated based on three components: quizzes, assignments, and the final exam. The weights for each component are as follows: quizzes - 30%, assignments - 40%, and the final exam - 30%. If the student scores 80 out of 100 in quizzes, 90 out of 100 in assignments, and 70 out of 100 in the final exam, what is the student's weighted average grade?

Solution: To calculate the weighted average grade, we first need to find the weighted score for each component and then sum them up.

Weighted score for quizzes = (Quiz score) * (Quiz weight) = 80 * 0.30 = 24 Weighted score for assignments = (Assignment score) * (Assignment weight) = 90 * 0.40 = 36 Weighted score for the final exam = (Final exam score) * (Final exam weight) = 70 * 0.30 = 21

Weighted average grade = (Sum of weighted scores) / (Sum of weights) Weighted average grade = (24 + 36 + 21) / (0.30 + 0.40 + 0.30) = 81 / 1 = 81

The student's weighted average grade is 81.

Example 2: A store sells three types of fruits: apples, oranges, and bananas. The prices and quantities sold for each fruit are as follows: apples - $1.50 per piece (30 pieces sold), oranges - $2.00 per piece (20 pieces sold), and bananas - $0.75 per piece (50 pieces sold). Calculate the weighted average price.

Solution: To find the weighted average price, we first need to find the total revenue generated by each fruit and then sum them up.

Total revenue from apples = (Price per apple) * (Quantity sold) = $1.50 * 30 = $45 Total revenue from oranges = (Price per orange) * (Quantity sold) = $2.00 * 20 = $40 Total revenue from bananas = (Price per banana) * (Quantity sold) = $0.75 * 50 = $37.50

Total revenue from all fruits = $45 + $40 + $37.50 = $122.50

Total quantity sold = 30 + 20 + 50 = 100 pieces

Weighted average price of fruits = (Total revenue from all fruits) / (Total quantity sold) = $122.50 / 100 = $1.225

The weighted average price of the fruits is $1.225 (rounded to two decimal places).

__Average Speed:__

Finding the average speed of a moving object over a given distance and time.

Example 1:

A car travels from City A to City B, a distance of 250 kilometers, in 5 hours. Find the average speed.

Solution:

Average Speed of the car = (Total Distance covered by the car) / (Total Time taken)

Average Speed = 250 km / 5 hours

Average Speed = 50 km/h

The average speed of the car is 50 kilometers per hour.

Example 2:

A cyclist covers a distance of 60 kilometers in 3 hours, and then he covers another distance of 80 kilometers in 4 hours. Find the overall average speed of the cyclist for the entire journey.

Solution:

To find the overall average speed, we need to consider the total distance covered and the total time taken for the entire journey.

Total Distance = 60 km + 80 km = 140 km

Total Time = 3 hours + 4 hours = 7 hours

Overall Average Speed = (Total Distance covered by cyclist) / (Total Time taken)

Overall Average Speed = 140 km / 7 hours

Overall Average Speed = 20 km/h

The overall average speed of the cyclist for the entire journey is 20 kilometers per hour.

Note: Average speed is different from the average of speeds. Average speed is the total distance covered divided by the total time taken for the entire journey, while the average of speeds would involve finding the average of individual speeds at different points during the journey.

__Average Age:__

Problems related to the average age of a group of individuals.

Example 1:

In a family, there are five members with ages as follows: 20, 22, 24, 18, and 26 years. Calculate their average age

Solution:

Average Age = (Sum of ages) / (Number of members)

Average Age = (20 + 22 + 24 + 18 + 26) / 5

Average Age = 110 / 5

Average Age = 22 years

The average age of the family members is 22 years.

Example 2:

In a school classroom, there are 30 students. The ages of 29 students are as follows: 10, 11, 9, 12, 11, 13, 10, 9, 12, 11, 14, 10, 12, 9, 11, 10, 13, 14, 12, 9, 11, 12, 10, 9, 13, 14, 12, 10, 11, 9. Find the age of the 30th student, so that the average age of all students in the classroom becomes 11 years.

Solution:

To find the age of the 30th student, we need to consider the sum of the ages of all 30 students and then calculate the age of the 30th student to achieve an average of 11 years.

Let the age of the 30th student be 'x'.

Average Age = (Sum of ages of students) / (Number of students)

11 = (Sum of ages of 29 students + x) / 30

Now, substitute the given ages of the 29 students:

11 = (10 + 11 + 9 + 12 + 11 + 13 + 10 + 9 + 12 + 11 + 14 + 10 + 12 + 9 + 11 + 10 + 13 + 14 + 12 + 9 + 11 + 12 + 10 + 9 + 13 + 14 + 12 + 10 + 11 + 9 + x) / 30

Now, solve for 'x':

330 = (Sum of ages of 29 students) + x

330 = 319 + x

x = 330 - 319

x = 11

The age of the 30th student is 11 years to maintain an average age of 11 years for all students in the classroom.

__Averages in Combination with Other Concepts:__

Averages may be combined with other mathematical concepts such as percentages, ratios, and proportions.

Example: The average salary of a group of employees increased by 10%. If the average salary before the increase was $40,000, what is the average salary after the increase?

These are just some examples of the types of average-related questions you might encounter in the TCS NQT Quantitative Ability section. Practicing with various sample questions and previous year's papers will help you become familiar with the types of problems and improve your performance in this section.