Mathematics Chapter on Averages and Profit

What is the average of the first 57 natural numbers?

What price should the item have been sold for to earn a profit of 20% after losing 30% when sold for Rs. 8442?

What is the volume of metal used for a spherical shell with an outer radius of 13 cm and thickness of 3 cm? Use $rac{22}{7}$ for $ ext{pi}$.

If an item worth Rs. 10000 is sold for Rs. 8442, what is the loss percentage?

What will be the additional volume of a spherical shell if the thickness is increased to 5 cm while keeping the outer radius at 13 cm?

Finding the Average

The average of the first 57 natural numbers can be calculated by using the formula: (n+1)/2 , where n is the number of natural numbers.
In this case, n is 57, so the average would be (57+1)/2 = 29.

Calculating Profit and Loss

The selling price of the item is Rs. 8442, and the loss suffered is 30%.
To calculate the cost price, we can use the formula: Cost Price = Selling Price / (1 - Loss Percentage/100)
The cost price would be Rs. 8442 / (1-30/100) = Rs. 12060
To earn a profit of 20%, the selling price should be: Selling Price = Cost Price * (1 + Profit Percentage/100)
The required selling price would be Rs. 12060 * (1+20/100) = Rs. 14472.

Calculating Volume of a Spherical Shell

The outer radius of the shell is 13 cm, and the thickness is 3 cm.
This means the inner radius of the shell is 13 cm - 3 cm = 10 cm.
The volume of the metal used can be calculated by finding the difference between the volume of the outer sphere and the volume of the inner sphere.
The formula for the volume of a sphere is (4/3)πr³, where r is the radius.
The volume of the outer sphere is (4/3)π(13)³ = 9203 cm³
The volume of the inner sphere is (4/3)π(10)³ = 4190 cm³
The volume of the metal used is 9203 cm³ - 4190 cm³ = 5013 cm³

This quiz covers key concepts related to calculating averages, profit and loss, and the volume of a spherical shell. It includes essential formulas and calculations, providing a practical application of mathematical principles. Test your understanding and calculation skills through various problems.