At the beginning of day 1 of a month, Rajesh has 500 eggs. He sells 20% of the eggs by the end of the day and added b% of the eggs at the beginning of the next day and sold 20% of the eggs at the end of the next day. This pattern continued up to the end of the third day of the month when he is left with 1024 eggs. The value of b is equal to:

Given:

Rajesh starts with 500 eggs. He sells 20% of the eggs at the end of each day and adds b% of the remaining eggs at the beginning of the next day. At the end of the third day, he is left with 1024 eggs.

Formula used:

At the end of each day:

Remaining eggs after selling 20% = 80% of the eggs

New number of eggs at the beginning of the next day = 80% of the remaining eggs + b% of the remaining eggs

Calculation:

Day 1:

Rajesh starts with 500 eggs.

After selling 20%, remaining eggs = 80% of 500 = (80 / 100) × 500 = 400 eggs.

Day 2:

At the beginning of day 2, Rajesh adds b% of the remaining eggs:

Number of eggs at the start of day 2 = 400 + (b / 100) × 400 = 400(1 + b / 100)

After selling 20% of the eggs on day 2, remaining eggs = 80% of [400(1 + b / 100)] = (80 / 100) × 400(1 + b / 100) = 320(1 + b / 100)

Day 3:

At the beginning of day 3, Rajesh adds b% of the remaining eggs:

Number of eggs at the start of day 3 = 320(1 + b / 100) + (b / 100) × 320(1 + b / 100) = 320(1 + b / 100)(1 + b / 100) = 320(1 + b / 100)²

After selling 20% of the eggs on day 3, remaining eggs = 80% of 320(1 + b / 100)² = (80 / 100) × 320(1 + b / 100)² = 256(1 + b / 100)²

At the end of the third day, he is left with 1024 eggs:

256(1 + b / 100)² = 1024

Dividing both sides by 256:

(1 + b / 100)² = 4

Taking the square root of both sides:

1 + b / 100 = 2

b / 100 = 1

b = 100%

∴ The value of b is 100%.