Sure, let's go through these step by step. We'll break the problem into three parts as given in the question:
### Part 1: Writing the numbers in the Indian system
a. Sixteen thousand, four hundred and forty-two
- This number in figures is: 16,442
b. Five lakh, eighty-two thousand, seven hundred and eighty-one
- This number in figures is: 5,82,781
c. Ten lakh, fifty-six thousand, seven hundred and twenty-four
- This number in figures is: 10,56,724
d. Twelve lakh, seventy-five thousand, nine hundred and thirty-seven
- This number in figures is: 12,75,937
### Part 2: Writing each of the given expansions in short form
a. [tex]\( 10,000 + 3,000 + 500 + 70 + 8 \)[/tex]
- Adding these component values results in: 13,578
b. [tex]\( 30,000 + 700 + 80 + 4 \)[/tex]
- Adding these component values results in: 30,784
c. [tex]\( 60,000 + 3,000 + 900 + 10 + 5 \)[/tex]
- Adding these component values results in: 63,915
d. [tex]\( 40,000 + 0 + 300 + 80 + 2 \)[/tex]
- Adding these component values results in: 40,382
### Part 3: Writing the number names for each of the provided numbers
a. 2,31,42,541
- The number name for this is: Two crore, thirty-one lakh, forty-two thousand, five hundred and forty-one
b. 2,12,40,132
- The number name for this is: Two crore, twelve lakh, forty thousand, one hundred and thirty-two
c. 16,20,35,211
- The number name for this is: Sixteen crore, twenty lakh, thirty-five thousand, two hundred and eleven
d. 2,45,20,368
- The number name for this is: Two crore, forty-five lakh, twenty thousand, three hundred and sixty-eight
I hope this helps you understand how to write these numbers in the Indian system, convert expanded forms to their short forms, and write number names for the given figures. If you have any further questions, please feel free to ask!
