Suppose you write down all the numbers 1, 2, 3, 4, ..., 9, 10, 11,... The tenth digit you write is '1’ and the eleventh digit Class 7

Suppose you write down all the numbers 1, 2, 3, 4, ..., 9, 10, 11,... The tenth digit you write is ‘1’ and the eleventh digit Class 7

Question 1.

Suppose you write down all the numbers 1, 2, 3, 4, ..., 9, 10, 11,... The tenth digit you write is ‘1’ and the eleventh digit is ‘0’, as part of the number 10. Class 7

(a) What would the 1000th digit be? At which number would it occur?

Solution:

9 + 2 × 90 + 3 × 270 = 999

Also, 9 + 90 + 270 = 369.

So, (369 + (1000 - 999))th number will have the 1000th digit; i.e., number 370 will have the 1000th digit and this digit will be 3.

(b) What number would contain the millionth digit?

Solution:

9 + 2 × 90 + 3 × 900 + 4 × 9000 + 5 × 90000 + 6 × 85185

= 999999

Also, 9 + 90 + 900 + 9000 + 90000 + 85185

= 185184

So, [185184 + (1000000 - 999999)]th number will have 1000000th (millionth) digit, i.e., the number will be 185185 and the digit will be 1.

(c) When would you have written the digit ‘5’ for the 5000th time?

Solution:

Digit 5 occurs 1 time from numbers 1 to 9. Digit 5 will occur 1 × 10 + 10 = 20 times from number 1 to 100. Digit 5 will occur (20 × 10 + 100) = 300 times from 1 to 999. Digit 5 will occur (300 ×× 10 + 1000) = 4000 times from numbers 1 to 9999. So, digit 5 will be written 5000th times, while writing a number more than 9999 and less than 99999.

Now, the digit 5 will occur 100 times at unit place in numbers from 10000 to 10999, 100 times at tens place in 10000 to 10999 and 100 times at hundred place in 10000 to 10999 total 300 times. Similarly, digit 5 will occur 300 times in numbers from 11000 to 11999 and also in from numbers 12000 to 12999. Then, the digit 5 will occur 100 times at units place in numbers from 13000 to 13999. Now, 4000 + 300 + 300 + 300 + 100 = 5000. So, the digit 5 will occur 5000th time at the number 13995.

Question 2.

A calculator has only ‘+10,000’ and ‘+100’ buttons. Write an expression describing the number of button clicks to be made for the following numbers : Class 7

(a) 20,800

(b) 92,100

(c) 1,20,500

(d) 65,30,000

(e) 70,25,700

Solution:

(a) 20,800 = 2 × 10000 + 8 × 100

(b) 92,100 = 9 × 10000 + 21 × 100

(c) 1,20,500 = 12 × 10000 + 5 × 100

(d) 65,30,000 = 653 × 10000

(e) 70,25,700 = 702 × 10000 + 57 × 100

Question 3.

How many lakhs make a billion?

Solution:

10000 lakhs.