Question 1

The number 10,30,285 in words is ten lakhs thirty thousand two hundred eighty-five, which has 42 letters. Give a 7-digit number name which has the maximum number of letters.

Accepted Answer

One such number is 77,77,777, which has 60 letters. — The question asks for a 7-digit number whose name has the maximum number of letters. The example given is 77,77,777, which when written in words has 60 letters, more than the given example of 10,30,285.

Question 2

Write a 9-digit number where exchanging any two digits results in a bigger number. How many such numbers exist?

Accepted Answer

One such number is 987654312. Exchanging last two digits, we get 987654321, which is bigger number than the initial number. Try more! — The question requires a 9-digit number such that swapping any two digits results in a bigger number. The example 987654312 satisfies this because swapping the last two digits gives 987654321, which is larger.

Question 3

Strike out 10 digits from the number 12345123451234512345 so that the remaining number is as large as possible.

Accepted Answer

5534512345 — By carefully striking out 10 digits from the given 20-digit number, the largest possible remaining number is 5534512345.

Question 4

Suppose you write down all the numbers 1, 2, 3, 4, ..., 9, 10, 11, and so on. The tenth digit you write is ‘1’ and the eleventh digit is ‘0’, as part of the number 10. (This question can be solved in different ways. One of the way is —)
(a) What would the 1000th digit be? At which number would it occur?
(b) What number would contain the millionth digit?
(c) When would you have written the digit '5' for the 5000th time?

Accepted Answer

(a) Digits from 1 to 9 = 9 digits
Digits from 10 to 99 = 90 numbers × 2 = 180 digits
Total digits from 1 to 99 = 189 digits
Remaining digits to reach 1000th digit = 1000 – 189 = 811
Number of 3-digit numbers = 811 ÷ 3 = 270 full numbers + 1 digit left over
The first 3-digit number is 100
270th 3-digit number is 100 + 270 – 1 = 369
The next number is 370.
The first digit of 370, which is 3, is the 1000th digit.

(b) The millionth digit occurs in the number 185184 + 1 = 185185.

(c) The digit '5' would have been written for the 5000th time at 13995. — The solution involves counting digits in ranges:
- 1 to 9: 9 digits
- 10 to 99: 90 numbers × 2 digits = 180 digits
- Total digits up to 99: 189
- Remaining digits to reach 1000: 811
- Each 3-digit number contributes 3 digits, so 811/3 = 270 full numbers + 1 digit
- The 1000th digit is the first digit of 370, which is 3.
For the millionth digit, similar counting leads to number 185185.
For the 5000th '5', the count reaches at number 13995.

Question 5

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:
(a) 20,800
(b) 92,100
(c) 1,20,500
(d) 65,30,000
(e) 70,25,700

Accepted Answer

(a) 20,800 = (2 × 10,000) + (8 × 100)
Total clicks = 2 + 8 = 10 button clicks.

(b) 92,100 = (9 × 10,000) + (21 × 100)
Total clicks = 9 + 21 = 30 button clicks.

(c) 1,20,500 = (12 × 10,000) + (5 × 100)
Total clicks = 12 + 5 = 17 button clicks.

(d) 65,30,000 = (653 × 10,000)
Total clicks = 653 button clicks.

(e) 70,25,700 = (702 × 10,000) + (57 × 100)
Total clicks = 759 button clicks. — The calculator can only add 10,000 or 100 per click. So, the number is expressed as a sum of multiples of 10,000 and 100. The total clicks are the sum of these multiples.

Question 6

How many lakhs make a billion?

Accepted Answer

10,000 lakh make a billion. — 1 billion = 100 crores = 10,000 lakhs.

Question 7

You are given two sets of number cards numbered from 1 - 9. Place a number card in each box below to get the (a) largest possible sum (b) smallest possible difference of two resulting numbers.

Accepted Answer

(a) Largest possible sum:
First number: 9 9 8 8 7 7 6
Second number: 6 5 5 4 4
Sum = 1,00,54,320

(b) Smallest possible difference:
First number: 1 1 2 2 3 3 4
Second number: 9 9 8 8 7
Difference = 10,22,447 — By arranging the digits to maximize the sum, the largest digits are placed in the highest place values. For the smallest difference, digits are arranged to make the numbers as close as possible.

Question 8

You are given some number cards; 4000, 13000, 300, 70000, 150000, 20, 5. Using the cards get as close as you can to the numbers below using any operation you want. Each card can be used only once for making a particular number.
(a) 1,10,000
(b) 2,00,000
(c) 5,80,000
(d) 12,45,000
(e) 20,90,800

Accepted Answer

(a) 1,10,000: 4000 × (20 + 5) + 13000 = 1,13,000
(b) 2,00,000: closest estimate = 1,50,000 + 70,000 - (4000 × 5) = 2,00,000
(c) 5,80,000: closest estimate = (1,50,000 × 4) - (4000 × 5) = 5,80,000
(d) 12,45,000: closest estimate = (70,000 × 20) - 1,50,000 - 4,000 - (300 × 5) = 12,44,500
(e) 20,90,800: closest estimate = (1,50,000 × 14) + 4,000 - 13,000 = 20,91,000 — Using the given cards and arithmetic operations, the numbers are approximated as close as possible to the target numbers, each card used only once per calculation.

Question 9

Find out how many coins should be stacked to match the height of the Statue of Unity. Assume each coin is 1 mm thick.

Accepted Answer

Height of each coin = 1 mm
Height of Statue of Unity = 180 m = 180 × 100 × 10 mm = 180,000 mm
Number of coins to be stacked = 180,000 mm ÷ 1 mm = 180,000 coins. — Convert height of Statue of Unity from meters to millimeters and divide by thickness of one coin to find the number of coins needed.

Question 10

Grey-headed Albatrosses have a roughly 7-feet wide wingspan. They are known to migrate across several oceans. Albatrosses can cover about 900 - 1000 km in a day. One of the longest single trips recorded is about 12,000 km. How many days would such a trip take to cross an ocean approximately?

Accepted Answer

Estimated number of days if it flies 900 km/day = 12,000 ÷ 900 = 13.3 days
Estimated number of days if it flies 1000 km/day = 12,000 ÷ 1000 = 12 days
Hence it would take approximately 12 to 14 days for a grey headed albatross to complete a 12,000 km trip to cross an ocean. — Divide total distance by minimum and maximum daily distance to get range of days required.

Question 11

A bar-tailed godwit holds the record for the longest recorded non-stop flight. It travelled 13,560 km from Alaska to Australia without stopping. Its journey started on 13 October 2022 and continued for about 11 days. Find out the approximate distance it covered every day. Find out the approximate distance it covered every hour.

Accepted Answer

Distance covered per day = 13,560 km ÷ 11 days = 1232.73 km or 1233 km.
Distance covered per hour = 1233 ÷ 24 = 51.36 km or 51 km. — Divide total distance by number of days to get daily distance, then divide daily distance by 24 to get hourly distance.

Question 12

Bald eagles are known to fly as high as 4500 – 6000 m above the ground level. Mount Everest is about 8850 m high. Aeroplanes can fly as high as 10,000 – 12,800 m. How many times bigger are these heights compared to Somu’s building?

Accepted Answer

Height of Somu’s building = 40 m
(i) Bald eagles’ flight height = 4500 – 6000 m
Lower estimate = 4500 ÷ 40 = 112.5 times
Upper estimate = 6000 ÷ 40 = 150 times
Bald eagles fly about 112 to 150 times higher than Somu’s building.

(ii) Mount Everest height = 8850 m
8850 ÷ 40 = 221.25
Mount Everest is approximately 221 times taller than Somu’s building.

(iii) Aeroplanes flight height = 10,000 – 12,800 m
Lower estimate = 10,000 ÷ 40 = 250
Upper estimate = 12,800 ÷ 40 = 320
Airplanes fly about 250 to 320 times as high as Somu’s building. — Divide each height by the height of Somu’s building to find how many times bigger each is.

Question 13

In the Indian numbering system, which of the following correctly represents the place value immediately after thousands?

Accepted Answer

Ten thousands — In the Indian numbering system, after thousands comes ten thousands, followed by lakhs and then crores. This system groups digits differently from the international system, using terms like lakh and crore.

Question 14

What is the correct way to place commas in the number 123456789 according to the Indian numbering system?

Accepted Answer

12,34,56,789 — The Indian numbering system places commas after the first three digits from the right and then after every two digits. So, 123456789 is written as 12,34,56,789.

Question 15

Write the number 7,89,45,321 in words using the Indian numbering system.

Accepted Answer

Seven crore eighty-nine lakh forty-five thousand three hundred twenty-one — The number 7,89,45,321 is read as seven crore eighty-nine lakh forty-five thousand three hundred twenty-one in the Indian numbering system. This involves breaking the number into crore, lakh, thousand, hundred, and remaining digits and reading each part accordingly.

Large Numbers

Large Numbers — Study Notes

Introduction

Reading and Writing Large Numbers

Comparing Large Numbers

Practice Questions — Large Numbers

All 8 Chapters in Ganita Prakash