NCERT Solutions for Class 7 Maths Chapter 1 Large Numbers Around Us – step-by-step answers to every exercise and in-text question of the chapter, as per the latest NCERT syllabus for 2027. Read them online or download the free question-answer PDF.
Chapter 1 of Ganita Prakash (Grade 7, Part I) starts with a farmer buying rice seeds and ends with numbers in the crores. On the way you meet lakhs, crores and arabs, the Indian and American comma systems, rounding, quick multiplication tricks and a set of calculators called Thoughtful Thousands, Tedious Tens, Handy Hundreds, Creative Chitti and Systematic Sippy.
NCERT Solutions for Class 7 Maths Chapter 1 Large Numbers Around Us PDF
All the answers are on this page below. The PDF also explains every answer – why it is right and how to check it – so download it for revision.
NCERT Solutions for Class 7 Maths Chapter 1 Large Numbers Around Us – Questions and Answers
All 83 questions of the chapter (textbook pages 1 – 22) are answered below, in the order of the book – in-text questions, activities and the end-of-chapter exercises. Each answer is written the way you would write it in the exam.
In-text Questions — Pages 1–21.1 A Lakh Varieties!
Q1Estu wondered, “One lakh! So far I have only tasted 3 varieties. If we tried a new variety each day, would we even come close to tasting all the varieties in a lifetime of 100 years?” What do you think? Guess.
Answer
No — not even close. One new variety a day for 100 years gives only about 36,500 varieties, which is roughly a third of a lakh.
Days in 1 year = 365 (ignoring leap years) Days in 100 years = 365 × 100 = 36,500 Varieties tasted = 36,500 × 1 = 36,500 Still needed = 1,00,000 − 36,500 = 63,500 varieties
Q2But how much is one lakh? Observe the pattern and fill in the boxes given below.Page 2 — the chain of statements, and the ten boxes that count on from 99,995.
Answer
Each arrow is simply “+ 1”, so every largest number is one step below the next smallest number.
Statement
Number
The largest 3-digit number is
999
The smallest 4-digit number is
1,000
The largest 4-digit number is
9,999
The smallest 5-digit number is
10,000
The largest 5-digit number is
99,999
The smallest 6-digit number is
1,00,000
The chain of ten boxes below it is just counting on by 1:
Q3Roxie suggests, “What if we ate 2 varieties of rice every day? Would we then be able to eat 1 lakh varieties of rice in 100 years?” What if a person ate 3 varieties of rice every day? Will they be able to taste all the lakh varieties in a 100 year lifetime? Find out.
Answer
2 varieties a day: No — only 73,000 varieties in 100 years. 3 varieties a day: Yes — 1,09,500 varieties, which is more than one lakh.
Days in 100 years = 365 × 100 = 36,500 2 a day: 36,500 × 2 = 73,000 < 1,00,000 (27,000 short) 3 a day: 36,500 × 3 = 1,09,500 > 1,00,000 (9,500 extra)
In-text Questions — Page 31.1 A Lakh Varieties!
Q1Choose a number for y. How close to one lakh is the number of days in y years, for the y of your choice?
Answer
Take y = 274 — it is the value that lands almost exactly on one lakh.
Days in y years = 365 × y 365 × 274 = 365 × 200 + 365 × 74 = 73,000 + 27,010 = 1,00,010 days 1,00,010 − 1,00,000 = only 10 days more than a lakh
A few other choices:
y (years)
Days = 365 × y
Compared with 1,00,000
50
18,250
81,750 less
100
36,500
63,500 less
200
73,000
27,000 less
274
1,00,010
only 10 more
300
1,09,500
9,500 more
Figure it Out — Page 31.1 A Lakh Varieties!
Q1According to the 2011 Census, the population of the town of Chintamani was about 75,000. How much less than one lakh is 75,000?
Answer
75,000 is 25,000 less than one lakh.
One lakh = 1,00,000 1,00,000 − 75,000 = 25,000
Q2The estimated population of Chintamani in the year 2024 is 1,06,000. How much more than one lakh is 1,06,000?
Answer
1,06,000 is 6,000 more than one lakh.
1,06,000 − 1,00,000 = 6,000
Q3By how much did the population of Chintamani increase from 2011 to 2024?
Answer
The population went up by 31,000.
Population in 2024 = 1,06,000 Population in 2011 = 75,000 Increase = 1,06,000 − 75,000 = 31,000 people
In-text Questions — Page 3 (Getting a Feel of Large Numbers)1.1 A Lakh Varieties!
Q1Look at the picture on the right. Somu is 1 metre tall. If each floor is about four times his height, what is the approximate height of the building?Page 3 — the building beside the text; Somu waves from a balcony.
Answer
The building in the picture has 10 floors, so its height is about 40 metres.
Somu’s height = 1 m Height of one floor = 4 × 1 m = 4 m Number of floors in the picture = 10 Height of building = 10 × 4 m = 40 m
Q2Which is taller — The Statue of Unity or this building? How much taller? ____________ m. (The Statue of Unity is about 180 m tall.)
Answer
The Statue of Unity is taller, by 140 m.
Height of Statue of Unity = 180 m Height of Somu’s building = 40 m Difference = 180 − 40 = 140 m
Q3How much taller is the Kunchikal waterfall than Somu’s building? ____________ m. (The Kunchikal waterfall drops about 450 m.)
Answer
The waterfall is 410 m taller than the building.
Height of Kunchikal waterfall = 450 m Height of Somu’s building = 40 m Difference = 450 − 40 = 410 m
Q4How many floors should Somu’s building have to be as high as the waterfall? ____________ .
Answer
About 113 floors.
Height of waterfall = 450 m Height of one floor = 4 m Number of floors = 450 ÷ 4 = 112.5 = 113 floors (you cannot build half a floor, so round up)
In-text Questions — Pages 4–5Reading and Writing Numbers
Q1How do you view a lakh — is a lakh big or small?
Answer
Both — it depends on what you are counting. A lakh is big for slow things and small for crowded things.
A lakh feels BIG when…
A lakh feels SMALL when…
1 lakh days = 274 years of living
1 lakh people fit in the Ahmedabad cricket stadium
1 lakh people in a line stretch 38 km
80,000–1,20,000 hairs sit on one head
1 lakh varieties of rice is a huge treasure of seeds
One female fish can lay nearly a lakh eggs at a time
Q2Write each of the numbers given below in words: (a) 3,00,600 (b) 5,04,085 (c) 27,30,000 (d) 70,53,138
Answer
Read the groups from the left: crore, lakh, thousand, then the last three digits.
Number
Number name (Indian system)
(a) 3,00,600
Three lakh six hundred
(b) 5,04,085
Five lakh four thousand eighty five
(c) 27,30,000
Twenty seven lakh thirty thousand
(d) 70,53,138
Seventy lakh fifty three thousand one hundred thirty eight
Q3Write the corresponding number in the Indian place value system for each of the following: (a) One lakh twenty three thousand four hundred and fifty six (b) Four lakh seven thousand seven hundred and four (c) Fifty lakhs five thousand and fifty (d) Ten lakhs two hundred and thirty five
Answer
Write each group in its own place, and fill any missing group with zeroes.
Number name
Number
(a) One lakh twenty three thousand four hundred and fifty six
1,23,456
(b) Four lakh seven thousand seven hundred and four
4,07,704
(c) Fifty lakhs five thousand and fifty
50,05,050
(d) Ten lakhs two hundred and thirty five
10,00,235
(d) Ten lakhs = 10,00,000 two hundred thirty five = 235 10,00,000 + 235 = 10,00,235
Land of Tens — Pages 5–61.2 Land of Tens
Q1The Thoughtful Thousands only has a +1000 button. How many times should it be pressed to show: (a) Three thousand? 3 times (b) 10,000? (c) Fifty three thousand? (d) 90,000? (e) One Lakh? (f) ______? 153 times (g) How many thousands are required to make one lakh?
Answer
Every press adds 1000, so the number of presses is simply the number divided by 1000.
Q2The Tedious Tens only has a +10 button. How many times should it be pressed to show: (a) Five hundred? (b) 780? (c) 1000? (d) 3700? (e) 10,000? (f) One lakh? (g) ______? 435 times
Answer
Divide each number by 10.
Target
Working
Presses
(a) Five hundred
500 ÷ 10
50 times
(b) 780
780 ÷ 10
78 times
(c) 1000
1000 ÷ 10
100 times
(d) 3700
3700 ÷ 10
370 times
(e) 10,000
10000 ÷ 10
1000 times
(f) One lakh
1,00,000 ÷ 10
10,000 times
(g) 4,350
435 × 10
435 times
Q3The Handy Hundreds only has a +100 button. How many times should it be pressed to show: (a) Four hundred? (b) 3,700? (c) 10,000? (d) Fifty three thousand? (e) 90,000? (f) 97,600? (g) 1,00,000? (h) ______? 582 times (i) How many hundreds are required to make ten thousand? (j) How many hundreds are required to make one lakh? (k) Handy Hundreds says, “There are some numbers which Tedious Tens and Thoughtful Thousands can’t show but I can.” Is this statement true? Think and explore.
Answer
Divide each number by 100.
Target
Working
Presses
(a) Four hundred
400 ÷ 100
4 times
(b) 3,700
3700 ÷ 100
37 times
(c) 10,000
10000 ÷ 100
100 times
(d) Fifty three thousand
53000 ÷ 100
530 times
(e) 90,000
90000 ÷ 100
900 times
(f) 97,600
97600 ÷ 100
976 times
(g) 1,00,000
100000 ÷ 100
1000 times
(h) 58,200
582 × 100
582 times
(i) 10,000 ÷ 100 = 100 hundreds make ten thousand. (j) 1,00,000 ÷ 100 = 1000 hundreds make one lakh.
(k)No, the statement is not true.
Handy Hundreds can show only multiples of 100. Every multiple of 100 is also a multiple of 10. e.g. 3,700 = 37 × 100 = 370 × 10 So Tedious Tens can show every number Handy Hundreds can show.
Q4Creative Chitti is a different kind of calculator. It has the following buttons: +1, +10, +100, +1000, +10000, +100000 and +1000000. It always has multiple ways of doing things. To get the number 321, it presses +10 thirty two times and +1 once. Will it get 321? Alternatively, it can press +100 two times and +10 twelve times and +1 once.
Q5Two of the many different ways to get 5072 are shown below.
Buttons
5072
+10,00,000
+1,00,000
+10,000
+1,000
3
+100
50
20
+10
7
+1
2
72
These two ways can be expressed as: (a) (50 × 100) + (7 × 10) + (2 × 1) = 5072 (b) (3 × 1000) + (20 × 100) + (72 × 1) = 5072. Find a different way to get 5072 and write an expression for the same.
Answer
Here are three fresh ways, each different from the two given.
Q1For each number given below, write expressions for at least two different ways to obtain the number through button clicks. Think like Chitti and be creative. (a) 8300 (b) 40629 (c) 56354 (d) 66666 (e) 367813
Answer
The first way in each pair uses the digits of the number itself; the second trades one big button for ten smaller ones.
Q1Creative Chitti has some questions for you — (a) You have to make exactly 30 button presses. What is the largest 3-digit number you can make? What is the smallest 3-digit number you can make?
Answer
Largest = 993. Smallest = 102.
Largest: put as many presses as possible on +100. (9 × 100) + (8 × 10) + (13 × 1) = 900 + 80 + 13 = 993 Presses = 9 + 8 + 13 = 30 ✓
Smallest: put as few presses as possible on +100. (0 × 100) + (8 × 10) + (22 × 1) = 0 + 80 + 22 = 102 Presses = 0 + 8 + 22 = 30 ✓
Q2997 can be made using 25 clicks. Can you make 997 with a different number of clicks?
Answer
Yes — 997 can be made with many different click counts.
Q3Systematic Sippy has the buttons +1, +10, +100, +1000, +10000, +100000. It wants to be used as minimally as possible. How can we get the numbers (a) 5072, (b) 8300 using as few button clicks as possible? Also: the table below shows one way to get 5072 using 23 button clicks.
Buttons
5072
+10,00,000
+1,00,000
+10,000
+1,000
5
+100
0
+10
6
+1
12
Is there another way to get 5072 using less than 23 button clicks? Write the expression for the same.
Answer
Yes. 5072 can be made in just 14 clicks, and 8300 in 11 clicks.
Q1Read the following numbers in Indian place value notation and write their number names in both the Indian and American systems: (a) 4050678 (b) 48121620 (c) 20022002 (d) 246813579 (e) 345000543 (f) 1020304050
Answer
Place the commas first — 3-2-2-2 from the right for the Indian system, 3-3-3 for the American system.
Number
Indian system
American system
(a) 4050678
40,50,678 Forty lakh fifty thousand six hundred seventy eight
4,050,678 Four million fifty thousand six hundred seventy eight
(b) 48121620
4,81,21,620 Four crore eighty one lakh twenty one thousand six hundred twenty
48,121,620 Forty eight million one hundred twenty one thousand six hundred twenty
(c) 20022002
2,00,22,002 Two crore twenty two thousand two
20,022,002 Twenty million twenty two thousand two
(d) 246813579
24,68,13,579 Twenty four crore sixty eight lakh thirteen thousand five hundred seventy nine
246,813,579 Two hundred forty six million eight hundred thirteen thousand five hundred seventy nine
(e) 345000543
34,50,00,543 Thirty four crore fifty lakh five hundred forty three
345,000,543 Three hundred forty five million five hundred forty three
(f) 1020304050
1,02,03,04,050 One arab two crore three lakh four thousand fifty
1,020,304,050 One billion twenty million three hundred four thousand fifty
Q2Write the following numbers in Indian place value notation: (a) One crore one lakh one thousand ten (b) One billion one million one thousand one (c) Ten crore twenty lakh thirty thousand forty (d) Nine billion eighty million seven hundred thousand six hundred
Answer
For (b) and (d), first build the number in the American system, then re-comma it the Indian way.
Number name
Indian place value notation
(a) One crore one lakh one thousand ten
1,01,01,010
(b) One billion one million one thousand one
1,00,10,01,001
(c) Ten crore twenty lakh thirty thousand forty
10,20,30,040
(d) Nine billion eighty million seven hundred thousand six hundred
9,08,07,00,600
(b) 1 billion = 1,000,000,000 1 million = 1,000,000 1 thousand = 1,000, and one = 1 Total = 1,001,001,001 Indian commas → 1,00,10,01,001
In-text Questions — Page 101.4 Exact and Approximate Values
Q1What do you think of this conversation? Have you read or heard such headlines or statements? (“1 lakh people visited the book fair.” “If I had not gone, they would have written 99,999 people visited the book fair last week.”)
Answer
The joke is that the headline number was never meant to be exact — it is a rounded figure.
Headlines of this kind are everywhere:
“Over 50,000 devotees at the temple on Ekadashi.”
“Nearly 1 crore people watched the match.”
“About 2 lakh vehicles cross the bridge every day.”
Q2Think and share situations where it is appropriate to (a) round up, (b) round down, (c) either rounding up or rounding down is okay and (d) when exact numbers are needed.
Answer
Round the way that keeps you safe from a shortfall.
Situation
Example
Why
(a) Round up
A school with 732 people orders 750 sweets; buying chairs for 118 guests → order 125
Running short is worse than having a little extra
(b) Round down
A shopkeeper says an item costing ₹470 is “around ₹450”; a bag holds 27 kg → “about 25 kg”
A modest claim is safer than an exaggerated one
(c) Either is fine
Distance between two towns, 268 km → “about 270 km”; population of a city
Only a rough idea of size is needed
(d) Exact numbers
Money in a bank account, marks in an exam, votes counted, a phone number, medicine dose
Even a difference of 1 changes the meaning completely
Q1Similarly, write the five nearest neighbours for these numbers: (a) 3,87,69,957 (b) 29,05,32,481
Answer
Look only at the digits to the right of the place you are rounding to: 5 or more rounds up, less than 5 rounds down.
Nearest…
(a) 3,87,69,957
(b) 29,05,32,481
thousand
3,87,70,000
29,05,32,000
ten thousand
3,87,70,000
29,05,30,000
lakh
3,88,00,000
29,05,00,000
ten lakh
3,90,00,000
29,10,00,000
crore
4,00,00,000
29,00,00,000
(a) last 3 digits 957 ≥ 500 → thousand rounds up to 3,87,70,000 last 4 digits 9957 ≥ 5000 → ten thousand also 3,87,70,000 last 5 digits 69,957 ≥ 50,000 → lakh up to 3,88,00,000 last 6 digits 7,69,957 ≥ 5,00,000 → ten lakh up to 3,90,00,000 last 7 digits 87,69,957 ≥ 50,00,000 → crore up to 4,00,00,000
Q2I have a number for which all five nearest neighbours are 5,00,00,000. What could the number be? How many such numbers are there?
Answer
The number must lie between 4,99,99,500 and 5,00,00,499, and there are exactly 1000 such whole numbers.
Nearest crore = 5,00,00,000 → number lies in 4,50,00,000 to 5,49,99,999 Nearest ten lakh = 5,00,00,000 → 4,95,00,000 to 5,04,99,999 Nearest lakh = 5,00,00,000 → 4,99,50,000 to 5,00,49,999 Nearest ten thousand = 5,00,00,000 → 4,99,95,000 to 5,00,04,999 Nearest thousand = 5,00,00,000 → 4,99,99,500 to 5,00,00,499
Estimating Sums and Differences — Pages 11–121.4 Exact and Approximate Values
Q14,63,128 + 4,19,682. Roxie: “The sum is near 8,00,000 and is more than 8,00,000.” Estu: “The sum is near 9,00,000 and is less than 9,00,000.” (a) Are these estimates correct? Whose estimate is closer to the sum? (b) Will the sum be greater than 8,50,000 or less than 8,50,000? Why do you think so? (c) Will the sum be greater than 8,83,128 or less than 8,83,128? Why do you think so? (d) Exact value of 4,63,128 + 4,19,682 = ___________
Answer
(a) Both estimates are correct, but Estu’s is closer.
Exact sum = 4,63,128 + 4,19,682 = 8,82,810 Distance from 8,00,000 = 82,810 (Roxie) Distance from 9,00,000 = 17,190 (Estu) 17,190 < 82,810 → Estu is closer
(b) The sum is greater than 8,50,000.
4,63,128 > 4,50,000 and 4,19,682 > 4,00,000 So the sum > 4,50,000 + 4,00,000 = 8,50,000 ✓
(c) The sum is less than 8,83,128.
8,83,128 = 4,63,128 + 4,20,000 But the second number is 4,19,682, which is 318 less than 4,20,000 So the sum = 8,83,128 − 318 = 8,82,810 < 8,83,128
(d) Exact value = 8,82,810.
Q214,63,128 – 4,90,020. Roxie: “The difference is near 10,00,000 and is less than 10,00,000.” Estu: “The difference is near 9,00,000 and is more than 9,00,000.” (a) Are these estimates correct? Whose estimate is closer to the difference? (b) Will the difference be greater than 9,50,000 or less than 9,50,000? Why do you think so? (c) Will the difference be greater than 9,63,128 or less than 9,63,128? Why do you think so? (d) Exact value of 14,63,128 – 4,90,020 = __________
Answer
(a) Both are correct, but Roxie’s estimate is closer.
Exact difference = 14,63,128 − 4,90,020 = 9,73,108 Distance from 10,00,000 = 26,892 (Roxie) Distance from 9,00,000 = 73,108 (Estu) 26,892 < 73,108 → Roxie is closer
(b) The difference is greater than 9,50,000.
14,63,128 − 5,00,000 = 9,63,128 We are actually subtracting 4,90,020, which is less than 5,00,000 Subtracting less leaves more, so difference > 9,63,128 > 9,50,000 ✓
(c) The difference is greater than 9,63,128.
9,63,128 = 14,63,128 − 5,00,000 5,00,000 − 4,90,020 = 9,980 So the difference = 9,63,128 + 9,980 = 9,73,108
(d) Exact value = 9,73,108.
Populations of Cities — Page 131.4 Exact and Approximate Values
Q1Observe the populations of some Indian cities in the table below.
Rank
City
Population (2011)
Population (2001)
1
Mumbai
1,24,42,373
1,19,78,450
2
New Delhi
1,10,07,835
98,79,172
3
Bengaluru
84,25,970
43,01,326
4
Hyderabad
68,09,970
36,37,483
5
Ahmedabad
55,70,585
35,20,085
6
Chennai
46,81,087
43,43,645
7
Kolkata
44,86,679
45,72,876
8
Surat
44,67,797
24,33,835
9
Vadodara
35,52,371
16,90,000
10
Pune
31,15,431
25,38,473
11
Jaipur
30,46,163
23,22,575
12
Lucknow
28,15,601
21,85,927
13
Kanpur
27,67,031
25,51,337
14
Nagpur
24,05,665
20,52,066
15
Indore
19,60,631
14,74,968
16
Thane
18,18,872
12,62,551
17
Bhopal
17,98,218
14,37,354
18
Visakhapatnam
17,28,128
13,45,938
19
Pimpri-Chinchwad
17,27,692
10,12,472
20
Patna
16,84,222
13,66,444
From the information given in the table, answer the following questions by approximation: What is your general observation about this data? Share it with the class.
Answer
Almost every city in the table grew between 2001 and 2011 — several of them enormously — and only one shrank.
The top two, Mumbai and New Delhi, are the only cities above 1 crore (1,24,42,373 and 1,10,07,835 in 2011).
Bengaluru, Hyderabad, Surat and Vadodara nearly doubled in ten years.
Kolkata is the only city that lost people — it fell from 45,72,876 to 44,86,679, a drop of about 86,000.
Mumbai grew by only about 4.6 lakh, far less than Bengaluru’s 41 lakh, so being the biggest does not mean growing the fastest.
All 20 cities in the table had more than 16 lakh people in 2011.
Q2
Rank
City
Population (2011)
Population (2001)
1
Mumbai
1,24,42,373
1,19,78,450
2
New Delhi
1,10,07,835
98,79,172
3
Bengaluru
84,25,970
43,01,326
4
Hyderabad
68,09,970
36,37,483
5
Ahmedabad
55,70,585
35,20,085
6
Chennai
46,81,087
43,43,645
7
Kolkata
44,86,679
45,72,876
8
Surat
44,67,797
24,33,835
9
Vadodara
35,52,371
16,90,000
10
Pune
31,15,431
25,38,473
11
Jaipur
30,46,163
23,22,575
12
Lucknow
28,15,601
21,85,927
13
Kanpur
27,67,031
25,51,337
14
Nagpur
24,05,665
20,52,066
15
Indore
19,60,631
14,74,968
16
Thane
18,18,872
12,62,551
17
Bhopal
17,98,218
14,37,354
18
Visakhapatnam
17,28,128
13,45,938
19
Pimpri-Chinchwad
17,27,692
10,12,472
20
Patna
16,84,222
13,66,444
What is an appropriate title for the above table?
Answer
A good title names the what, the where and the when. For example:
“Population of the 20 largest Indian cities, 2001 and 2011”
“Census populations of major Indian cities (2001 vs 2011)”
“Twenty most populous Indian cities and how they grew, 2001–2011”
Q3How much is the population of Pune in 2011? Approximately, by how much has it increased compared to 2001?
Answer
Pune’s 2011 population is 31,15,431, an increase of about 6 lakh over 2001.
Population in 2011 = 31,15,431 Population in 2001 = 25,38,473 Increase = 31,15,431 − 25,38,473 = 5,76,958 Rounded to the nearest lakh = about 6,00,000 (6 lakh)
Q4Which city’s population increased the most between 2001 and 2011?
Answer
Bengaluru, by 41,24,644 people — more than 41 lakh.
City
2011
2001
Increase
Bengaluru
84,25,970
43,01,326
41,24,644
Hyderabad
68,09,970
36,37,483
31,72,487
Ahmedabad
55,70,585
35,20,085
20,50,500
Surat
44,67,797
24,33,835
20,33,962
Vadodara
35,52,371
16,90,000
18,62,371
Mumbai
1,24,42,373
1,19,78,450
4,63,923
Q5Are there cities whose population has almost doubled? Which are they?
Answer
Yes — Bengaluru, Hyderabad, Surat and Vadodara. Vadodara has more than doubled.
City
2001
2011
Roughly how many times
Vadodara
16,90,000
35,52,371
about 2.1 times
Bengaluru
43,01,326
84,25,970
about 2 times
Hyderabad
36,37,483
68,09,970
about 1.9 times
Surat
24,33,835
44,67,797
about 1.8 times
Pimpri-Chinchwad
10,12,472
17,27,692
about 1.7 times
Q6By what number should we multiply Patna’s population to get a number/population close to that of Mumbai?
In-text Questions — Pages 14–15 (How Long is the Product?)1.5 Patterns in Products
Math Talk
Q1In each of the following boxes, the multiplications produce interesting patterns. Evaluate them to find the pattern. Extend the multiplications based on the observed pattern.Page 14 — the four boxes of multiplications.
Answer
Work each one out and the pattern jumps out.
Multiplication
Product
Multiplication
Product
11 × 11
121
66 × 61
4,026
111 × 111
12,321
666 × 661
4,40,226
1111 × 1111
12,34,321
6666 × 6661
4,44,02,226
Extend: 11111 × 11111
12,34,54,321
Extend: 66666 × 66661
4,44,40,22,226
3 × 5
15
101 × 101
10,201
33 × 35
1,155
102 × 102
10,404
333 × 335
1,11,555
103 × 103
10,609
Extend: 3333 × 3335
1,11,15,555
Extend: 104 × 104
10,816
Patterns: 1’s: the product counts up and back down — 1, 121, 12321, 1234321, 123454321 6’s: 4026 → 440226 → 44402226 — one extra 4, one extra 2 each time 3 & 5: 15 → 1155 → 111555 — as many 1’s as 5’s 101, 102, 103: 10201, 10404, 10609 — the middle jumps 2, 4, 6 and the end 1, 4, 9 (the squares)
Q2Observe the number of digits in the two numbers being multiplied and their product in each case. Is there any connection between the numbers being multiplied and the number of digits in their product?
Answer
Yes. The number of digits in the product is either the total number of digits in the two numbers, or one less than that. If one number has m digits and the other has n digits, the product has (m + n − 1) or (m + n) digits.
Multiplication
m + n
Digits in product
11 × 11 = 121
2 + 2 = 4
3 = m + n − 1
1111 × 1111 = 12,34,321
4 + 4 = 8
7 = m + n − 1
66 × 61 = 4,026
2 + 2 = 4
4 = m + n
666 × 661 = 4,40,226
3 + 3 = 6
6 = m + n
3 × 5 = 15
1 + 1 = 2
2 = m + n
Q3Roxie says that the product of two 2-digit numbers can only be a 3- or a 4-digit number. Is she correct?
Answer
Yes, Roxie is correct.
Smallest 2-digit × smallest 2-digit = 10 × 10 = 100 → 3 digits Largest 2-digit × largest 2-digit = 99 × 99 = 9,801 → 4 digits So every such product lies between 100 and 9,801 → it must have 3 or 4 digits
Q4Should we try all possible multiplications with 2-digit numbers to tell whether Roxie’s claim is true? Or is there a better way to find out?
Answer
There is a far better way — check only the smallest and the largest case. There is no need to test all 8,100 pairs.
All 2-digit numbers lie between 10 and 99. So every product lies between 10 × 10 = 100 and 99 × 99 = 9,801. Both ends have 3 or 4 digits → all products do
Q5Can multiplying a 3-digit number with another 3-digit number give a 4-digit number?
Answer
No. The product always has 5 or 6 digits.
Smallest case: 100 × 100 = 10,000 → 5 digits Largest case: 999 × 999 = 9,98,001 → 6 digits The smallest possible product, 10,000, is already a 5-digit number.
Q6Can multiplying a 4-digit number with a 2-digit number give a 5-digit number?
Answer
Yes.
Smallest case: 1000 × 10 = 10,000 → 5 digits ✓ Largest case: 9999 × 99 = 9,89,901 → 6 digits Example: 1234 × 12 = 14,808, a 5-digit number
Q7Observe the multiplication statements below. Do you notice any patterns? See if this pattern extends for other numbers as well.
1-digit
×
1-digit
=
1-digit
or
2-digit
2-digit
×
1-digit
=
2-digit
or
3-digit
2-digit
×
2-digit
=
3-digit
or
4-digit
3-digit
×
3-digit
=
5-digit
or
6-digit
5-digit
×
5-digit
=
or
8-digit
×
3-digit
=
or
12-digit
×
13-digit
=
or
Answer
The blanks follow the same rule: the answer has (m + n − 1) or (m + n) digits.
Fascinating Facts about Large Numbers — Pages 16–181.5 Patterns in Products
Q11250 × 380 = ______________ is the number of kīrtanas composed by Purandaradāsa according to legends. How many years did he live to compose so many songs? At what age did he start composing songs? If he composed 4,75,000 songs, how many songs per year did he have to compose?
Indian system: four lakh seventy five thousand American system: four hundred seventy five thousand
How many songs per year? Purandaradāsa is said to have lived about 80 years (1484–1564).
If he composed for all 80 years: 4,75,000 ÷ 80 = about 5,938 songs a year = about 16 songs every single day
If he began at about age 30 and composed for 50 years: 4,75,000 ÷ 50 = 9,500 songs a year = about 26 songs every day
Q22100 × 70,000 = _______________ is the approximate distance in kilometers, between the Earth and the Sun. Roxie asks, “How did they measure the distance between the Earth and the Sun?”
Answer
The hidden number is 14,70,00,000 km — about 14 crore 70 lakh kilometres.
Indian system: fourteen crore seventy lakh American system: one hundred forty seven million
How was it measured? Astronomers first worked it out with geometry, by watching the planet Venus pass across the face of the Sun from places far apart on the Earth. Today it is measured very precisely by sending radar signals to planets and timing the echoes.
Q36400 × 62,500 = _________________ is the average number of litres of water the Amazon river discharges into the Atlantic Ocean every second.
Answer
The hidden number is 40,00,00,000 litres every second — 40 crore litres.
Population of Mumbai = 1,24,42,373 1,24,42,373 < 1,25,00,000 ✓ Space to spare = 57,627 places
Q3Roxie wondered, “If I could travel 100 kilometers every day, could I reach the Moon in 10 years?” (The distance between the Earth and the Moon is 3,84,400 km.) How far would she have travelled in a year? How far would she have travelled in 10 years?
Answer
No — after 10 years she would still be about 19,400 km short.
Distance in 1 year = 100 × 365 = 36,500 km
Distance in 10 years = 36,500 × 10 = 3,65,000 km
Distance to the Moon = 3,84,400 km Short by = 3,84,400 − 3,65,000 = 19,400 km
Q4Find out if you can reach the Sun in a lifetime, if you travel 1000 kilometers every day. (You had written down the distance between the Earth and the Sun in a previous exercise.)
Answer
No. The journey would take about 403 years — far more than a lifetime.
Distance to the Sun = 14,70,00,000 km (from page 16) Distance in 1 year = 1000 × 365 = 3,65,000 km
Years needed = 14,70,00,000 ÷ 3,65,000 = 1,47,000 ÷ 365 ≈ 402.7 years
Q5Make necessary reasonable assumptions and answer the questions below: (a) If a single sheet of paper weighs 5 grams, could you lift one lakh sheets of paper together at the same time? (b) If 250 babies are born every minute across the world, will a million babies be born in a day? (c) Can you count 1 million coins in a day? Assume you can count 1 coin every second.
Answer
All three answers are no.
(a) Weight = 1,00,000 × 5 g = 5,00,000 g = 500 kg A person can lift perhaps 40–50 kg, so no — 500 kg is about the weight of 8 adults.
(b) Minutes in a day = 60 × 24 = 1,440 Babies in a day = 250 × 1,440 = 3,60,000 1 million = 10,00,000 3,60,000 < 10,00,000 → no, only about a third of a million.
(c) Seconds in a day = 60 × 60 × 24 = 86,400 So at most 86,400 coins a day. 86,400 < 10,00,000 → no, not even a tenth of a million.
Figure it Out — Pages 19–211.6 Did You Ever Wonder…?
Try This
Q1Using all digits from 0 – 9 exactly once (the first digit cannot be 0) to create a 10-digit number, write the — (a) Largest multiple of 5 (b) Smallest even number
Answer
(a) 98,76,54,32,10(b) 1,02,34,56,798
(a) A multiple of 5 must end in 0 or 5. Ending in 0 is better — it frees the 5 for a higher place. Put the rest in decreasing order: 9,8,7,6,5,4,3,2,1 Largest = 9876543210
(b) An even number must end in 0, 2, 4, 6 or 8. To be small, the front should read 1,0,2,3,4,… Ending in 8 keeps the front smallest: 102345679 | 8 Smallest = 1023456798
Q2The 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.
Answer
77,77,777 — “seventy seven lakh seventy seven thousand seven hundred seventy seven” — with 60 letters (61 if you write “lakhs”, as the book does in its example).
Q3Write a 9-digit number where exchanging any two digits results in a bigger number. How many such numbers exist?
Answer
The number is 123456789, and there is exactly one such number.
Swap the 1 and the 2 → 213456789 > 123456789 ✓ Swap the 1 and the 9 → 923456781 > 123456789 ✓ Swap the 8 and the 9 → 123456798 > 123456789 ✓ Every swap makes it bigger.
Q4Strike out 10 digits from the number 12345123451234512345 so that the remaining number is as large as possible.
Answer
The largest number left is 5534512345.
12345 12345 12345 12345 (20 digits, keep 10)
Step 1: for the 1st kept digit, scan positions 1–11 → biggest is 5 (position 5) Step 2: for the 2nd, scan positions 6–12 → biggest is 5 (position 10) Step 3: for the 3rd, scan positions 11–13 → biggest is 3 (position 13) Step 4: only 7 digits are left (positions 14–20) and 7 are still needed → keep them all: 4512345
Result: 5 5 3 4512345 = 5534512345
Q5The words ‘zero’ and ‘one’ share letters ‘e’ and ‘o’. The words ‘one’ and ‘two’ share a letter ‘o’, and the words ‘two’ and ‘three’ also share a letter ‘t’. How far do you have to count to find two consecutive numbers which do not share an English letter in common?
Answer
You would have to count for ever — no such pair exists, however far you go.
Pair
Shared letter(s)
Pair
Shared letter(s)
one, two
o
six, seven
s
two, three
t
seven, eight
e
three, four
r
eight, nine
e, i
four, five
f
nine, ten
n, e
five, six
i
ten, eleven
e, n
Q6Suppose 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. (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?
Answer
(a) The 1000th digit is 3, the first digit of the number 370.
1–9 use 9 digits; 10–99 use 90 × 2 = 180 digits (total 189) Digits still needed = 1000 − 189 = 811 811 ÷ 3 = 270 remainder 1 270 three-digit numbers: 100 to 369 → uses 810 digits (total 999) The next digit, the 1000th, is the 1st digit of 370, which is 3
(b) The millionth digit lies in the number 1,85,185.
Digits used by 1 to 99,999: 9 + 180 + 2,700 + 36,000 + 4,50,000 = 4,88,889 Digits still needed = 10,00,000 − 4,88,889 = 5,11,111 5,11,111 ÷ 6 = 85,185 remainder 1 85,185 six-digit numbers: 1,00,000 to 1,85,184 → 5,11,110 digits (total 9,99,999) The next digit, the millionth, is the 1st digit of 1,85,185
(c) You write the digit ‘5’ for the 5000th time at the number 13,495 — it is the units digit of 13,495.
Count of 5’s written from 1 up to 13,494 = 4,999 13,495 contains one 5 (in the units place) 4,999 + 1 = 5,000 ✓
Q7A 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
Answer
Take as many ten-thousands as you can, then make up the rest in hundreds.
Q9You 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 the two resulting numbers. (The boxes make a 7-digit number above a 5-digit number.)Page 20 — seven boxes in the top row and five boxes in the bottom row, lined up on the right.
Answer
(a) Largest sum = 1,00,75,308(b) Smallest difference = 10,22,447
(a) 99,87,654 + 87,654 = 1,00,75,308
(b) 11,22,334 − 99,887 = 10,22,447
Q10You 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
Answer
Four of the five targets can be hit exactly; for (e) the closest we found is 20,90,750, only 50 away.
Q11Find out how many coins should be stacked to match the height of the Statue of Unity. Assume each coin is 1 mm thick.
Answer
1,80,000 coins — one lakh eighty thousand.
Height of Statue of Unity = 180 m 1 m = 100 cm and 1 cm = 10 mm So 180 m = 180 × 100 × 10 mm = 1,80,000 mm
Thickness of one coin = 1 mm Number of coins = 1,80,000 ÷ 1 = 1,80,000 coins
Q12Grey-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 the Pacific Ocean approximately?
Answer
About 12 to 14 days.
At 1000 km a day: 12,000 ÷ 1000 = 12 days At 900 km a day: 12,000 ÷ 900 = 13.3 → about 14 days (13 full days cover only 11,700 km, so a 14th day is needed)
So the trip takes roughly 12 to 14 days
Q13A 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.
Answer
About 1,233 km a day and about 51 km an hour.
Distance per day = 13,560 ÷ 11 = 1232.7… ≈ 1,233 km per day
Hours in the journey = 11 × 24 = 264 Distance per hour = 13,560 ÷ 264 = 51.36… ≈ 51 km per hour
(Or: 1,233 ÷ 24 ≈ 51 km per hour)
Q14Bald 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?
Answer
Somu’s building is 40 m tall, so divide every height by 40.
Height
Working
Times Somu’s building
Bald eagle (lower) 4,500 m
4500 ÷ 40
112.5 ≈ 112 times
Bald eagle (upper) 6,000 m
6000 ÷ 40
150 times
Mount Everest 8,850 m
8850 ÷ 40
221.25 ≈ 221 times
Aeroplane (lower) 10,000 m
10000 ÷ 40
250 times
Aeroplane (upper) 12,800 m
12800 ÷ 40
320 times
So: bald eagles fly about 112 to 150 times the height of the building, Mount Everest is about 221 times as tall, and aeroplanes fly about 250 to 320 times as high.
Puzzle Time: Toothpick Digits — Page 23Puzzle Time
Try This
Q1We can write digits using sticks as shown in the image.Page 23 — the ten digits written with sticks.
To make the digit 7, three sticks are needed. Write or make the number 5108. How many sticks are required?
Q21. Make or write the number 42,019. It would require exactly 23 sticks. 2. Starting with 42,019, add or write two more sticks, and make a bigger number. One example is 42,078. What other numbers bigger than 42,019 can you make in this way? 3. Preetham wants to insert the digit ‘1’ somewhere among the digits ‘4’, ‘2’, ‘0’, ‘1’ and ‘9’. Where should he place the digit ‘1’ to get the biggest possible number? 4. What other numbers can he make by placing the digit ‘1’?
Answer
1. 42,019 needs 4 + 5 + 6 + 2 + 6 = 23 sticks ✓
2. Adding two sticks (without moving any) can turn digits into bigger digits, or make a new ‘1’. The numbers bigger than 42,019 are:
1 stick on each of two digits (1 → 7, 9 → 8, 0 → 8): 42,078 (the book’s example) · 42,818 · 42,879 2 sticks on one digit (1 → 4, 2 → 8, 4 → 9): 42,049 · 48,019 · 92,019 A new ‘1’ (2 sticks) written in: 1,42,019 · 4,12,019 · 4,20,119 · 4,20,191 · 4,21,019 (the biggest)
3. He should place the ‘1’ between the ‘2’ and the ‘0’, giving 4,21,019.
4. The other numbers he can make are 1,42,019 · 4,12,019 · 4,20,119 · 4,20,191.
Q31. Make or write the number 63,890. 2. Starting with 63,890, rearrange exactly four sticks and make a bigger number. One example is 88,078. What other numbers bigger than 63,890 can you make in this way?
Answer
1. 63,890 needs 6 + 5 + 7 + 6 + 6 = 30 sticks.
2. Moving sticks keeps the total at 30, and exactly four sticks must leave their places. Some bigger numbers made this way:
88,078 (the book’s example) 66,066 · 66,069 · 90,000 · 90,999 · 92,008 99,985 — the largest possible
Q41. Make any number using exactly 24 sticks or lines. 2. What is the biggest number that can be made using 24 sticks or lines? 3. What is the smallest number that can be made using 24 sticks or lines?
Answer
1. Many numbers use exactly 24 sticks. Three easy ones:
2. The biggest number is 1,11,11,11,11,111 — twelve 1’s. The digit 1 uses the fewest sticks (2), and 24 ÷ 2 = 12, so no number made with 24 sticks can have more than 12 digits.
3. The smallest number is 2008 (5 + 6 + 6 + 7 = 24). Three digits use at most 3 × 7 = 21 sticks, so 4 digits are needed. The first digit is 2 (a first digit 1 would leave 22 sticks for three digits — too many), and then 0, 0 and 8 use up the remaining 19 sticks.