[NCERT] Class 7 Maths Chapter 1 Large Numbers Around Us Solutions

Large Numbers Around Us Class 7 Solutions Ganita Prakash Maths Chapter 1

Class 7 Maths Ganita Prakash Chapter 1 Large Numbers Around Us NCERT Solutions provide you the best study material available online, covering the exercises and questions of Chapter 1 Large Numbers Around Us.

NCERT Class 7 Maths Chapter 1 Large Numbers Around Us Solutions Question Answer

Ganita Prakash Class 7 Chapter 1 Solutions Large Numbers Around Us

NCERT Class 7 Maths Ganita Prakash Chapter 1 Large Numbers Around Us Solutions Question Answer

NCERT Text Question Page (1 – 2)

Question 1. Eshwarappa is a farmer in Chintamani, a town in Karnataka. He visits the market regularly to buy seeds for his rice field. During one such visit he overheard a conversation between Ramanna and Lakshmamma. Ramanna said, “Earlier our country had about a lakh varieties of rice. Farmers used to preserve different varieties of seeds and use them to grow rice. Now, we only have a handful of varieties. Also, farmers have to come to the market to buy seeds”. Lakshmamma said, “There is a seed bank near my house. So far, they have collected about a hundred indigenous varieties of rice seeds from different places. You can also buy seeds from there.”

Eshwarappa shared this incident with his daughter Roxie and son Estu . Estu was surprised to know that there were about one lakh varieties of rice in this country. He 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.

Solution:-

Q. 2. But how much is one lakh? Observe the pattern and fill in the boxes given below.

Solution:-

Q. 3. Roxie and Estu found that if they ate one variety of rice a day, they would come nowhere close to a lakh in a lifetime! Roxie 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?”

Solution:-

Q. 4 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.

Solution:-

NCERT Text Question Page – 3

Q. 5. Estu said, “We know how many days there are in a year – 365, if we ignore leap years. If we live for y years, the number of days in our lifetime will be 365 x y.”

Choose a number for y. How close to one lakh is the number of days in y years, for the y of your choice?

Solution:-

Figure it out (Page 3)

Q. 1. According to the 2022 Census, the population of the town of Chintamani was about 75,000. How much less than one lakh is 75,000?

Solution:-

Q. 2. The estimated population of Chintamani in the year 2024 is 1,06,000. How much more than one lakh is 1,06,000?

Solution:-

Q.3. By how much did the population of Chintamani increase from 2022 to 2024?

Solution:-

Getting a Feel of Large Numbers

You may have come across intersecting facts like these:

The world’s tallest statue is the ‘Statue of Unity‘ in Gujarat depicting Sardar Vallabhbhai Patel. Its height is about 180 meteres.

Kunchikal Waterfall in Karnataka is said to drop from a height of about 450 metres.

Q. 1. Loot 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?

Solution:-

Q. 2. Which is taller – The Statue of Unity or this building? How much taller? __ m.

Solution:-

Q. 2. How much taller is the kunchikal waterfall than Somu’s building? __ m.

Solution:-

Q. 3. How many floor should Somun’s Building have to be as high as the waterfall?

Solution:-

NCERT Text Questions (Page – 4)

Is one Lakh a Very Large Number?

Q. How do you view a lakh – is a lakh big or small?

Solution:-

For a Class 7 student, a lakh can be understood like this:

A lakh = 1,00,000 (one hundred thousand).

Is a lakh big or small?

It depends on what we are talking about.

Big:

• 1 lakh steps
• 1 lakh pages in a book
• 1 lakh grains of rice

These are very large and hard to imagine!

Small (in some situations):

• ₹1 lakh for building a big bridge
• 1 lakh people in a big city

Compared to these, a lakh feels small.

Simple way to remember

• 10 thousands = 1 lakh
• Counting to a lakh one by one would take a very long time

Conclusion

A lakh is a big number in daily life, but in math and science, it can sometimes be just the beginning

So, a lakh is big — until you meet even bigger numbers!

NCERT Text Question (Pages 4)

Reading and Writing Numbers:

Q. Write each of the numbers given below in words:

(a) 3,00,600
(b) 5,04,085
(c) 27,30,000
(d) 70,53,138

Solution:-

Q. Write 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

Solution:-

NCERT Text Question (Pages – 5)

1.2 Land of Tens

In the Land of Tens, there are special calculators with special buttons.

Q. 1. The 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?

Solution:-

(b) 10,000 ÷ 1,000 = 10 times

(c) 53,000 ÷ 1,000 = 53 times

(d) 90,000 ÷ 1,000 = 90 times

(e) 1,00,000 ÷ 1,000 = 100 times

(f) 153 x 1,000 = 1,53,000

(g) 1,00,000 ÷ 1,000 = 100 thousands

Q. 2. The 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

Solution:-

(a) 500 ÷ 10 = 50 times.

(b) 780 ÷ 10 = 78 times.

(c) 1000 ÷ 10 = 100 times.

(d) 3700 ÷ 10 = 370 times.

(e) 10,000 ÷ 10 = 1,000 times.

(f) 1,00,000 ÷ 10 = 10,000 times.

(g) 435 × 10 = 4350.

Q. 3. The Handy Hundreds only has a +100 button. How many times should it be pressed to show:

(a) Four hundred?_______ times
(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.

Solution:-

(a) 400 ÷ 100 = 4 times.

(b) 3700 ÷ 100 = 37 times.

(c) 10,000 ÷ 100 = 100 times.

(d) 53,000 ÷ 100 = 530 times.

(e) 90,000 ÷ 100 = 900 times.

(f) 97,600 ÷ 100 = 976 times.

(g) 1,00,000 ÷ 100 = 1,000 times.

(h) 582 × 100 = 58,200.

(i) 10,000 ÷ 100 = 100 hundreds.

(j) 1,00,000 ÷ 100 = 1000 hundreds.

(k) Yes — Handy Hundreds is correct, and here’s why

Who are they?

• Tedious Tens shows numbers by counting 10 at a time
• Thoughtful Thousands shows numbers by counting 1000 at a time
• Handy Hundreds shows numbers by counting 100 at a time

Let’s explore with examples

• Handy Hundreds can show numbers like:
  100, 200, 300, 450, 700, 900
• Tedious Tens can show only numbers that are multiples of 10
  (10, 20, 30, 40, …)
• Thoughtful Thousands can show only numbers that are multiples of 1000
  (1000, 2000, 3000, …)

Now the key idea is, There are many numbers that are:

• multiples of 100
• but not multiples of 1000

For example:

• 100
• 200
• 300
• 900
• 1500

These can be shown by Handy Hundreds, but cannot be shown by Thoughtful Thousands.

Also, some numbers like 150:

• cannot be shown by Tedious Tens or Thoughtful Thousands
• but can be shown by Handy Hundreds

Conclusion

✅ Yes, the statement is true.

Handy Hundreds can show some numbers that Tedious Tens and Thoughtful Thousands cannot.

So Handy Hundreds is very useful — right in the middle!

Q. 4. Creative Chitti is a different kind of calculator. It has the following buttons:

+1, +10, +100, +1000, +10,000, +100000 and +1000000. It always has multiple ways of doing things. “How so?’, you might ask. 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.

Solution:-

Q. 5. Two of the many different ways to get 5072 are shown below. These two ways can be expressed as:

(a) (50 x 100) + (7 x 10) + (2 x 1) = 5072

(b) (3 x 1000) + (20 x 100) + (72 x 1) = 5072

Find a different way to get 5072 and write an expression for the same.

Solution:-

NCERT Text Question Page (6)

Figure it Out

Q. For 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

Solution:-

Q. Creative 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?

Solution:-

(b) 997 can be made using 25 clicks. Can you make 997 with a different
number of clicks?

Solution:-

Q. Systematic Sippy is a different kind of calculator. It has the following 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?

Find out which buttons should be clicked and how many times to get the desired numbers given in the table. The aim is to click as few buttons as possible. Here is one way to get the number 5072. This method uses 23 button clicks in total. Is there another way to get 5072 using less than 23 button clicks? Write the expression for the same.

Solution:-

NCERT Text Question Page (7)

Figure it Out

Q. 1. For the numbers in the previous exercise, find out how to get each number by making the smallest number of button clicks and write the expression.

Solution:-

Q. 2. Do you see any connection between each number and the corresponding smallest number of button clicks?

Solution:-

Q. 3. If you notice, the expressions for the least button clicks also give the Indian place value notation of the numbers. Think about why this is so.

What if we press the +10,00,000 button ten times? What number will come up? How many zeros will it have? What should we call it? The number will be 100 lakhs, which is also called a crore. 1 crore is written as 1,00,00,000 – it is 1 followed by seven zeroes.

Solution:-

NCERT Question Page (8)

Figure it out

Q. 1. How many zeros does a thousand lakh have? –

Solution:-

Thousand Lakh = 1,000 x 1,00,000 = 10,00,00,000 and it has 8 zeros.

NCERT Question Page (9)

Q. 2. How many zeros does a hundred thousand have? –

Solution:-

Hundred thousand = 100 x 1,000 = 1,00,000 and it has 5 zeros.

NCERT Question Page (9)

Figure it Out

Q. 1. Read 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

Solution:-

Q. 2. Write 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

Solution:-

(a) 1,01,01,010

(b) 1,001,001,001

(c) 10,20,30,040

(d) 9,080,700,600

Q. 3. Compare and write ‘<‘, ‘>’ or ‘=’.

(a) 30 thousand __ 3 lakhs

(b) 500 lakhs _ 5 million

(c) 800 thousand_____8 million

(d) 640 crore______60 billion

Solution:-

Let’s compare step by step

(a) 30 thousand < 3 lakh
(30,000 < 3,00,000)

(b) 500 lakh > 5 million
(500 lakh = 50 million > 5 million)

(c) 800 thousand < 8 million
(8,00,000 < 80,00,000)

(d) 640 crore < 60 billion
(640 crore = 6.4 billion < 60 billion)

1.4 Exact and Approximate Values

NCERT Question Page (10)

Q. Think 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

Solution:-

Here are some real-life situations explained simply

(a) Situations where it is appropriate to round up

When we need to make sure there is enough of something.

• Buying bus tickets: If 48.2 km is shown, we say 49 km for planning fuel.
• Packing items: If 32 students need notebooks, we buy 33 notebooks.
• Time: If a movie is 1 hour 52 minutes, we may say 2 hours.

(b) Situations where it is appropriate to round down

When we want to stay within a limit or avoid overestimating.

• Pocket money: If you have ₹99.75, you may say you have ₹99.
• Age: A child of 9.7 years is said to be 9 years old.
• Distance covered: If you ran 4.8 km, you may say 4 km completed.

(c) Situations where either rounding up or rounding down is okay

When an estimate is enough, not the exact value.

• Estimating the number of people in a crowd.
• Estimating the cost of shopping before billing.
• Estimating time taken for a journey.

(d) Situations where exact numbers are needed

When accuracy is very important.

• Bank transactions and money transfers.
• Marks obtained in exams.
• Medicine dosage.
• Measurements in science experiments.

✨ Key idea to remember:

• Round up → to be safe
• Round down → to stay within limits
• Either → estimation is fine
• Exact → no guessing allowed

NCERT Question Page (11)

Nearest Neighbours

Q. 1. Similarly, write the five nearest neighbours for these numbers:

(a) 3,87,69,957

(b) 29,05,32,481

Solution:-

(a) 3,87,69,957

(b) 29,05,32,481

Q. 2. I 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?

Solution:-

We are talking about rounding to the nearest crore.

Step 1: When does a number round to 5,00,00,000?

Any number from
4,50,00,000 to 5,49,99,999
(rounding rule: 50 lakh or more → round up)

So, every number in this range has nearest crore = 5,00,00,000.

Step 2: Condition about five nearest neighbours

Five nearest neighbours means:

n − 2, n − 1, n, n + 1, n + 2

All five numbers must still lie in the rounding range
4,50,00,000 to 5,49,99,999.

So the number n cannot be too close to the ends.

✔ Smallest possible number: 4,50,00,002
✔ Largest possible number: 5,49,99,997

Step 3: What could the number be?

One possible number is:
5,00,00,000 itself 😊
(But many others are possible.)

Step 4: How many such numbers are there?

All numbers from
4,50,00,002 to 5,49,99,997

Total count:
[5,49,99,997 – 4,50,00,002 + 1 = \boxed{9,999,996}]

✅ Final Answer

• The number could be any number between 4,50,00,002 and 5,49,99,997
• There are 9,999,996 such numbers

Roxie and Estu are estimating the values of simple expressions.

Question – 1

4,63,128 + 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 greaterthan 8,83,128 or less than 8,83,128? Why do you think so?

(d) Exact value of 4,63,128 + 4,19,682 = __.

Solution:-

Let’s work through this step by step, just like Roxie and Estu 😊

First, find the exact sum

[4,63,128 + 4,19,682 = 8,82,810]

(a) Are the estimates correct? Whose estimate is closer?

• Roxie says: near 8,00,000 and more than 8,00,000
  ✔ Correct, because 8,82,810 is more than 8,00,000.
• Estu says: near 9,00,000 and less than 9,00,000
  ✔ Correct, because 8,82,810 is less than 9,00,000.

Whose estimate is closer?

• Distance from 8,00,000 = 82,810
• Distance from 9,00,000 = 17,190

👉 Estu’s estimate is closer.

(b) Will the sum be greater than 8,50,000 or less than 8,50,000? Why?

• 4,63,128 ≈ 4,60,000
• 4,19,682 ≈ 4,20,000
• Estimated sum ≈ 8,80,000

So, the sum is greater than 8,50,000.

(c) Will the sum be greater than 8,83,128 or less than 8,83,128? Why?

Compare:

• 4,19,682 is less than 4,20,000
• So the total will be less than
  (4,63,128 + 4,20,000 = 8,83,128)

👉 Therefore, the sum is less than 8,83,128.

(d) Exact value

Exact value of 4,63,128 + 4,19,682 = [8,82,810]

✨ Summary:

• Both estimates are correct
• Estu’s estimate is closer
• The sum is > 8,50,000
• The sum is < 8,83,128
• Exact sum = 8,82,810

Question – 2

14,63,128 – 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 = _

Solution:-

Let’s solve it step by step

First, find the exact difference

[14,63,128 – 4,90,020 = 9,73,108]

(a) Are the estimates correct? Whose estimate is closer?

• Roxie says: near 10,00,000 and less than 10,00,000
  ✔ Correct, because 9,73,108 is less than 10,00,000.
• Estu says: near 9,00,000 and more than 9,00,000
  ✔ Correct, because 9,73,108 is more than 9,00,000.

Whose estimate is closer?

• Distance from 10,00,000 = 26,892
• Distance from 9,00,000 = 73,108

👉 Roxie’s estimate is closer.

(b) Will the difference be greater than 9,50,000 or less than 9,50,000? Why?

Estimate:

• 14,63,128 ≈ 14,60,000
• 4,90,020 ≈ 4,90,000
• Difference ≈ 9,70,000

So, the difference is greater than 9,50,000.

(c) Will the difference be greater than 9,63,128 or less than 9,63,128? Why?

Compare with:

[14,63,128 – 5,00,000 = 9,63,128]

But 4,90,020 is less than 5,00,000, so we are subtracting a smaller number.

👉 Therefore, the difference will be greater than 9,63,128.

(d) Exact value

[14,63,128 – 4,90,020 = 9,73,108]

✨ Summary:

• Both estimates are correct
• Roxie’s estimate is closer
• Difference > 9,50,000
• Difference > 9,63,128
• Exact difference = 9,73,108

NCERT Text Question Page (13)

Population of the Cities

From the information given in the table, answer the following questions by approximation:

Q. 1. What is your general observation about this data? Share it with the class.

Solution:-

1️⃣ General observation about the data

• The population of most big cities has increased from 2001 to 2011.
• Metro cities like Mumbai, Delhi, Bengaluru, Hyderabad grew very fast.
• A few cities (like Kolkata) show very little growth or slight decrease.
• Overall, it shows rapid urbanisation in India.

👉 Observation: India’s large cities are becoming more crowded over time.

Q. 2. What is an appropriate title for the above table?

Solution:-

2️⃣ Appropriate title for the table

“Population of Major Indian Cities (2001 and 2011 Census)”

Q. 3. How much is the population of Pune in 2011? Approxiamtely, by how much has it increased compared to 2001?

Solution:-

3️⃣ Population of Pune in 2011 and increase since 2001

• Pune (2011) = 31,15,431 ≈ 31 lakh
• Pune (2001) = 25,38,473 ≈ 25 lakh

Increase ≈ 31 lakh − 25 lakh = 6 lakh

👉 Population increased by about 6 lakh.

Q. 4. Which city’s population increased the most between 2001 and 2011?

Solution:-

4️⃣ Which city’s population increased the most (2001–2011)?

Let’s approximate:

• Bengaluru: 43 lakh → 84 lakh → increase ≈ 41 lakh
• Hyderabad: 36 lakh → 68 lakh → increase ≈ 32 lakh
• Ahmedabad: 35 lakh → 56 lakh → increase ≈ 21 lakh
• Surat: 24 lakh → 45 lakh → increase ≈ 21 lakh

👉 Bengaluru has the maximum increase.

Q. 5. Are there cities whose population has almost doubled? Which are they?

Solution:-

5️⃣ Cities whose population has almost doubled

Check where 2011 population is about twice of 2001:

• Bengaluru: 43 lakh → 84 lakh ✔
• Hyderabad: 36 lakh → 68 lakh ✔
• Surat: 24 lakh → 45 lakh ✔
• Vadodara: 17 lakh → 35 lakh ✔

👉 Yes, these cities have almost doubled.

Q. 6. By what number should we multiply Patna’s population to get a number/population close to that of Mumbai?

Solution:-

6️⃣ By what number should we multiply Patna’s population to get close to Mumbai’s?

Approximate values:

• Mumbai ≈ 1.25 crore
• Patna ≈ 17 lakh

Convert:

• 1.25 crore = 125 lakh

• 125 ÷17 = approx 7

👉 Multiply Patna’s population by about 7 to get close to Mumbai’s.

✨ Summary (Exam-friendly):

1. Most cities show rapid population growth.
2. Title: Population of Major Indian Cities (2001 & 2011)
3. Pune ≈ 31 lakh; increase ≈ 6 lakh
4. Bengaluru increased the most
5. Bengaluru, Hyderabad, Surat, Vadodara almost doubled
6. Patna × 7 ≈ Mumbai

1.5 Patterns in Products

A Multiplication Shortcut

NCERT In-Text Questions (page 14)

Q. Using the meaning of multiplication and division, can you explain why multiplying by 5 is the same as dividing by 2 and multiplying by 10?

Solution:-

Yes — let’s explain this using meanings of multiplication and division, in a simple way 😊

Key idea:-

Multiplication and division are related operations.

Dividing by 2 and then multiplying by 10 means:
[Number ÷ 2 x10]

But
[ ÷ 2 x 10 = x 10/2 = x 5]

So, multiplying by 5 is the same as dividing by 2 and multiplying by 10.

Explanation using meaning (group’s idea)

Suppose we have a number n.

Method 1: Multiply by 5

• Multiplying by 5 means making 5 equal groups of the number.
  [n x 5]

Method 2: Divide by 2, then multiply by 10

• Dividing by 2 means taking half of the number.
• Multiplying by 10 means making 10 equal groups of that half.

So: (n/2) x 10 = n x 5

Both give the same total.

Simple number example

Take 20:

• Multiply by 5:
  (20 x 5 = 100)
• Divide by 2, then multiply by 10:
  (20/2 = 10)
  (10 x 10 = 100)

👉 Same answer!

Visual way (for Class 7)

• Dividing by 2 splits a quantity into two equal parts.
• Multiplying by 10 makes ten copies of one part.
• Ten halves = five wholes.

⭐ Conclusion

Multiplying by 5 = dividing by 2 and multiplying by 10,
because (10/2 = 5 )

Figure it Out

Q. 1. Find quick ways to calculate these products:

(a) 2 x 1768 x 50

(b) 72 x 125 [Hint: 125 = 1000/8]

(c) 125 x 40 x 8 x 25

Solution:-

Here are quick and smart ways to calculate each product by rearranging and regrouping numbers 😊

(a) (2 x 1768 x 50)

Rearrange:
[(2 x 50) x 1768 = 100 x 1768] = 176800

✔ Fast because multiplying by 100 is easy.

(b) (72 x 125)

Hint used: (125 = 1000/8)

[72 x 125 = 72 x 1000/8]

[(72 ÷ 8) x 1000 = 9 x 1000] = 9000

(c) (125 x 40 x 8 x 25)

Rearrange smartly:
[(125 x 8) x (40 x 25)] = [1000 x1000] = 10,00,000

✨ Key trick to remember:

• Rearrange numbers to make 10, 100, or 1000
• Use factors like 2, 4, 5, 8, 25, 125

Q. 2. Calculate these products quickly.

(a) 25 x 12 =

(b) 25 x 240 =

(c) 250 x 120 =

(d) 2500 x 12 =

(e) _ x _ = 120000000

Solution:-

Let’s do these quickly using smart regrouping 😊

(a) (25 x 12)

[25 x 12 = (25 x 4) x 3 = 100 x 3 = 300]

(b) (25 x 240)

[25 x 240 = 25 x (4 x 60) = (25 x 4) x 60 = (100 x 60) = 6000]

(c) (250 x 120)

[250 x 120 = (25 x 12) x (10 x 10) = (300 x 100) = 30,000]

(d) (2500 x 12)

[2500 x 12 = (25 x 100) x 12) = (25 x 12 ) x 100 = 300 x 100 = 30000

(e) ( ____ x ____ = 120000000)

One quick way:
[25 x 4,800,000] = 120,000,000

(There are many correct answers — this is one of them.)

🌟 Pattern to notice:

• 25 × 4 = 100
• Add zeros at the end → calculation becomes easy

How Long is the product?

Q. In each of the following boxes, the multiplications produce interesting patterns. Evaluate them to find the pattern. Extend the multiplications based on the observed pattern.

Solution:-

Sure 😊 Here is the data neatly represented in tables, one for each box:

📊 Box 1: Squares of numbers made of 1s

Pattern:-

Digits go up step by step, then come down.

Extension:-

11111 × 11111 = 123454321

📊 Box 2: Numbers with 6s and 1

Pattern:-

• Many 4s, then 0, then many 2s, then 6.

Extension:-

66666 × 66661 = 4444022226

📊 Box 3: Numbers with 3s and 5s

Pattern

• Product has all 1s first, then all 5s.
• Number of 1s = number of digits in the first number.

Extension

3333 × 3335 = 11115555

📊 Box 4: Squares of numbers near 100

Pattern:-

• Square of numbers just above 100
• Middle number increases regularly.

Extension:-

104 × 104 = 10816

✨ Tip for exams:

Tables like these help you see patterns clearly, which is exactly what such questions test.

Page – 15

Q. Observe 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?

Solution:-

Yes, when we multiply:

• 1-digit number with a 1-digit number, we get a 1-digit or 2-digit product
• 2-digit number with a 2-digit number, we get a 3-digit or 4-digit product
• 3-digit number with a 3-digit number, we get a 5-digit or 6-digit product
• 4-digit number with a 4-digit number, we get a 7-digit or 8-digit product

Q. Roxie says that the product of two 2-digit numbers can only be a 3 or a 4 digit number. Is She Correct?

Solution:-

Yes — Roxie is correct ✅

Here’s a clear explanation:

• The smallest 2-digit number is 10
• The largest 2-digit number is 99

Smallest possible product

[10 x 10 = 100 (3 digits)]

Largest possible product

[99 x 99 = 9801 (4 digits)]

So, the product of two 2-digit numbers lies between 100 and 9801.

👉 That means the product can have only 3 or 4 digits, never 2 digits or 5 digits.

✅ Final answer (exam-friendly):

Yes, Roxie is correct. The product of two 2-digit numbers can only be a 3-digit or a 4-digit number.

Q. Should 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?

Solution:-

No, we don’t need to do all possible multiplications, check extremes: 10×10 and 99×99 give the minimum and maximum digits possible.

Q. Can multiplying a 3-digit number with another 3-digit number give a 4-digit number?

Solution:-

No, it cannot. ❌
A 3-digit × 3-digit multiplication can never give a 4-digit number.

Why?

• Smallest 3-digit number = 100
• Largest 3-digit number = 999

Smallest possible product:

[100 x 100 = 10,000]
This is already a 5-digit number.

Largest possible product:

[999 x 999 = 9,98,001]
That’s 6 digits.

Conclusion (exam-ready):

The product of two 3-digit numbers is always a 5-digit or a 6-digit number, never a 4-digit number.

So the answer is No — a 4-digit product is not possible 👍

Q. Can multiplying a 4-digit number with a 2-digit number give a 5-digit number?

Solution:-

Yes, it can. ✅

Let’s see why with clear bounds.

Smallest possible product

• Smallest 4-digit number = 1000
• Smallest 2-digit number = 10

[1000 x 10 = 10,000]
This is a 5-digit number.

Largest possible product

• Largest 4-digit number = 9999
• Largest 2-digit number = 99

[9999 x 99 = 989,901]
This is a 6-digit number.

Conclusion (exam-ready):

The product of a 4-digit number and a 2-digit number can be 5 digits or 6 digits.

So, yes, multiplying a 4-digit number with a 2-digit number can give a 5-digit number 👍

Q. Observe the multiplication statements below. Do you notice any patterns? See if this pattern extends for other numbers as well.

Solution:-

Yes 😊 there is a very clear and consistent pattern, and it works for all numbers.

Let’s observe and complete the table using that pattern.

🔍 The pattern

If a number with m digits is multiplied by a number with n digits, then the product will have either$$(m + n – 1)\ \text{digits} \quad \text{or} \quad (m + n)\ \text{digits}$$

It can never have fewer or more digits than this.

📊 Pattern in Number of Digits in Products

🧠 Pattern to remember (one line)

If an m-digit number is multiplied by an n-digit number, the product has (m + n − 1) or (m + n) digits.

This rule works for all numbers, big or small 👍

Page – 16

Q. 1250 × 380 is the number of kīrtanas composed by Purandaradāsa according to legends. Purandaradāsa was a composer and singer in the 15th century. His kīrtanas spanned social reform, bhakti and spirituality. He systematised methods for teaching Carnatic music which is followed to the present day

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?

Solution:-

This is a thinking + estimation question, not a history test, so we use reasonable assumptions 😊

First, understand the number

[1250 x 380 = 4,75,000]

So, according to legend, Purandaradāsa composed about 4,75,000 kīrtanas.

1️⃣ How many years did he live?

From traditional accounts, Purandaradāsa lived for about 80 years. (This is an approximate, commonly accepted figure.)

2️⃣ At what age did he start composing?

It is believed that he turned towards devotion and music after a major change in his life, around the age of 30 years.

So, he composed for about:
[80 – 30 = 50 years]

3️⃣ How many songs per year did he have to compose?

Total songs ≈ 4,75,000
Years of composing ≈ 50 years

[4,75,000 ÷ 50 = 9,500]

✅ Final Answers (approximate):

• He lived: about 80 years
• He started composing: around 30 years of age
• Songs composed per year: about 9,500 songs

🌟 Reflection

This shows why the number is considered legendary — composing nearly 26 songs every day for decades is extraordinary!

Q. 2100 × 70,000 is the approximate distance in kilometers, between the Earth
and the Sun. This distance keeps varying throughout the year. The farthest distance is about 152 million kilometers. How did they measure the distance between the Earth and the Sun?

Solution:-

Great question — this gets into how scientists measure huge distances 😊

Short answer

They did not measure it with a ruler.
The Earth–Sun distance was found using astronomy, geometry, and later radar.

1️⃣ Early method: Geometry and transits

Early astronomers used geometry.

• They observed transits of Venus (Venus passing in front of the Sun).
• Observers at different places on Earth saw Venus at slightly different positions.
• Using this difference (called parallax) and geometry, they calculated the Earth–Sun distance, called 1 Astronomical Unit (AU).

👉 This gave a fairly accurate value.

2️⃣ Parallax method (key idea)

• When you look at an object from two different points, its position seems to shift.
• The bigger the shift, the closer the object.
• Astronomers measured tiny shifts of planets and used triangles to calculate distances.

📐 Math + angles = distance

3️⃣ Modern method: Radar signals

Today, scientists use radar.

• A radar signal is sent from Earth to a nearby planet (like Venus).
• The signal bounces back.
• Since the speed of light is known, the time taken gives the distance.

Distance = Speed of light x Time divided by 2

This gives very accurate results.

4️⃣ Why the distance changes

• Earth’s orbit is elliptical, not perfectly circular.
• So the Earth–Sun distance varies from about
  147 million km (nearest) to 152 million km (farthest).

✅ Final answer (student-friendly):

Scientists measured the distance between the Earth and the Sun using geometry and parallax in early times, and radar signals in modern times. The distance is not fixed because Earth moves in an elliptical orbit around the Sun.

Page – 17

Q. 6400 × 62,500 is the average number of litres of water the Amazon river discharges into the Atlantic Ocean every second. The river’s flow into the Atlantic is so much that drinkable freshwater is found even 160 kilometers into the open sea.

Solution:-

Let’s calculate it quickly and make sense of the number 😊

Given:

[6400 x 62,500]

Quick method:

• (6400 = 64 x 100)
• (62,500 = 625 x 100)

[(64 x 625) x (100 x 100)]

Now,
[64 x 625 = 40,000]

So,
[40,000 x10,000 = [400,000,000]

✅ Final Answer

The Amazon River discharges about 400 million litres of water every second into the Atlantic Ocean.

🌍 Why this is amazing

• That’s 4 × 10⁸ litres per second
• In just 1 minute → 24 billion litres
• No wonder freshwater can be found 160 km into the sea!

Q. 13,95,000 ÷ 150 is the distance (in kms) of the longest single-train journey in the world. The train runs in Russia between Moscow and Vladivostok. The duration of this journey is about 7 days. The longest train route in India is from Dibrugarh in Assam to Kanyakumari in Tamil Nadu; it covers 4219 kms in about 76 hours.

As you did before, divide the given numbers to uncover interesting facts about division. Share your thoughts and questions with the class after you uncover each number.

Solution:-

Let’s divide step by step, then think about what the numbers tell us 😊

🔢 Step 1: Do the division

Given:
[13,95,000 ÷150]

Break it smartly:
[13,95,000 ÷ 150 = (13,95,000 ÷ 15) ÷ 10]

First:
[13,95,000 ÷ 15 = 93,000]

Then:
[93,000 ÷ 10 = 9,300]

✅ Result

The longest single-train journey in the world is about 9,300 km long.

🤔 Interesting facts we uncover (to share with the class)

1️⃣ About distance

• Russia train journey: ≈ 9,300 km
• India’s longest train route: 4,219 km

👉 The Russian route is more than twice as long as India’s longest route.

2️⃣ About time and speed

• Russia journey: 7 days ≈ 168 hours
• India journey: 76 hours

Approximate speeds:

• Russia: (9300 ÷ 168 ≈ 55 \text{ km/h})
• India: (4219 ÷ 76 ≈ 55 \text{ km/h})

👉 Interesting surprise:
Both trains run at almost the same average speed!

3️⃣ What division helps us understand

• Division helps convert big numbers into meaningful facts
• It helps us compare:
  • distances
  • time
  • speed
  • scale of countries

💬 Thoughts and questions to share with the class

• How can trains travel such long distances without stopping for days?
• Why is the average speed similar even in different countries?
• How does geography affect train routes?
• If a train ran nonstop, how much faster could it be?

🌟 Big idea

Division is not just about answers — it helps us understand the real world.

Page – 18

Q. Adult blue whales can weigh more than 10,50,00,000 ÷ 700 kilograms. A newborn blue whale weighs around 2,700 kg, which is similar to the weight of an adult hippopotamus. The heart of a blue whale was recorded to be nearly 700 kg. The tongue of a blue whale weighs as much as an elephant. Blue whales can eat up to 3500 kg of krill every day. The largest known land animal, Argentinosauras, is estimated to weigh 90,000 kgs.

Solution:-

Let’s uncover the number by division and then share the interesting facts it reveals, just like before 😊

🔢 Step 1: Do the division

Given:
[10,50,00,000 ÷ 700]

First rewrite:
[10,50,00,000 = 105,000,000]

Now divide:
[105,000,000 ÷ 700 = 150,000

✅ Result

An adult blue whale can weigh about 1,50,000 kg (that is 1.5 lakh kilograms or 150 tonnes!)

🐋 Interesting facts we uncover (to share with the class)

1️⃣ Comparing weights

• Adult blue whale: ≈ 1,50,000 kg
• Newborn blue whale: ≈ 2,700 kg
• Adult hippopotamus: ≈ 2,700 kg

👉 A newborn blue whale already weighs as much as a fully grown hippo!

2️⃣ Body parts heavier than animals

• Heart of blue whale: ≈ 700 kg
  (as heavy as a small car)
• Tongue of blue whale: ≈ one elephant

👉 Just one body part of a blue whale can weigh as much as an entire large animal.

3️⃣ Eating capacity

• Food eaten per day: 3,500 kg of krill

👉 That is more than the weight of a hippopotamus every day.

4️⃣ Comparison with the largest land animal

• Argentinosaurus: ≈ 90,000 kg
• Blue whale: ≈ 1,50,000 kg

👉 The blue whale is much heavier than the largest dinosaur to ever walk on land.

💬 Thoughts and questions to discuss in class

• How does the ocean support animals so much larger than land animals?
• Why can’t land animals grow as heavy as whales?
• How much food would a blue whale eat in a year?
• How strong must a heart be to pump blood through such a huge body?

🌟 Big idea

Division helps us break down huge numbers and truly understand how massive these animals are.

Q. 52,00,00,00,000 ÷ 130 was the weight, in tonnes, of global plastic waste generated in the year 2021.

Solution:-

Let’s divide smartly and then see what this huge number tells us 😊

🔢 Step 1: Do the division

Given:
[52,00,00,00,000 ÷ 130]

First rewrite in international form:
[52,00,00,00,000 = 52,000,000,000]

Now divide:
[52,000,000,000 ÷ 130 = 400,000,000]

✅ Result

The global plastic waste generated in 2021 was about

400,000,000 tonnes

(that is 40 crore tonnes)

🌍 Interesting facts we uncover (to share with the class)

1️⃣ Understanding the size of the number

• 400 million tonnes of plastic in just one year
• That means over 1 million tonnes every single day

👉 This shows how serious the plastic waste problem is.

2️⃣ Comparing with familiar weights

• Blue whale: ~150 tonnes
• Plastic waste in 2021 = weight of about
  [400,000,000/150} = approx 2.7 million blue whales]

😮 That’s mind-blowing!

3️⃣ Why division is useful here

• Division helps convert a huge number into a meaningful quantity
• It allows us to compare, estimate, and understand impact

💬 Thoughts and questions for class discussion

• Where does most of this plastic waste end up?
• How much plastic waste does one person generate in a year?
• What small steps can reduce this number?
• How will this number look after 10 years if nothing changes?

🌟 Big idea

Division helps us understand the real-world meaning hidden inside very large numbers.

Did You Ever Wonder

NCERT In-Text Questions (Page 19)

Q. The population of Mumbai is more than 1 crore 24 lakhs. The RMS Titanic Ship carried about 2500 passengers. Can the population of Mumbai fit into 5000 such ships?

Solution:-

Let’s check this with simple estimation 😊

Given:

• Population of Mumbai > 1 crore 24 lakh ≈ 1,24,00,000
• Passengers per Titanic ship ≈ 2,500
• Number of ships = 5,000

Step 1: Capacity of 5,000 ships

[2,500 x 5,000 = 12,500,000]

That is:
[1,25,00,000 People (1 crore 25 lakh)]

Step 2: Compare

• Mumbai population ≈ 1,24,00,000
• Capacity of ships ≈ 1,25,00,000

👉 1,25,00,000 is slightly more than 1,24,00,000

✅ Final Answer:

Yes, the population of Mumbai can fit into 5,000 such ships, approximately.

💡 Interesting thought to share in class:

• Just one city’s population would need thousands of giant ships like the Titanic!
• This shows how huge city populations really are when compared to familiar objects.

Q. Inspired by this strange question, Roxie 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)

1 How far would she have travelled in a year?

2. How far would she have travelled in 10 years?

3. Is it not easier to perform these calculations in stages? You can use this method for all large calculations.

Solution:-

Let’s answer this step by step, just like Roxie is thinking 😊 (And yes — doing it in stages makes it much easier!)

Given:

• Distance travelled per day = 100 km
• Distance to the Moon = 3,84,400 km
• 1 year ≈ 365 days

1️⃣ How far would she have travelled in one year?

[100 x 365 = 36,500 km]

👉 In 1 year, she would travel 36,500 km.

2️⃣ How far would she have travelled in 10 years?

[36,500 x 10 = 3,65,000 km]

👉 In 10 years, she would travel 3,65,000 km.

3️⃣ Can she reach the Moon in 10 years?

• Distance to Moon = 3,84,400 km
• Distance travelled in 10 years = 3,65,000 km

Since:
[3,65,000 < 3,84,400]

👉 No, she would not quite reach the Moon — she would fall short by:

[3,84,400 – 3,65,000 = 19,400 km]

4️⃣ Is it easier to calculate in stages? Why?

Yes, definitely! ✅

Why this method helps:

• Large problems become small, manageable steps
• Fewer mistakes
• Easier to estimate and check
• Works well for big numbers and real-life problems

🌟 Final takeaway (to share in class):

Breaking large calculations into stages makes them easier, faster, and more accurate. This method is very useful when working with large numbers.

Q. Find out if you can reach the Sun in a lifetime, if you travel 1000 kilometers every day.  (Distance between the Sun and the Earth from page 16 calculation = 147,000,000 km)

Solution:-

Distance between the Sun and the Earth = 14,70,00,000 km
Distance travelled in one day = 1000 km
Distance travelled in one year (365 days) = 365 × 1000 = 3,65,000 km
Time (in years) to reach the Sun = 14,70,00,000/3,65,000 = 403 years
Since a man can’t live up to 403 years. So, you can’t reach the Sun in a lifetime.

Q. Make 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?

Solution:-

Let’s find out step by step 😊

Given:

• Weight of 1 sheet = 5 grams
• Number of sheets = 1 lakh = 1,00,000

1️⃣ Total weight of 1 lakh sheets

[1,00,000 x 5 = 5,00,000 grams]

Convert grams to kilograms:
[5,00,000 ÷ 1,000 = 500 kg]

2️⃣ Can a person lift 500 kg at the same time?

👉 No, a human cannot lift 500 kg.

• Even very strong weightlifters can lift around 250–300 kg, and that too only briefly and with special equipment.
• 500 kg is roughly the weight of:
  • a small car, or
  • 6–7 adult people together.

✅ Final Answer:

No, you could not lift one lakh sheets of paper together at the same time, because together they weigh about 500 kg, which is far too heavy for a person to lift.

(b) If 250 babies are born every minute across the world, will a million babies be born in a day?

Solution:-

Let’s check it step by step 😊

Given:

• Babies born = 250 per minute
• Time = 1 day = 24 hours
• 1 hour = 60 minutes

1️⃣ Babies born in one hour

[250 x 60 = 15,000]

2️⃣ Babies born in one day

[15,000 x 24 = 3,60,000]

3️⃣ Compare with one million

• Babies born in a day = 3,60,000
• One million = 10,00,000

Since:
[3,60,000 < 10,00,000]

✅ Final Answer:

No, a million babies will not be born in a day.
Only about 3.6 lakh babies are born in a day at this rate.

🌟 Interesting thought:

At this speed, it would take almost 3 days to reach one million births.

(c) Can you count 1 million coins in a day? Assume you can count 1 coin every second.

Solution:-

Let’s find out step by step 😊

Given:

• Speed of counting = 1 coin per second
• Total coins = 1,000,000
• Time in a day = 24 hours

1️⃣ How many coins can you count in one day?

First, find seconds in a day:
[24 x 60 x 60 = 86,400 seconds]

So, in one day you can count:
[86,400 coins]

2️⃣ Compare with 1 million coins

• Coins counted in a day = 86,400
• Coins needed = 1,000,000

Since:
[86,400 < 1,000,000]

3️⃣ How long would it actually take?

[1,000,000 ÷ 86,400 = approx 11.6 days]

✅ Final Answer:

No, you cannot count 1 million coins in a day if you count 1 coin every second.
It would take about 12 days.

NCERT Text Questions Page (19 – 21)

Figure it Out

Q. 1. Using 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

Solution:-

(a) Largest multiple of 5 Key rules

A multiple of 5 must end in 0 or 5.

To make the number largest, we want the largest digits in front.

Best choice for last digit

Ending with 0 is better than 5, because 0 is smaller and can be pushed to the end, leaving larger digits in front.

Arrange remaining digits (largest to smallest)

Digits left: 1,2,3,4,5,6,7,8,9 Arrange in descending order.

✅ Answer – 9876543210

(b) Smallest even number

Good question — let’s do it carefully and correctly 👍

Conditions

• Use all digits 0–9 exactly once
• First digit ≠ 0
• Number must be even (last digit even)
• We want the smallest possible number

Step-by-step thinking

1. To make the number as small as possible, the first digit should be 1 (smallest non-zero digit).
2. Next, we place 0 as early as possible (this makes the number smaller).
3. After that, we arrange digits in increasing order.
4. The last digit must be even.

If we write:

1 0 2 3 4 5 6 7 8 9

The last digit is 9 (odd) ❌

So, we make the smallest possible change: swap the last two digits so the number ends with an even digit.

✅ Smallest even 10-digit number – [1023456798]

Q. 2. The number 10,30,285 in words is Ten lakhs thirty thousand two hundred eighty five, which has 43 letters. Give a 7-digit number name which has the maximum number of letters.

Solution:-

Nice puzzle 🙂
We’re using the Indian number system and counting letters only (no spaces or hyphens).

To maximise letters, we should use the longest number-words:

• “seventy” (7 letters) is the longest tens word
• “seven” (5 letters) is longer than one–nine alternatives

A 7-digit number with the maximum letters

77,77,777
In words:

Seventy seven lakhs seventy seven thousand seven hundred seventy seven

Letter count

✅ Final Answer

77,77,777 — “Seventy seven lakh seventy seven thousand seven hundred seventy seven”
Number of letters = 60

That’s more than Ten lakhs thirty thousand two hundred eighty five (43 letters).

Q. 3. Write a 9-digit number where exchanging any two digits results in a bigger number. How many such numbers exist?

Solution:-

Neat logical question 👌
Let’s reason it out carefully.

Condition

Exchanging any two digits results in a bigger number.

Take a 9-digit number with digits
( d₁, d₂, d₃__________ d₉ )

Pick any two positions ( i < j ).

• When we swap (d_i) and (d_j), the first place where the number changes is position (i).
• For the new number to be bigger, the digit coming to position ( i ) must be larger than the original digit there.

So we must have:

[dj > di for all i < j]

What does this mean?

The digits must be strictly increasing from left to right.

Also:

• A 9-digit number cannot start with 0
• Digits must be distinct

Verification

Any swap (say 1 and 9, or 3 and 7, or any pair):

• A larger digit moves to a more significant position
• The number always increases

✅ Condition satisfied.

Final Answers

• One such 9-digit number:
  123456789
• Number of such numbers: is always 1

Only one number in the entire universe of 9-digit numbers has this property! 🎯

Q. 4. Strike out 10 digits from the number 12345123451234512345 so that the remaining number is as large as possible.

Solution:-

Given number : 12345123451234512345

We have to strike out 10 digits so that the remaining number is as large as possible.

To be a number as large as possible, we would strike out smaller number from left side keeping largest number.

So we would strike out 1,2,3,4 from left most side. 1234 5123451234512345 taking largest number 5 at the left-most side.

Now we would again strike out smaller digits from left as follows 5 1234 51234512345 keeping largest digits at the left-most side

Now we would strike further 2 smaller digits: – 55 12 34512345

In this process we get required number as: 5534512345

Q. 5. The 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?

Solution:-

Let us examine consecutive numbers written in words:

• one and two share the letter o
• two and three share t and e
• three and four share r
• four and five share f
• five and six share i
• six and seven share s
• seven and eight share e
• eight and nine share e, i, n
• nine and ten share n, e

From this pattern, we see that every pair of consecutive numbers shares at least one English letter.

Conclusion

There is no pair of consecutive numbers whose names do not have a letter in common.

Q. 6. 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.

(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?

Solution:-

Nice set of counting problems — all about digit positions. Let’s go step by step.

(a) What is the 1000th digit? At which number does it occur?

Step 1: Count digits by number of digits

• 1 to 9 → 9 numbers × 1 digit = 9 digits
• 10 to 99 → 90 numbers × 2 digits = 180 digits
  • Total so far = 9 + 180 = 189 digits
• 100 to 999 → 900 numbers × 3 digits = 2700 digits

Since
[189 < 1000 ≤ 189 + 2700]
the 1000th digit lies in the 3-digit numbers.

Step 2: Position inside the 3-digit block

Digits needed after 99:
[1000 – 189 = 811]

Each 3-digit number contributes 3 digits:
[811 ÷ 3 = 270 remainder 1]

So:

• After 270 three-digit numbers → 810 digits
• The next digit is the first digit of the next number

That number is:
[100 + 270 = 370]

The first digit of 370 is 3.

✅ Answer (a)

• 1000th digit: 3
• It occurs in the number: 370

(b) What number would contain the millionth digit?

Step 1: Total digits before 6-digit numbers

• 1-digit numbers: 9
• 2-digit numbers: 180
• 3-digit numbers: 2700
• 4-digit numbers: 36,000
• 5-digit numbers: 450,000

Total so far:
[9 + 180 + 2700 + 36000 + 450000 = 488,889]

Step 2: Position inside 6-digit numbers

Digits remaining:
[1,000,000 – 488,889 = 511,111]

Each 6-digit number gives 6 digits:
[511,111 ÷ 6 = 85,185 remainder 1]

So the digit lies in:
[100,000 + 85,185 = 185,185]

✅ Answer (b)

• The millionth digit occurs in the number 185,185

(c) When is the digit ‘5’ written for the 5000th time?

Key observation

In long runs of numbers, each digit (0–9) appears about equally often.

So when the total number of digits written reaches about:
[5000 x 10 = 50,000]
the digit 5 will have appeared roughly 5000 times.

Step 1: Total digits up to certain points

• Up to 99 → 189 digits
• Up to 999 → 2,889 digits
• Digits needed after 999:
  [50,000 – 2,889 = 47,111]

Each 4-digit number contributes 4 digits:
[47,111 ÷ 4 = ≈11,778]

So the digit ‘5’ appears for the 5000th time during the number:
[1000 + 11,778 = 12,778]

✅ Answer (c)

• The digit ‘5’ is written for the 5000th time while writing the number 12,778

🌟 Final Answers Summary

Q. 7. 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

Solution:-

We can reach a number by pressing only +10,000 and +100.

So each number is written as:

(number of +10,000 clicks) x 10,000 + (number of +100 clicks) x 100

(a) 20,800

20, 800 = 2 x 10,000 + 8 x 100
]

Expression:
2(+10,000) + 8(+100)
Button clicks: 2 + 8 = 10

(b) 92,100

[92100 = 9 x 10,000 + 21 x 100]

Expression:
9(+10,000) + 21(+100)
Button clicks: 9 + 21 = 30

(c) 1,20,500

[1,20,500 = 12 x 10,000 + 5 x 100]

Expression:
12(+10,000) + 5(+100)
Button clicks: 12 + 5 = 17

(d) 65,30,000

[65,30,000 = 653 x 10,000 + 0 x 100]

Expression:
653(+10,000)
Button clicks: 653

(e) 70,25,700

[70,25,700 = 702 x 10,000 + 57 x 100]

Expression:
702(+10,000) + 57(+100)
Button clicks: 702 + 57 = 759

✅ Final Answers (Expressions)

(a) 2(+10,000) + 8(+100)
(b) 9(+10,000) + 21(+100)
(c) 12(+10,000) + 5(+100)
(d) 653(+10,000)
(e) 702(+10,000) + 57(+100)

Q. 8. How many lakhs make a billion?

Solution:-

In the Indian system:

• 1 lakh = 1,00,000
• 1 billion = 1,000,000,000

[1,000,000,000 ÷ 1,00,000} = 10,000]

✅ Answer

1 billion = 10,000 lakhs

(Equivalently, 1 billion = 100 crores.)

Q. 9. 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 the two resulting numbers.

Solution:-

(a) Largest Possible Sum:

Top number: 9,987,654
Bottom number: 87,654

✅ Largest possible sum

9,987,654 + 87,654 = 10,075,308

(b) Smallest Possible Sum:

Result

Top number: 1,122,334
Bottom number: 99,887

✅ Smallest possible difference

11,22,334 – 99,887 = 10,22,447

Q. 10. 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: Closest I could make is 4000 × (20 + 5) + 13000 = 1,13,000

(b) 2,00,000:

(c) 5,80,000:

(d) 12,45,000:

(e) 20,90,800:

Solution:-

Q. 11. Find out how many coins should be stacked to match the height of the Statue of Unity. Assume each coin is 1 mm thick (Statue of Unity Height = 180M).

Solution:-

Let’s convert everything to the same unit

• Height of Statue of Unity = 180 m
• 1 m = 1000 mm

So,
[180 m = 180 x 1000 = 1,80,000 mm]

Each coin is 1 mm thick.

Number of coins needed

[1,80,000 mm ÷ 1 mm per coin = 1,80,000 coins]

✅ Final Answer

You would need 1,80,000 coins stacked one on top of another to match the height of the Statue of Unity.

Q. 12. 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 the Pacific Ocean approximately?

Solution:-

We are given that a grey-headed albatross can fly about 900–1000 km per day, and one of the longest recorded single trips is about 12,000 km.

To estimate the number of days:

• At 1000 km per day:

[12,000 ÷ 1000 = 12 days]

• At 900 km per day:

[12,000 ÷ 900 = approx 13.3 days]

✅ Approximate Answer

Crossing the Pacific Ocean over a distance of about 12,000 km would take roughly 12–14 days, or about 13 days, for a grey-headed albatross.

Q. 13. 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.

Solution:-

Let’s find the average speed step by step 😊

Given

• Total distance = 13,560 km
• Total time = about 11 days

Distance covered per day

[13,560 ÷ 11 = approx 1,233 km/day]

So, the bird covered about 1,200–1,250 km per day.

Distance covered per hour

There are 24 hours in a day:

[1,233 ÷ 24 = approx 51.4 km/hour]

So, it covered about 50–52 km per hour on average.

✅ Final Answers

• Approximate distance per day: ≈ 1,230 km
• Approximate distance per hour: ≈ 51 km

Pretty incredible stamina for a bird—flying nonstop for 11 days straight!

Q. 14. 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 (40m)?

Solution:-

Let’s compare each height with Somu’s building = 40 m by finding how many times bigger it is.

1️⃣ Bald eagles (4,500 – 6,000 m)

• ( 4,500 ÷ 40 = 112.5 )
• ( 6,000 ÷ 40 = 150 )

👉 Bald eagles fly about 113 to 150 times higher than Somu’s building.

2️⃣ Mount Everest (8,850 m)

[8,850 ÷ 40 = 221.25]

👉 Mount Everest is about 221 times taller than Somu’s building.

3️⃣ Aeroplanes (10,000 – 12,800 m)

• ( 10,000 ÷ 40 = 250 )
• ( 12,800 ÷ 40 = 320 )

👉 Aeroplanes fly about 250 to 320 times higher than Somu’s building.

✅ Summary Table

This really shows how tiny a 40 m building looks compared to birds, mountains, and planes.