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

Page 2

1. 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?
See SolutionTo find out, we calculate the total number of varieties tasted over 100 years.
First, find the number of days in 100 years (ignoring leap years as suggested in the text):
100 years × 365 days/year
= 36500 days.
Then, multiply the number of days by the varieties eaten per day:
36500 days × 2 varieties/day
= 73000 varieties.
Since 73,000 is less than one lakh (100,000), no, you would not be able to taste 1 lakh varieties in 100 years if you ate 2 varieties per day.

2. 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.
See SolutionSimilar to the first question, we calculate the total varieties tasted over 100 years at a rate of 3 per day.
– Number of days in 100 years:
100 years × 365 days/year
= 36500 days.
Total varieties tasted:
36500 days × 3 varieties/day
= 109500 varieties.
Since 109,500 is more than one lakh (100,000), yes, you would be able to taste all the lakh varieties in a 100-year lifetime if you ate 3 varieties per day.

Page 3

1. Choose a number for y. How close to one lakh is the number of days in y years, for the y of your choice?
See SolutionLet’s choose two values for y and calculate the number of days, comparing it to one lakh (100,000). We use 365 days per year.
Choice 1: y = 100 years
– Number of days = y years x 365 days/year = 100 x 365 = 36500 days.
– How close to 1 lakh? 100000 – 36500 = 63500 days.
For y=100 years, the number of days (36,500) is 63,500 days less than one lakh.
Choice 2: y = 274 years (mentioned on page 4 as roughly 1 lakh days)
– Number of days = y years x 365 days/year = 274 x 365 = 100010 days.
– How close to 1 lakh? 100010 – 100000 = 10 days.
For y = 274 years, the number of days (100,010) is very close, just 10 days more than one lakh.

Figure it Out

1. According to the 2011 Census, the population of the town of Chintamani was about 75,000. How much less than one lakh is 75,000?
See SolutionTo find out, we subtract the population from one lakh (100,000).
Calculation: 100000 – 75000 = 25000.
So, 75,000 is 25,000 less than one lakh.

2. The estimated population of Chintamani in the year 2024 is 1,06,000. How much more than one lakh is 1,06,000?
See SolutionTo find out, we subtract one lakh (100,000) from the estimated 2024 population.
– Calculation: 106000 – 100000 = 6000.
So, 1,06,000 is 6,000 more than one lakh.

3. By how much did the population of Chintamani increase from 2011 to 2024?
See SolutionTo find the increase, we subtract the 2011 population from the 2024 estimated population.
– Calculation: 106000 (2024 pop) – 75000 (2011 pop) = 31000.
The population increased by 31,000 from 2011 to 2024.

4. Look 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?
See SolutionFirst, determine the height of one floor.
– Height of one floor = 4 x Somu’s height = 4 x 1 metre = 4 metres.
The image comparing the building to the Statue of Unity and the waterfall suggests the building is approximately 40 metres tall. (Assuming 10 floors visually: 10 floors x 4 metres/floor = 40 metres).
The approximate height of the building is 40 metres.

5. Which is taller – The Statue of Unity or this building? How much taller?
See SolutionCompare the height of the Statue of Unity (180 metres) to the building’s estimated height (40 metres).
– Comparison: 180 m > 40 m. The Statue of Unity is taller.
– To find how much taller, subtract the building’s height from the statue’s height: 180 m – 40 m = 140 m.
The Statue of Unity is 140 metres taller than the building.

6. How much taller is the Kunchikal waterfall than Somu’s building?
See SolutionCompare the height of the Kunchikal waterfall (450 metres) to the building’s estimated height (40 metres).
– To find how much taller, subtract the building’s height from the waterfall’s height: 450 m – 40 m = 410 m.
The Kunchikal waterfall is 410 metres taller than Somu’s building.

7. How many floors should Somu’s building have to be as high as the waterfall?
See SolutionTo find the number of floors, divide the waterfall’s height by the height of one floor.
– Height of waterfall = 450 metres. Height of one floor = 4 metres (calculated above).
– Number of floors needed = 450 m / 4 m/floor = 112.5 floors.
Somu’s building would need to have approximately 112.5 (or 113) floors to be as high as the Kunchikal waterfall.

Page 4

1. How do you view a lakh – is a lakh big or small?
See Solution– Roxie feels it’s large, citing the large number of rice varieties (one lakh), the long time span of one lakh days (274 years) and the length of a line of one lakh people (38 km).
– Estu thinks it’s not that big, mentioning a stadium capacity greater than one lakh, the number of hairs on a human head (around one lakh), and fish laying one lakh or more eggs at once.
Therefore, the perception of “one lakh” being big or small is subjective and depends on what it’s being compared to.

2. Write each of the numbers given below in words:
(a) 3,00,600
See AnswerThree lakh six hundred.
(b) 5,04,085
See AnswerFive lakh four thousand eighty-five.
(c) 27,30,000
See AnswerTwenty-seven lakh thirty thousand.
(d) 70,53,138
See AnswerSeventy lakh fifty-three thousand one hundred thirty-eight.

3. 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
See Answer1,23,456
(b) Four lakh seven thousand seven hundred and four
See Answer4,07,704
(c) Fifty lakhs five thousand and fifty
See Answer50,05,050
(d) Ten lakhs two hundred and thirty five
See Answer10,00,235

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Section 1.2: Land of Tens
1. The Thoughtful Thousands only has a +1000 button. How many times should it be pressed to show:
(a) Three thousand?
See Solution3 times (3 x 1000 = 3000)
(b) 10,000?
See Solution10 times (10 x 1000 = 10,000)
(c) Fifty three thousand?
See Solution53 times (53 x 1000 = 53,000)
(d) 90,000?
See Solution90 times (90 x 1000 = 90,000)
(e) One Lakh?
See Solution100 times (100 x 1000 = 1,00,000)
(f) What number is shown if the button is pressed 153 times?
See Solution153,000 (153 x 1000 = 153,000)
(g) How many thousands are required to make one lakh?
See Solution100 thousands (100,000 / 1000 = 100)

2. The Tedious Tens only has a +10 button. How many times should it be pressed to show:
(a) Five hundred?
See Solution50 times (50 x 10 = 500)
(b) 780?
See Solution78 times (78 x 10 = 780)
(c) 1000?
See Solution100 times (100 x 10 = 1000)
(d) 3700?
See Solution370 times (370 x 10 = 3700)
(e) 10,000?
See Solution1000 times (1000 x 10 = 10,000)
(f) One lakh?
See Solution10,000 times (10,000 x 10 = 1,00,000)
(g) What number is shown if the button is pressed 435 times?
See Solution4,350 (435 x 10 = 4,350)

3. The Handy Hundreds only has a +100 button. How many times should it be pressed to show:
(a) Four hundred?
See Answer4 times (4 x 100 = 400)
(b) 3,700?
See Answer37 times (37 x 100 = 3,700)

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(c) 10,000?
See Answer100 times (100 x 100 = 10,000)
(d) Fifty three thousand?
See Answer530 times (530 x 100 = 53,000)
(e) 90,000?
See Answer900 times (900 x 100 = 90,000)
(f) 97,600?
See Answer976 times (976 x 100 = 97,600)
(g) 1,00,000?
See Answer1000 times (1000 x 100 = 1,00,000)
(h) What number is shown if the button is pressed 582 times?
See Answer58,200 (582 x 100 = 58,200)
(i) How many hundreds are required to make ten thousand?
See Answer100 hundreds (10,000 / 100 = 100)
(j) How many hundreds are required to make one lakh?
See Answer1000 hundreds (100,000 / 100 = 1000)
(k) Handy Hundreds says, “There are some numbers which Tedious Tens and Thoughtful Thousands can’t show but I can.” Is this statement true?
See AnswerYes, the statement is true. Handy Hundreds (+100) can show multiples of 100 (like 3,700). Tedious Tens (+10) can only show multiples of 10. Thoughtful Thousands (+1000) can only show multiples of 1000. Therefore, Handy Hundreds can show numbers like 3,700 which the other two calculators cannot create directly (though Tedious Tens could eventually reach it with 370 presses).

5. Two ways to get 5072 are shown:
(a) (50×100)+(7×10)+(2×1)=5072 and
(b) (3×1000)+(20×100)+(72×1)=5072. Find a different way to get 5072 and write an expression for the same.
See SolutionHere is another way: Use the standard place value decomposition.
Expression: (5 x 1000) + (0 x 100) + (7 x 10) + (2 x 1)
= 5000 + 0 + 70 + 2
= 5072
Another example: (4 x 1000) + (10 x 100) + (7 x 10) + (2 x 1)
= 4000 + 1000 + 70 + 2
= 5072

6. Figure it Out:
For each number below, write expressions for at least two different ways to obtain the number through button clicks (like Chitti).
(a) 8300
See SolutionWay 1: (8 x 1000) + (3 x 100) = 8000 + 300 = 8300
Way 2: (83 x 100) = 8300

(b) 40629
See SolutionWay 1: (4 x 10000) + (6 x 100) + (2 x 10) + (9 x 1) = 40000 + 600 + 20 + 9 = 40629
Way 2: (40 x 1000) + (62 x 10) + (9 x 1) = 40000 + 620 + 9 = 40629

(c) 56354
See SolutionWay 1: (5 x 10000) + (6 x 1000) + (3 x 100) + (5 x 10) + (4 x 1) = 50000 + 6000 + 300 + 50 + 4 = 56354
Way 2: (56 x 1000) + (35 x 10) + (4 x 1) = 56000 + 350 + 4 = 56354

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(d) 66666
See SolutionWay 1: (6 x 10000) + (6 x 1000) + (6 x 100) + (6 x 10) + (6 x 1) = 60000 + 6000 + 600 + 60 + 6 = 66666
Way 2: (66 x 1000) + (66 x 10) + (6 x 1) = 66000 + 660 + 6 = 66666

(e) 367813
See SolutionWay 1: (3 x 100000) + (6 x 10000) + (7 x 1000) + (8 x 100) + (1 x 10) + (3 x 1) = 300000 + 60000 + 7000 + 800 + 10 + 3 = 367813
Way 2: (367 x 1000) + (81 x 10) + (3 x 1) = 367000 + 810 + 3 = 367813

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?
See SolutionLargest 3-digit number: To maximize the number, use the largest place value buttons (+100) as much as possible while staying within 3 digits and 30 clicks.
– (9 x 100) uses 9 clicks = 900. Remaining clicks = 30 – 9 = 21.
– (9 x 10) uses 9 clicks = 90. Remaining clicks = 21 – 9 = 12.
– (12 x 1) uses 12 clicks = 12.
– Total = 900 + 90 + 12 = 1002 (This is 4 digits).
Let’s reduce the hundreds clicks by 1.
– (8 x 100) uses 8 clicks = 800. Remaining clicks = 30 – 8 = 22.
– (9 x 10) uses 9 clicks = 90. Remaining clicks = 22 – 9 = 13.
– (13 x 1) uses 13 clicks = 13.
– Total = 800 + 90 + 13 = 903. Total clicks = 8 + 9 + 13 = 30.
Let’s try maximizing 100s and 10s differently:
– (9 x 100) uses 9 clicks = 900. Remaining clicks = 21.
– (8 x 10) uses 8 clicks = 80. Remaining clicks = 21 – 8 = 13.
– (13 x 1) uses 13 clicks = 13.
– Total = 900 + 80 + 13 = 993. Total clicks = 9 + 8 + 13 = 30. This is the largest possible 3-digit number.
Smallest 3-digit number: To minimize the number, use the smallest place value buttons (+1) as much as possible, ensuring the number is at least 100 and uses 30 clicks.
– Must use at least one +100 button (1 click). Value = 100. Remaining clicks = 29.
– Using only +1 for the rest: (29 x 1) uses 29 clicks. Value = 29.
– Total = 100 + 29 = 129. Total clicks = 1 + 29 = 30.
Can we make it smaller using +10?
– (1 x 100) uses 1 click = 100. Remaining clicks = 29.
– (1 x 10) uses 1 click = 10. Remaining clicks = 28.
– (28 x 1) uses 28 clicks = 28.
– Total = 100 + 10 + 28 = 138. Total clicks = 1 + 1 + 28 = 30.
The smallest seems to be 129.
Largest 3-digit number is 993. Smallest 3-digit number is 129.

(b) 997 can be made using 25 clicks. Can you make 997 with a different number of clicks?
See SolutionYes. The minimal way (using Systematic Sippy’s logic) is (9 x 100) + (9 x 10) + (7 x 1), which takes 9 + 9 + 7 = 25 clicks.
Other ways are possible:
– Example: (8 x 100) + (19 x 10) + (7 x 1) = 800 + 190 + 7 = 997. Clicks = 8 + 19 + 7 = 34 clicks.
– Example: (99 x 10) + (7 x 1) = 990 + 7 = 997. Clicks = 99 + 7 = 106 clicks.
So, yes, 997 can be made with different numbers of clicks (more than 25).
Systematic Sippy is a different kind of calculator. It has buttons: +1, +10, +100, +1000, +10000, +100000. It wants to be used as minimally as possible.

1. How can we get the numbers (a) 5072, (b) 8300 using as few button clicks as possible?
See SolutionUse the standard place value decomposition, pressing each place value button corresponding to the digit in that place.
(a) 5072: (5 x 1000) + (0 x 100) + (7 x 10) + (2 x 1). Minimum clicks = 5 + 0 + 7 + 2 = 14 clicks.
(b) 8300: (8 x 1000) + (3 x 100) + (0 x 10) + (0 x 1). Minimum clicks = 8 + 3 + 0 + 0 = 11 clicks.

2. Is there another way to get 5072 using less than 23 button clicks? Write the expression for the same.
See SolutionYes. The example in the chapter showed a way using 23 clicks ((5 x 1000) + (6 x 10) + (12 x 1)). The minimal way uses fewer clicks.
Minimal clicks expression: (5 x 1000) + (7 x 10) + (2 x 1).
Number of clicks = 5 + 7 + 2 = 14 clicks, which is less than 23.

Figure it Out:

1. For the numbers in the previous exercise (8300, 40629, 56354, 66666, 367813), find out how to get each number by making the smallest number of button clicks and write the expression.
See SolutionThe smallest number of clicks corresponds to the standard place value expansion:
– 8300: Expression: (8 x 1000) + (3 x 100). Clicks = 8 + 3 = 11.
– 40629: Expression: (4 x 10000) + (6 x 100) + (2 x 10) + (9 x 1). Clicks = 4 + 6 + 2 + 9 = 21.
– 56354: Expression: (5 x 10000) + (6 x 1000) + (3 x 100) + (5 x 10) + (4 x 1). Clicks = 5 + 6 + 3 + 5 + 4 = 23.
– 66666: Expression: (6 x 10000) + (6 x 1000) + (6 x 100) + (6 x 10) + (6 x 1). Clicks = 6 + 6 + 6 + 6 + 6 = 30.
– 367813: Expression: (3 x 100000) + (6 x 10000) + (7 x 1000) + (8 x 100) + (1 x 10) + (3 x 1). Clicks = 3 + 6 + 7 + 8 + 1 + 3 = 28.

2. Do you see any connection between each number and the corresponding smallest number of button clicks?
See SolutionYes. The smallest number of button clicks needed to make a number using Systematic Sippy’s method is equal to the sum of the digits of that number when written in standard form.

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.
See SolutionThe statement is slightly imprecise. The expressions for the least button clicks represent the standard base-10 place value expansion (used internationally), not specifically the Indian grouping system (lakhs, crores). This method works because the calculator buttons correspond directly to powers of 10 (1, 10, 100, 1000, etc.). The standard way of writing numbers is a shorthand for expressing the number as a sum of its digits multiplied by these powers of 10. To minimize clicks, we use each place value button (like +1000) no more than 9 times, which perfectly matches how we write numbers in base-10.

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1. What if we press the +10,00,000 button ten times? What number will come up?
See SolutionTo find the number, multiply 10 by 10,00,000.
– Calculation: 10 x 10,00,000 = 1,00,00,000.
The number will be 1,00,00,000.

2. How many zeroes will it have?
See SolutionThe number 1,00,00,000 has 7 zeroes.

3. What should we call it?
See SolutionThe number 1,00,00,000 (1 followed by 7 zeroes) is called One Crore in the Indian system or Ten Million in the American/International system.

Section 1.3: Of Crores and Crores!

How many zeros does a thousand lakh have?
See SolutionFirst, calculate the value of a thousand lakh.
– Calculation: 1000 (thousand) x 1,00,000 (lakh) = 10,00,00,000.
This number is 10 crore (Indian System) or 100 million (American System).
The number 10,00,00,000 has 8 zeroes.

Page 9

1. How many zeros does a hundred thousand have?
See SolutionA hundred thousand is written as 100,000.
It has 5 zeroes.

Figure it Out

1. Read the following numbers in Indian place value notation and write their number names in both the Indian and American systems:
(a) 4050678
See SolutionIndian Notation: 40,50,678
Indian Name: Forty lakh fifty thousand six hundred seventy-eight.
American Name: Four million fifty thousand six hundred seventy-eight.

(b) 48121620
See SolutionIndian Notation: 4,81,21,620
Indian Name: Four crore eighty-one lakh twenty-one thousand six hundred twenty.
American Name: Forty-eight million one hundred twenty-one thousand six hundred twenty.

(c) 20022002
See SolutionIndian Notation: 2,00,22,002
Indian Name: Two crore twenty-two thousand two.
American Name: Twenty million twenty-two thousand two.

(d) 246813579
See SolutionIndian Notation: 24,68,13,579
Indian Name: Twenty-four crore sixty-eight lakh thirteen thousand five hundred seventy-nine.
American Name: Two hundred forty-six million eight hundred thirteen thousand five hundred seventy-nine.

(e) 345000543
See SolutionIndian Notation: 34,50,00,543
Indian Name: Thirty-four crore fifty lakh five hundred forty-three.
American Name: Three hundred forty-five million five hundred forty-three.

(f) 1020304050
See SolutionIndian Notation: 1,02,03,04,050
Indian Name: One arab two crore three lakh four thousand fifty (or One hundred two crore three lakh four thousand fifty).
American Name: One billion twenty million three hundred four thousand fifty.

2. Write the following numbers in Indian place value notation:
(a) One crore one lakh one thousand ten
See Answers1,01,01,010

(b) One billion one million one thousand one
See SolutionAmerican notation: 1,001,001,001
Equivalent Indian value: 1 arab 10 lakh 1 thousand 1 (since 1 billion = 1 arab = 100 crore; 1 million = 10 lakh).
Indian Notation: 1,00,10,01,001

(c) Ten crore twenty lakh thirty thousand forty
See Answer10,20,30,040

(d) Nine billion eighty million seven hundred thousand six hundred
See SolutionAmerican notation: 9,080,700,600
Equivalent Indian value: 9 arab 8 crore 7 lakh 6 hundred.
Indian Notation: 9,08,07,00,600

3. Compare and write ‘<‘, ‘>’ or ‘=’:
(a) 30 thousand ___ 3 lakhs
See Solution30 thousand = 30,000. 3 lakhs = 3,00,000.
30,000 < 3,00,000. So, 30 thousand < 3 lakhs.

(b) 500 lakhs ___ 5 million
See Solution500 lakhs = 500 x 1,00,000 = 5,00,00,000. 5 million = 5,000,000.
5,00,00,000 > 5,000,000. So, 500 lakhs > 5 million.

(c) 800 thousand ___ 8 million
See Solution800 thousand = 800,000. 8 million = 8,000,000.
800,000 < 8,000,000. So, 800 thousand < 8 million.

(d) 640 crore ___ 60 billion
See Solution640 crore = 640 x 1,00,00,000 = 6,40,00,00,000.
60 billion = 60 x 1,00,00,00,000 (American billion) = 60,00,00,00,000.
6,40,00,00,000 < 60,00,00,00,000. So, 640 crore < 60 billion.

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Section 1.4: Exact and Approximate Values

1. (Regarding the cartoon conversation about 1 lakh visitors vs 99,999) What do you think of this conversation? Have you read or heard such headlines or statements?
See SolutionThe conversation highlights the difference between exact numbers and approximations. Headlines often use rounded numbers (like “1 lakh visitors”) because they are easier to grasp and the exact number isn’t always necessary to convey the general scale. It’s common practice in news reporting and everyday conversation to use approximate values when precision isn’t critical. The boy’s comment about “99,999 visitors” if he hadn’t gone is humorous but technically incorrect, as the rounded figure “1 lakh” likely represents a number somewhere around 100,000, not exactly 100,000.

2. 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.
See Solution(a) Round Up: Ordering supplies for an event (like the school ordering 750 sweets for 732 people) to ensure there’s enough; estimating project completion time to provide a buffer; calculating material needed for construction to avoid shortages.
(b) Round Down: A shopkeeper estimating the cost for a customer might round down slightly (e.g., saying Rs 450 for a Rs 470 item) to make it sound more attractive; estimating remaining fuel in a car to be cautious.
(c) Either is Okay: Casual conversation about large numbers where precision isn’t important (e.g., “The city has about 10 lakh people”); rough estimations of distance or travel time for planning.
(d) Exact Numbers Needed: Calculating financial transactions (bank balances, salaries, taxes); scientific measurements and experiments; engineering specifications; emergency contact numbers (like dialing 139 for Railway Enquiry, not a rounded number); recipes requiring precise ingredient amounts.

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Nearest Neighbours

Write the five nearest neighbours for these numbers:
(a) 3,87,69,957
See SolutionNearest thousand: 3,87,70,000 (since 957 is >= 500, round up the thousands digit)
Nearest ten thousand: 3,87,70,000 (since 69,957 is >= 5,000, round up the ten thousands digit)
Nearest lakh: 3,88,00,000 (since 69,957 is >= 50,000, round up the lakhs digit)
Nearest ten lakh: 3,90,00,000 (since 87,69,957 is >= 5,00,000, round up the ten lakhs digit)
Nearest crore: 4,00,00,000 (since 87,69,957 is >= 50,00,000, round up the crores digit).

(b) 29,05,32,481
See SolutionNearest thousand: 29,05,32,000 (since 481 is < 500, round down)
Nearest ten thousand: 29,05,30,000 (since 2,481 is < 5,000, round down)
Nearest lakh: 29,05,00,000 (since 32,481 is < 50,000, round down) Nearest ten lakh: 29,10,00,000 (since 5,32,481 is >= 5,00,000, round up)
Nearest crore: 29,00,00,000 (since 05,32,481 is < 50,00,000, round down)

I have a number for which all five nearest neighbours (thousand, ten thousand, lakh, ten lakh, crore) are 5,00,00,000. What could the number be? How many such numbers are there?
See SolutionFor all approximations to round to 5,00,00,000, the number must satisfy the rounding condition for the smallest place value (nearest thousand) and be within the range for all larger place values.
– Nearest thousand is 5,00,00,000: Number must be in the range [4,99,99,500 to 5,00,00,499].
– Nearest ten thousand is 5,00,00,000: Number must be in the range [4,99,95,000 to 5,00,04,999].
– Nearest lakh is 5,00,00,000: Number must be in the range [4,99,50,000 to 5,00,49,999].
– Nearest ten lakh is 5,00,00,000: Number must be in the range [4,95,00,000 to 5,04,99,999].
– Nearest crore is 5,00,00,000: Number must be in the range [4,50,00,000 to 5,49,99,999].
The condition that satisfies all these simultaneously is the narrowest range, which is for the nearest thousand: [4,99,99,500 to 5,00,00,499].
Any integer number within this range would work.
How many such numbers? Count the integers from 4,99,99,500 to 5,00,00,499 inclusive.
– Number of integers = Last number – First number + 1
– Calculation: 50,000,499 – 49,999,500 + 1 = 999 + 1 = 1000.
There are 1000 such numbers.
Examples include 4,99,99,500; 5,00,00,000; 5,00,00,499.

Roxie and Estu are estimating the values of simple expressions.
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?
See SolutionExact sum = 463128 + 419682 = 882810.
– Roxie’s estimate (near 8L, >8L) is correct. Distance = 882810 – 800000 = 82810.
– Estu’s estimate (near 9L, <9L) is correct. Distance = 900000 – 882810 = 17190.
Both estimates are directionally correct. Estu’s estimate (9,00,000) is closer to the actual sum (8,82,810).

(b) Will the sum be greater than 8,50,000 or less than 8,50,000? Why do you think so?
See SolutionThe sum (8,82,810) is greater than 8,50,000.
Reasoning: 4,63,128 is greater than 4,50,000 and 4,19,682 is greater than 4,00,000. Adding the lower bounds 4.5L + 4L = 8.5L. Since both numbers are somewhat larger than these bounds (especially 463k vs 450k), the sum will be greater than 8,50,000. Alternatively, 460k + 410k = 870k, which is already > 850k.

(c) Will the sum be greater than 8,83,128 or less than 8,83,128? Why do you think so?
See SolutionThe sum (8,82,810) is less than 8,83,128.
Reasoning: Compare the second number (4,19,682) to the difference between 8,83,128 and the first number (4,63,128). Difference = 883128 – 463128 = 420000. Since 4,19,682 is less than 4,20,000, the actual sum will be less than 8,83,128.

(d) Exact value of 4,63,128 + 4,19,682 = ?
See Answer8,82,810

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?
See SolutionExact difference = 1463128 – 490020 = 973108.
– Roxie’s estimate (near 10L, <10L) is correct. Distance = 1000000 – 973108 = 26892. – Estu’s estimate (near 9L, >9L) is correct. Distance = 973108 – 900000 = 73108.
Both estimates are directionally correct. Roxie’s estimate (10,00,000) is closer to the actual difference (9,73,108).

(b) Will the difference be greater than 9,50,000 or less than 9,50,000? Why do you think so?
See SolutionThe difference (9,73,108) is greater than 9,50,000.
Reasoning: Estimate roughly: 14.6 lakhs – 4.9 lakhs. Since 4.9 lakhs is slightly less than 5 lakhs, subtracting it from 14.6 lakhs will leave slightly more than 14.6 – 5 = 9.6 lakhs. 9.6 lakhs (9,60,000) is greater than 9,50,000.

(c) Will the difference be greater than 9,63,128 or less than 9,63,128? Why do you think so?
See SolutionThe exact difference is 1463128 – 490020 = 973108.
This is greater than 9,63,128.
Reasoning: Let’s compare the number being subtracted (4,90,020) with the difference between the first number and the target number (14,63,128 – 9,63,128 = 5,00,000). Since 4,90,020 is less than 5,00,000, we are subtracting a smaller amount than would be needed to reach exactly 9,63,128. Therefore, the result will be greater than 9,63,128.

(d) Exact value of 14,63,128 – 4,90,020 = ?
See Answer9,73,108

Page 13

From the information given in the table, answer the following questions by approximation:
1. What is your general observation about this data? Share it with the class.
See SolutionA general observation is that the population of most, if not all, listed Indian cities increased significantly between the 2001 census and the 2011 census. The rate of growth varies between cities, with some showing much larger increases than others. The ranking of cities by population may also have changed slightly between the two census years.

2. What is an appropriate title for the above table?
See SolutionAn appropriate title could be: “Population of Selected Major Indian Cities (2001 & 2011 Census)” or “Comparison of City Populations in India: 2001 vs 2011”.

3. How much is the population of Pune in 2011? Approximately, by how much has it increased compared to 2001?
See SolutionPune’s population in 2011 was 31,15,431.
Its population in 2001 was 25,38,473.
Increase = 2011 Population – 2001 Population
– Calculation: 3,115,431 – 2,538,473 = 576,958.
Approximately, the population increased by about 5.8 lakhs or nearly 6 lakhs.

4. Which city’s population increased the most between 2001 and 2011?
See SolutionTo find the largest increase, we calculate the difference for a few top cities:
– Mumbai: 12,442,373 – 11,978,450 = 463,923
– New Delhi: 11,007,835 – 9,879,172 = 1,128,663
– Bengaluru: 8,425,970 – 4,301,326 = 4,124,644
– Hyderabad: 6,809,970 – 3,637,483 = 3,172,487
– Surat: 4,467,797 – 2,433,835 = 2,033,962
Comparing these increases, Bengaluru had the largest absolute population increase between 2001 and 2011.

5. Are there cities whose population has almost doubled? Which are they?
See Solution“Almost doubled” means the 2011 population is close to or greater than twice the 2001 population.
– Bengaluru: 2011 Pop = 84.3L; 2 x 2001 Pop = 2 x 43.0L = 86.0L. (Close, but slightly less than double).
– Hyderabad: 2011 Pop = 68.1L; 2 x 2001 Pop = 2 x 36.4L = 72.8L. (Less than double).
– Surat: 2011 Pop = 44.7L; 2 x 2001 Pop = 2 x 24.3L = 48.6L. (Less than double).
– Vadodara: 2011 Pop = 35.5L; 2 x 2001 Pop = 2 x 16.9L = 33.8L. (Yes, more than doubled).
Considering “almost doubled”, Bengaluru is very close. Vadodara actually more than doubled.

6. By what number should we multiply Patna’s population to get a number/population close to that of Mumbai?
See SolutionPatna 2011 Population approx. = 16.8 lakhs (1,684,222).
Mumbai 2011 Population approx. = 124.4 lakhs (1,24,42,373).
Multiplier approx. = Mumbai Pop / Patna Pop
– Calculation: 124.4 / 16.8 approx. = 7.4
We should multiply Patna’s population by approximately 7 to get close to Mumbai’s population.

Page 14

Section 1.5: Patterns in Products

A Multiplication Shortcut (Explanation of multiplying by 5 and 25)

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?
See SolutionMultiplying by 5 can be thought of as multiplying by (10 / 2).
So, Number x 5 = Number x (10 / 2).
Using the associative property of multiplication and division, this can be rewritten as (Number x 10) / 2 or (Number / 2) x 10.
The expression (Number / 2) x 10 (dividing by 2 then multiplying by 10) works easily if the number is even.
The expression (Number x 10) / 2 (multiplying by 10 then dividing by 2) always works. Both are equivalent to multiplying by 5.

Figure it Out

1. Find quick ways to calculate these products:
(a) 2 x 1768 x 50
See SolutionRearrange using commutative property: (2 x 50) x 1768 = 100 x 1768 = 176,800.
(b) 72 x 125 [Hint: 125 = 1000/8]
See Solution72 x (1000 / 8) = (72 / 8) x 1000 = 9 x 1000 = 9,000.
(c) 125 x 40 x 8 x 25
See SolutionRearrange and group: (125 x 8) x (40 x 25) = 1000 x 1000 = 1,000,000.

2. Calculate these products quickly.
(a) 25 x 12
See Solution25 x 12 = (100 / 4) x 12 = 100 x (12 / 4) = 100 x 3 = 300.
(b) 25 x 240
See Solution25 x 240 = (100 / 4) x 240 = 100 x (240 / 4) = 100 x 60 = 6,000.
(c) 250 x 120
See Solution250 x 120 = (1000 / 4) x 120 = 1000 x (120 / 4) = 1000 x 30 = 30,000.
(d) 2500 x 12
See Solution2500 x 12 = (10000 / 4) x 12 = 10000 x (12 / 4) = 10000 x 3 = 30,000.
(e) ___ x ___ = 12000000000
See SolutionThere are many possibilities. The number is 12 billion.
Example 1: 12 x 1,000,000,000 = 12,000,000,000
Example 2: 120,000 x 100,000 = 12,000,000,000
Example 3: 30,000 x 400,000 = 12,000,000,000

How Long is the Product?
(Evaluate the multiplications to find the pattern.)
Pattern Set 1:
11 x 11 = 121
111 x 111 = 12321
1111 x 1111 = 1234321
(Pattern: Digits ascend 1 to N, then descend N-1 to 1, where N is the number of 1s).
Pattern Set 2 (Assuming typo, should be 66×66, 666×666 etc.):
66 x 66 = 4356
666 x 666 = 443556
6666 x 6666 = 44435556
(Pattern: N 4s, followed by N-1 5s, followed by a 6, where N is the number of 6s, with the middle 5 replaced by a 3). OR (N-1 4s, a 3, N-1 5s, a 6).
Pattern Set 3 (Assuming 3×5, 33×35, 333×335):
3 x 5 = 15
33 x 35 = 1155
333 x 335 = 111555
(Pattern: N 1s followed by N 5s, where N is the number of 3s in the first factor).
Pattern Set 4:
101 x 101 = 10201
102 x 102 = 10404
103 x 103 = 10609
(Pattern: These are squares of numbers just above 100. (100+n)² = 10000 + 200n + n². The result is 1, followed by 2n, followed by n², padding with zeros as needed e.g., 10000+400+4=10404).

Page 15

1. Observe the number of digits in the two numbers being multiplied and their product in each case (from page 14 patterns). Is there any connection between the numbers being multiplied and the number of digits in their product?
See SolutionYes, there is a connection. If you multiply an M-digit number by an N-digit number, the product will have either (M + N – 1) digits or (M + N) digits.
– The minimum number of digits occurs when the product is relatively small (e.g., 10 x 10 = 100; M = 2, N = 2, M + N – 1 = 3 digits).
– The maximum number of digits occurs when the product is relatively large (e.g., 99 x 99 = 9801; M = 2, N = 2, M + N = 4 digits).

2. Roxie says that the product of two 2-digit numbers can only be a 3- or a 4-digit number. Is she correct?
See SolutionYes, Roxie is correct.
– The smallest product of two 2-digit numbers is 10 x 10 = 100 (which has 3 digits).
– The largest product of two 2-digit numbers is 99 x 99 = 9801 (which has 4 digits).
Therefore, the product must have either 3 or 4 digits.

3. 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?
See SolutionNo, we don’t need to try all possible multiplications. A better way is to check the extremes: calculate the product of the smallest possible 2-digit numbers (10 x 10) and the product of the largest possible 2-digit numbers (99 x 99). This establishes the minimum and maximum possible number of digits in the result.

4. Can multiplying a 3-digit number with another 3-digit number give a 4-digit number?
See SolutionNo.
– The smallest product of two 3-digit numbers is 100 x 100 = 10,000 (which has 5 digits).
– The largest product of two 3-digit numbers is 999 x 999 = 998,001 (which has 6 digits).
The product will always have 5 or 6 digits.

5. Can multiplying a 4-digit number with a 2-digit number give a 5-digit number?
See SolutionYes.
– The smallest product of a 4-digit and a 2-digit number is 1000 x 10 = 10,000 (which has 5 digits).
– The largest product is 9999 x 99 = 989,901 (which has 6 digits).
The product can have 5 or 6 digits.

6. Observe the multiplication statements below. Do you notice any patterns? See if this pattern extends for other numbers as well. (Fill in the blanks based on the M+N-1 or M+N digits rule)
See Solution1-digit X 1-digit = 1-digit or 2-digit
2-digit X 1-digit = 2-digit or 3-digit
2-digit X 2-digit = 3-digit or 4-digit
3-digit X 3-digit = 5-digit or 6-digit
5-digit X 5-digit = 9-digit or 10-digit (Min digits = 5 + 5 – 1 = 9; Max digits = 5 + 5 = 10)
8-digit X 3-digit = 10-digit or 11-digit (Min digits = 8 + 3 – 1 = 10; Max digits = 8 + 3 = 11)
12-digit X 13-digit = 24-digit or 25-digit (Min digits = 12 + 13 – 1  = 24; Max digits = 12 + 13 = 25)

Page 16

Fascinating Facts about Large Numbers
Calculate the product or quotient to uncover the facts.
1. 1250 x 380 = ? (Number of kirtanas composed by Purandaradāsa)
See SolutionCalculation: 1250 x 380 = 125 x 10 x 38 x 10 = (125 x 38) x 100
= (125 x (40 – 2)) x 100 = ((125 x 40) – (125 x 2)) x 100
= (5000 – 250) x 100 = 4750 x 100 = 475,000.
Result: 475,000 kirtanas (Four lakh seventy-five thousand kirtanas).

2. Question about Purandaradāsa: If he composed 4,75,000 songs, how many songs per year did he have to compose?
See SolutionThe text does not provide the number of years Purandaradāsa lived or composed for. Without knowing the time period over which these songs were composed, we cannot calculate the number of songs composed per year.

3. 2100 x 70,000 = ? (Approximate distance Earth-Sun in km)
See SolutionCalculation: 2100 x 70,000 = (21 x 100) x (7 x 10,000)
= (21 x 7) x (100 x 10,000) = 147 x 1,000,000 = 147,000,000.
Result: 147,000,000 km (One hundred forty-seven million km or Fourteen crore seventy lakh km).

Page 17

1. Question about Earth-Sun distance: How did they measure the distance between the Earth and the Sun?
See SolutionThe provided text does not explain the method used to measure the distance between the Earth and the Sun. (Methods historically include techniques like parallax, and modern methods involve radar measurements and Kepler’s laws of planetary motion).

2. 6400 x 62,500 = ? (Average litres/sec water discharge of Amazon river)
See SolutionCalculation: 6400 x 62,500 = (64 x 100) x (625 x 100)
= (64 x 625) x (100 x 100)
To calculate 64 x 625: 64 x 625 = 64 x (600 + 25) = (64 x 600) + (64 x 25) = 38400 + 1600 = 40,000.
Result = 40,000 x 10,000 = 400,000,000.
Result: 400,000,000 litres/sec (Four hundred million litres/sec or Forty crore litres/sec).

3. 13,95,000 / 150 = ? (Distance in km of longest single-train journey)
See SolutionCalculation: 13,95,000 / 150 = 139,500 / 15
= (135,000 + 4,500) / 15 = (135,000 / 15) + (4,500 / 15)
= 9,000 + 300 = 9,300.
Result: 9,300 km.

Page 18

1. 10,50,00,000 / 700 = ? (Weight in kg an adult blue whale can weigh more than)
See SolutionCalculation: 10,50,00,000 / 700 = 105,000,000 / 700 = 1,050,000 / 7
= (700,000 + 350,000) / 7 = (700,000 / 7) + (350,000 / 7)
= 100,000 + 50,000 = 150,000.
Result: 150,000 kg.

2. 52,00,00,00,000 / 130 = ? (Weight in tonnes of global plastic waste generated in 2021)
See SolutionCalculation: 52,00,00,00,000 / 130 = 5,200,000,000 / 13
Since 52 / 13 = 4, then 5,200,000,000 / 13 = 400,000,000.
Result: 400,000,000 tonnes (Four hundred million tonnes or Forty crore tonnes).

Page 19

Section 1.6: Did You Ever Wonder…?
1. Could the entire population of Mumbai fit into 1 lakh buses?
See SolutionThe text calculates this:
– Assume a bus capacity of 50 people.
– Total capacity of 1 lakh buses = 1,00,000 buses x 50 people/bus = 50,00,000 people (50 lakh).
– Mumbai’s population (from page 12) is 1,24,42,373 (more than 1 crore 24 lakh).
Since 50 lakh is much less than 1 crore 24 lakh, no, the entire population of Mumbai cannot fit into 1 lakh buses.

2. The RMS Titanic ship carried about 2500 passengers. Can the population of Mumbai fit into 5000 such ships?
See SolutionFirst, calculate the total capacity of 5000 Titanic-sized ships.
– Capacity = 5000 ships x 2500 passengers/ship = 12,500,000 passengers.
This capacity is 1 crore 25 lakh. Mumbai’s population is 1,24,42,373 (just over 1 crore 24 lakh).
Since the capacity (1.25 crore) is slightly larger than Mumbai’s population (approx. 1.24 crore), yes, the population of Mumbai could (just barely) fit into 5000 such ships.

3. If I could travel 100 kilometers every day, could I reach the Moon in 10 years? (Distance to Moon = 3,84,400 km)
See SolutionCalculate the total distance traveled in 10 years.
– Distance per year = 100 km/day x 365 days/year = 36,500 km.
– Distance in 10 years = 36,500 km/year x 10 years = 365,000 km.
Compare the distance traveled (365,000 km) to the distance to the Moon (384,400 km).
Since 365,000 km is less than 384,400 km, no, you would not reach the Moon in 10 years traveling at 100 km per day.

4. Find out if you can reach the Sun in a lifetime, if you travel 1000 kilometers every day. (Distance to Sun from page 16 calculation = 147,000,000 km)
See SolutionCalculate the distance traveled in a lifetime (assume 100 years).
– Distance per year = 1000 km/day x 365 days/year = 365,000 km.
– Distance in 100 years = 365,000 km/year x 100 years = 36,500,000 km (3 crore 65 lakh km).
Compare the distance traveled (approx. 3.65 crore km) to the distance to the Sun (14.7 crore km).
Since 36,500,000 km is much less than 147,000,000 km, no, you could not reach the Sun in a 100-year lifetime traveling at 1000 km per day.

5. 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?
See SolutionCalculate the total weight.
– Total weight = 100,000 sheets x 5 grams/sheet = 500,000 grams.
– Convert to kilograms: 500,000 grams / 1000 g/kg = 500 kg.
Assumption: A typical person cannot lift 500 kg.
Conclusion: No, you could not lift one lakh sheets of paper at once.

(b) If 250 babies are born every minute across the world, will a million babies be born in a day?
See SolutionCalculate the number of babies born in a day.
– Babies per hour = 250 babies/min x 60 min/hour = 15,000 babies/hour.
– Babies per day = 15,000 babies/hour x 24 hours/day = 360,000 babies/day.
Compare this to one million (1,000,000).
Since 360,000 is less than 1,000,000, no, a million babies will not be born in a day at this rate.

(c) Can you count 1 million coins in a day? Assume you can count 1 coin every second.
See SolutionCalculate the number of seconds in a day.
– Seconds per day = 60 sec/min x 60 min/hour x 24 hours/day = 86,400 seconds.
If you count 1 coin per second, you can count 86,400 coins in a day.
Compare this to one million (1,000,000).
Since 86,400 is much less than 1,000,000, no, you cannot count 1 million coins in a day at a rate of 1 per second.

Figure it Out

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
See SolutionTo make the largest number, arrange digits in descending order: 9876543210. To be a multiple of 5, it must end in 0 or 5. Ending in 0 gives the largest number.
Largest multiple of 5: 9,876,543,210.

(b) Smallest even number
See SolutionTo make the smallest number, arrange digits in ascending order, ensuring the first digit isn’t 0. The number must end in an even digit (0, 2, 4, 6, 8).
Smallest start: 10234… To make it even, the last digit must be even. We want the smallest number overall, so we use the smallest digits first. The available digits are 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
Smallest number starts with 10234567… The remaining digits are 8, 9. To make it even, it must end in 8.
Smallest even number: 1,023,456,798.

Page 20

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.
See SolutionThis requires counting letters in number names. Let’s test a candidate often cited for having many letters: 77,77,777.
Word form: Seventy seven lakhs seventy seven thousand seven hundred seventy seven.
Letter count (ignoring spaces/hyphens): Seventy(7) + seven(5) + lakhs(5) + seventy(7) + seven(5) + thousand(8) + seven(5) + hundred(7) + seventy(7) + seven(5) = 66 letters.
Trying others like eighty-eight… gives fewer letters. It’s likely that 77,77,777 (resulting in 66 letters) has the maximum number of letters for a 7-digit number name written out this way.
Example Answer: 77,77,777 (Seventy seven lakhs seventy seven thousand seven hundred seventy seven).

3. Write a 9-digit number where exchanging any two digits results in a bigger number. How many such numbers exist?
See SolutionFor any exchange of digits to result in a bigger number, the original number must have its digits sorted in strictly decreasing order from left to right.
Using distinct digits (presumably 1 through 9 for a 9-digit number): The only possibility is 987,654,321.
If digits could repeat, 999,999,999 would also satisfy this, but the context usually implies distinct digits if not specified.
Assuming distinct digits 1-9, there is only 1 such number: 987,654,321.

4. Strike out 10 digits from the number 12345123451234512345 so that the remaining number is as large as possible.
See SolutionThe original number has 20 digits. We need to keep 10 digits to form the largest possible number. We use a greedy approach: scan from left to right, keeping digits that maximize the resulting number.
Original: 12345 12345 12345 12345
– Keep the first ‘5’. Strike ‘1234’. (4 struck). Remaining digits needed: 9.
– From ‘12345 12345 12345’, keep the next ‘5’. Strike ‘1234’. (8 struck). Remaining digits needed: 8.
– From ‘12345 12345’, keep the next ‘5’. Strike ‘1234’. (12 struck) -> Too many. Backtrack.
Alternative strategy:
Keep the first ‘5’ (Strike 1234 – 4 struck). Need 9 more digits from ‘123451234512345’.
Keep the second ‘5’ (Strike 1234 – 8

5. The words ‘zero’ and ‘one’ share letters ‘e’ and ‘o’. ‘one’ and ‘two’ share ‘o’, ‘two’ and ‘three’ share ‘t’. How far do you have to count to find two consecutive numbers which do not share an English letter in common?
See SolutionThis is a known puzzle. Let’s check pairs:
ONE/TWO share ‘O’. TWO/THREE share ‘T’,’E’. THREE/FOUR share ‘R’. FOUR/FIVE share ‘F’. FIVE/SIX share ‘I’. SIX/SEVEN share ‘S’. SEVEN/EIGHT share ‘E’. EIGHT/NINE share ‘E’,’I’,’N’. NINE/TEN share ‘N’,’E’.
It appears that all consecutive number names share at least one letter. There might not be such a pair in standard English number names, or the answer lies in a less common interpretation or a much higher number not typically checked.

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?
See Solution– Digits for 1-9: 9 digits.
– Digits for 10-99: 90 numbers * 2 digits/number = 180 digits.
– Total digits up to 99: 9 + 180 = 189 digits.
– We need the 1000th digit. It falls within the 3-digit numbers.
– Position into 3-digit numbers = 1000 – 189 = 811th digit.
– Which 3-digit number? Each uses 3 digits. 811 / 3 = 270 with a remainder of 1.
– This means it’s the 1st digit of the (270 + 1) = 271st three-digit number.
– The first 3-digit number is 100. The 271st is 100 + 270 = 370.
– The 1000th digit is the first digit of 370, which is ‘3’.

(b) At which number would it occur?
See SolutionThe 1000th digit occurs within the number 370.

(c) What number would contain the millionth digit?
See Solution– Digits up to 99: 189.
– Digits for 100-999: 900 numbers * 3 digits/number = 2700 digits. Total up to 999 = 189 + 2700 = 2889.
– Digits for 1000-9999: 9000 * 4 = 36000 digits. Total up to 9999 = 2889 + 36000 = 38889.
– Digits for 10000-99999: 90000 * 5 = 450000 digits. Total up to 99999 = 38889 + 450000 = 488889.
– We need the 1,000,000th digit. It falls within the 6-digit numbers.
– Position into 6-digit numbers = 1,000,000 – 488,889 = 511,111th digit.
– Which 6-digit number? Each uses 6 digits. 511,111 / 6 = 85,185 with a remainder of 1.
– This means it’s the 1st digit of the (85185 + 1) = 85186th six-digit number.
– The first 6-digit number is 100,000. The 85186th is 100,000 + 85,185 = 185,185.
– The millionth digit is the first digit of 185,185, which is ‘1’. The number containing it is 185,185.

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
See Solution(2 x +10,000) + (8 x +100). Clicks = 2 + 8 = 10.

(b) 92,100
See Solution(9 x +10,000) + (21 x +100). Clicks = 9 + 21 = 30.

(c) 1,20,500
See Solution(12 x +10,000) + (5 x +100). Clicks = 12 + 5 = 17.

(d) 65,30,000
See Solution(653 x +10,000) + (0 x +100). Clicks = 653 + 0 = 653.

(e) 70,25,700
See Solution(702 x +10,000) + (57 x +100). Clicks = 702 + 57 = 759.

8. How many lakhs make a billion?
See SolutionAn American billion is 1,000,000,000 (one thousand million).
A lakh is 1,00,000.
Number of lakhs in a billion = 1,000,000,000 / 100,000
Calculation: Remove 5 zeros from both numbers: 10,000 / 1 = 10,000.
So, 10,000 lakhs make a billion.

Page 21

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.
(Note: Finding the absolute closest value can be complex. Below are attempts to get close.)
(a) 1,10,000: Example given is 1,13,000 using 4000 x (20+5) + 13000.
Another attempt: 70000 + (4000 × 5) + 13000 + 300 + 20 = 70000 + 20000 + 13000 + 300 + 20 = 103,320.
(b) 2,00,000:
Attempt: 150000 + 70000 – (4000 × 5) = 220000 – 20000 = 200000. (Exact match found). Uses: 150k, 70k, 4k, 5.
(c) 5,80,000:
Attempt: 150000 × (20 / 5) – 13000 – 4000 = (150000 * 4) – 13000 – 4000 = 600000 – 13000 – 4000 = 583000. (Very close). Uses: 150k, 20, 5, 13k, 4k.
(d) 12,45,000:
Attempt: (70000 × 20) – 150000 – 5 = 1,400,000 – 150000 – 5 = 1,250,000 – 5 = 1,249,995. (Very close). Uses: 70k, 20, 150k, 5.
(e) 20,90,800:
Attempt: (4000 × (150000 / 300)) + 70000 + 13000 + (20 × 5) = (4000 × 500) + 70000 + 13000 + 100 = 2,000,000 + 70000 + 13000 + 100 = 2,083,100. (Reasonably close). Uses: 4k, 150k, 300, 70k, 13k, 20, 5.

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 height from page 3 is ~180 metres).
See SolutionConvert statue height to millimeters.
– Height = 180 metres × 1000 mm/metre = 180,000 mm.
Calculate number of coins.
– Number of coins = Total Height / Coin Thickness = 180,000 mm / 1 mm/coin = 180,000 coins.
Approximately 180,000 coins are needed.

12. Grey-headed albatrosses … 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…?
See SolutionCalculate the time for the 12,000 km trip using the given daily distance range.
– Time = Total Distance / Distance per day
– Minimum days = 12,000 km / 1000 km/day = 12 days.
– Maximum days = 12,000 km / 900 km/day = 13.33 days.
Such a trip would take approximately 12 to 14 days.

13. A bar-tailed godwit… travelled 13,560 km from Alaska to Australia without stopping. Its journey … continued for about 11 days. Find out the approximate distance it covered every day. Find out the approximate distance it covered every hour.
See SolutionCalculate distance per day.
– Distance per day = Total Distance / Number of Days = 13,560 km / 11 days approx. = 1232.7 km/day.
Approximately 1233 km per day.
Calculate distance per hour.
– Distance per hour = Distance per day / 24 hours/day = 1232.7 km/day / 24 hours/day approx. = 51.36 km/hour.
Approximately 51 km per hour.

14. Bald eagles fly as high as 4500-6000 m… 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? (Building height estimated as 40m on page 3).
See SolutionCompare heights to Somu’s building (40 m).
– Eagle max height vs Building: 6000 m / 40 m = 150 times bigger.
– Everest height vs Building: 8850 m / 40 m = 221.25 times bigger (approx. 221 times).
– Aeroplane max height vs Building: 12800 m / 40 m = 320 times bigger.