NCERT Solutions Class 7 Mathematics Ganita Prakash 1 Chapter 01 Large Numbers Around Us

Question 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?
Answer: To find out, we need to work out how many varieties can be tasted in a 100-year period. First, we find the total number of days: 100 years times 365 days per year gives us 36,500 days. Next, we multiply this by the number of varieties eaten each day: 36,500 days times 2 varieties per day equals 73,000 varieties. Since 73,000 is less than one lakh (100,000), you would not be able to taste 1 lakh varieties in 100 years if you ate only 2 varieties per day.
In simple words: In 100 years, eating 2 types of rice daily, you can taste only 73,000 types. One lakh is 100,000, so you fall short.

Exam Tip: Always multiply the number of days by the daily rate to find the total. Check whether your answer is more or less than the target number being asked about.

Question 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?
Answer: To answer this, we calculate the total varieties tasted over 100 years at a rate of 3 per day. The number of days in 100 years is 100 times 365, which equals 36,500 days. If someone eats 3 varieties daily: 36,500 days times 3 varieties per day equals 109,500 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.
In simple words: In 100 years, eating 3 types of rice daily, you can taste 109,500 types. That is more than one lakh, so you succeed.

Exam Tip: Compare your final total with the target. If the problem asks "will they be able to," state yes or no clearly before giving supporting calculations.

Question 3. Choose a number for y. How close to one lakh is the number of days in y years, for the y of your choice?
Answer: Let's choose two different values for y and see how many days we get, comparing each to one lakh (100,000). We use 365 days per year. For the first choice, y = 100 years: number of days equals 100 times 365, which is 36,500 days. How far is this from one lakh? 100,000 minus 36,500 equals 63,500 days - so 100 years gives us 63,500 fewer days than one lakh. For the second choice, y = 274 years: number of days equals 274 times 365, which is 100,010 days. How far is this from one lakh? 100,010 minus 100,000 equals 10 days - so 274 years gives us just 10 extra days past one lakh. This second choice gets us very close to exactly one lakh days.
In simple words: If you pick 100 years, you get 36,500 days - that is far from one lakh. If you pick 274 years, you get 100,010 days - that is only 10 days more than one lakh.

Exam Tip: When comparing to a target number, always show both the exact count and how far away it is. Picking different values can help you find one that is very close to the target.

Question 4. 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?
Answer: To find the difference, we subtract the population from one lakh. Calculate: 100,000 minus 75,000 equals 25,000. Therefore, 75,000 is 25,000 less than one lakh.
In simple words: One lakh is 100,000. Chintamani had 75,000 people. The gap between them is 25,000.

Exam Tip: When asked "how much less," subtract the smaller number from the larger one. Always state both the numbers and the final difference.

Question 5. The estimated population of Chintamani in the year 2024 is 1,06,000. How much more than one lakh is 1,06,000?
Answer: To find how much more, we take the 2024 population and subtract one lakh from it. Calculate: 106,000 minus 100,000 equals 6,000. Therefore, 1,06,000 is 6,000 more than one lakh.
In simple words: The 2024 population is 106,000. One lakh is 100,000. The extra amount is 6,000 people.

Exam Tip: When asked "how much more," subtract the baseline (one lakh) from the new number. State the answer as a clear difference.

Question 6. By how much did the population of Chintamani increase from 2011 to 2024?
Answer: To find the increase, we subtract the earlier population from the later population. Calculate: 106,000 (2024 population) minus 75,000 (2011 population) equals 31,000. The population increased by 31,000 from 2011 to 2024.
In simple words: In 2011, there were 75,000 people. In 2024, there were 106,000. The gain is 31,000 people.

Exam Tip: When finding growth over time, always subtract the earlier year from the later year. Make sure to identify which is the start and which is the end.

Question 7. 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?
Answer: First, we find the height of one floor. Each floor is about 4 times Somu's height: 4 times 1 metre equals 4 metres per floor. From the image comparing the building to the Statue of Unity and the waterfall, we can see the building is roughly 40 metres tall. This suggests the building has about 10 floors: 10 floors times 4 metres per floor equals 40 metres. The approximate height of the building is 40 metres.
In simple words: Somu is 1 metre tall. Each floor is 4 times that height, so each floor is 4 metres. If there are about 10 floors, the building is around 40 metres high.

Exam Tip: Use the visual comparison in the image to estimate the number of floors. Then multiply the number of floors by the height per floor to get the total height.

Question 8. Which is taller - The Statue of Unity or this building? How much taller?
Answer: We compare the height of the Statue of Unity (180 metres) to the building's estimated height (40 metres). Since 180 metres is greater than 40 metres, the Statue of Unity is taller. To find how much taller, we subtract: 180 metres minus 40 metres equals 140 metres. The Statue of Unity is 140 metres taller than the building.
In simple words: The statue is 180 metres tall. The building is 40 metres tall. The statue is 140 metres taller.

Exam Tip: Always state which object is taller first, then give the difference. This makes your answer clear and complete.

Question 9. How much taller is the Kunchikal waterfall than Somu's building?
Answer: We compare the height of the Kunchikal waterfall (450 metres) to the building's estimated height (40 metres). To find how much taller the waterfall is, we subtract: 450 metres minus 40 metres equals 410 metres. The Kunchikal waterfall is 410 metres taller than Somu's building.
In simple words: The waterfall is 450 metres high. The building is 40 metres high. The difference is 410 metres.

Exam Tip: When comparing heights, always subtract the smaller from the larger to find the difference. State the comparison clearly.

Question 10. How many floors should Somu's building have to be as high as the waterfall?
Answer: We know the height of the waterfall is 450 metres and the height of one floor is 4 metres (calculated earlier). To find how many floors are needed, we divide the waterfall height by the height per floor: 450 metres divided by 4 metres per floor equals 112.5 floors. Somu's building would need to have approximately 112.5, or about 113 floors, to be as high as the Kunchikal waterfall.
In simple words: The waterfall is 450 metres tall. Each floor is 4 metres. Dividing 450 by 4 gives about 112 or 113 floors.

Exam Tip: When finding how many units fit into a total, divide the total by the size of one unit. Round your answer sensibly - in this case, you can't have half a floor, so round to the nearest whole number.

Question 11. How do you view a lakh - is a lakh big or small?
Answer: Whether one lakh seems big or small depends on what you are comparing it to. Roxie feels it is large, pointing to the example of one lakh rice varieties (which is hard to taste in a lifetime), the time span of one lakh days (which is 274 years), and the length of a line of one lakh people (38 kilometres). Estu thinks it is not that big, mentioning that a sports stadium can hold more than one lakh people, a human head has around one lakh hairs, and some fish lay one lakh or more eggs at once. So the view of whether "one lakh" is big or small is subjective and depends on what it is being measured against.
In simple words: One lakh can seem big when compared to rice types or people in a line, but small when compared to stadium capacity or fish eggs.

Exam Tip: Context matters when deciding if a number is "big" or "small." Always think about what the number is measuring and what you are comparing it to.

Question 12. Write each of the numbers given below in words: (a) 3,00,600
Answer: Three lakh six hundred.

Exam Tip: When writing numbers in words using the Indian place value system, identify the main groups (crore, lakh, thousand) and write each part separately.

Question 12. (b) 5,04,085
Answer: Five lakh four thousand eighty-five.

Exam Tip: Say each place value group in order - lakhs, then thousands, then hundreds/tens/ones. Skip any group that is zero.

Question 12. (c) 27,30,000
Answer: Twenty-seven lakh thirty thousand.

Exam Tip: When a group ends in zeros, just name that group and skip the zero part (e.g., "thirty thousand" instead of "thirty thousand zero hundred").

Question 12. (d) 70,53,138
Answer: Seventy lakh fifty-three thousand one hundred thirty-eight.

Exam Tip: Go through each group from left to right. For the final group (ones), say the full number (e.g., "one hundred thirty-eight").

Question 13. 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
Answer: 1,23,456

Exam Tip: When converting from words to numbers, place each group in its correct position using commas in the Indian system (lakhs, thousands, ones).

Question 13. (b) Four lakh seven thousand seven hundred and four
Answer: 4,07,704

Exam Tip: Remember to place zeros for missing groups or positions (e.g., if there are no hundreds, write 0 in that place).

Question 13. (c) Fifty lakhs five thousand and fifty
Answer: 50,05,050

Exam Tip: "Fifty lakhs" means 50 in the lakhs place. Fill in zeros for any missing intermediate groups.

Question 13. (d) Ten lakhs two hundred and thirty five
Answer: 10,00,235

Exam Tip: "Ten lakhs" equals 10 in the lakhs place. When there are no thousands, write 00 in that position.

Question 14. The Thoughtful Thousands only has a +1000 button. How many times should it be pressed to show: (a) Three thousand?
Answer: Press the +1000 button 3 times. The calculation is: 3 times 1000 equals 3000.

Exam Tip: To find how many times to press a button, divide the target number by the button value.

Question 14. (b) 10,000?
Answer: Press the +1000 button 10 times. The calculation is: 10 times 1000 equals 10,000.

Exam Tip: Make sure your division is exact - if it is not a whole number, that button alone cannot make the target.

Question 14. (c) Fifty three thousand?
Answer: Press the +1000 button 53 times. The calculation is: 53 times 1000 equals 53,000.

Exam Tip: Be careful to read large numbers correctly and to count the digits accurately.

Question 14. (d) 90,000?
Answer: Press the +1000 button 90 times. The calculation is: 90 times 1000 equals 90,000.

Exam Tip: Numbers ending in 000 are easy to divide by 1000 - just drop the three zeros and you have your answer.

Question 14. (e) One Lakh?
Answer: Press the +1000 button 100 times. The calculation is: 100 times 1000 equals 1,00,000 (one lakh).

Exam Tip: Remember that one lakh equals 100,000. It takes 100 thousands to make one lakh.

Question 14. (f) What number is shown if the button is pressed 153 times?
Answer: The number shown is 153,000. The calculation is: 153 times 1000 equals 153,000.

Exam Tip: When you know the number of presses, multiply that number by the button value to get the result.

Question 14. (g) How many thousands are required to make one lakh?
Answer: One hundred thousands are required to make one lakh. The calculation is: 100,000 divided by 1000 equals 100.

Exam Tip: This is a reverse question - you are finding how many groups of one size fit into a larger number.

Question 15. The Tedious Tens only has a +10 button. How many times should it be pressed to show: (a) Five hundred?
Answer: Press the +10 button 50 times. The calculation is: 50 times 10 equals 500.

Exam Tip: When dividing by 10, just remove one zero from the number you want to make.

Question 15. (b) 780?
Answer: Press the +10 button 78 times. The calculation is: 78 times 10 equals 780.

Exam Tip: Numbers ending in zero are easy to work with the +10 button.

Question 15. (c) 1000?
Answer: Press the +10 button 100 times. The calculation is: 100 times 10 equals 1000.

Exam Tip: One thousand equals 100 tens. You can remember this by dividing: 1000 divided by 10 equals 100.

Question 15. (d) 3700?
Answer: Press the +10 button 370 times. The calculation is: 370 times 10 equals 3700.

Exam Tip: For any number ending in zero, divide by 10 to find the number of button presses.

Question 15. (e) 10,000?
Answer: Press the +10 button 1000 times. The calculation is: 1000 times 10 equals 10,000.

Exam Tip: Ten thousand equals 1000 tens. This shows how place values build on each other.

Question 15. (f) One lakh?
Answer: Press the +10 button 10,000 times. The calculation is: 10,000 times 10 equals 1,00,000 (one lakh).

Exam Tip: One lakh equals 10,000 tens. This large number of presses shows why using larger buttons (like +1000) is much more efficient.

Question 15. (g) What number is shown if the button is pressed 435 times?
Answer: The number shown is 4,350. The calculation is: 435 times 10 equals 4,350.

Exam Tip: To multiply by 10, simply add one zero to the end of the number.

Question 16. The Handy Hundreds only has a +100 button. How many times should it be pressed to show: (a) Four hundred?
Answer: Press the +100 button 4 times. The calculation is: 4 times 100 equals 400.

Exam Tip: To find presses needed, divide the target number by 100.

Question 16. (b) 3,700?
Answer: Press the +100 button 37 times. The calculation is: 37 times 100 equals 3,700.

Exam Tip: For numbers ending in 00, just drop the two zeros to get the number of presses.

Question 16. (c) 10,000?
Answer: Press the +100 button 100 times. The calculation is: 100 times 100 equals 10,000.

Exam Tip: Ten thousand equals 100 hundreds. This is a key place value relationship.

Question 16. (d) Fifty three thousand?
Answer: Press the +100 button 530 times. The calculation is: 530 times 100 equals 53,000.

Exam Tip: Read the number carefully and divide by 100 to find the presses needed.

Question 16. (e) 90,000?
Answer: Press the +100 button 900 times. The calculation is: 900 times 100 equals 90,000.

Exam Tip: Large numbers of presses are needed when using smaller buttons - this shows why the larger buttons are more helpful.

Question 16. (f) 97,600?
Answer: Press the +100 button 976 times. The calculation is: 976 times 100 equals 97,600.

Exam Tip: Even when numbers do not end in round zeros, you can still divide by 100 if the number is a multiple of 100.

Question 16. (g) 1,00,000?
Answer: Press the +100 button 1000 times. The calculation is: 1000 times 100 equals 1,00,000 (one lakh).

Exam Tip: One lakh equals 1000 hundreds. Keep track of these place value relationships.

Question 16. (h) What number is shown if the button is pressed 582 times?
Answer: The number shown is 58,200. The calculation is: 582 times 100 equals 58,200.

Exam Tip: To multiply by 100, add two zeros to the end of the number.

Question 16. (i) How many hundreds are required to make ten thousand?
Answer: One hundred hundreds are required to make ten thousand. The calculation is: 10,000 divided by 100 equals 100.

Exam Tip: Division helps you find how many groups of one size fit into another number.

Question 16. (j) How many hundreds are required to make one lakh?
Answer: One thousand hundreds are required to make one lakh. The calculation is: 100,000 divided by 100 equals 1000.

Exam Tip: Practise these key relationships: 100 hundreds make 10,000, and 1000 hundreds make 1,00,000.

Question 16. (k) Handy Hundreds says, "There are some numbers which Tedious Tens and Thoughtful Thousands can't show but I can." Is this statement true?
Answer: Yes, the statement is true. Handy Hundreds with the +100 button can create any multiple of 100 (such as 3,700). Tedious Tens with the +10 button can only make multiples of 10. Thoughtful Thousands with the +1000 button can only make multiples of 1000. This means Handy Hundreds can show numbers like 3,700 that the other two machines cannot produce directly. Although Tedious Tens could eventually reach 3,700 with 370 presses, this would be much less efficient than using Handy Hundreds.
In simple words: The +100 button can make numbers like 3,700 because it goes in steps of 100. The +10 button skips these numbers (it stops at 3,700 but gets there through different steps). The +1000 button also cannot make 3,700 directly.

Exam Tip: Understanding what numbers each button can and cannot make depends on whether the target is a multiple of the button value.

Question 17. 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.
Answer: Here is another way using the standard place value breakdown. Expression: (5 times 1000) plus (0 times 100) plus (7 times 10) plus (2 times 1), which equals 5000 plus 0 plus 70 plus 2, which equals 5072. Another example would be: (4 times 1000) plus (10 times 100) plus (7 times 10) plus (2 times 1), which equals 4000 plus 1000 plus 70 plus 2, which equals 5072.
In simple words: You can break 5072 into 5000, 0, 70, and 2. You can also break it into 4000, 1000, 70, and 2. Both ways give the same answer.

Exam Tip: There are many ways to decompose a number. The standard way (using place values) is usually the simplest, but other groupings also work as long as they add up correctly.

Question 18. 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
Answer: Way 1: (8 times 1000) plus (3 times 100) equals 8000 plus 300 equals 8300. Way 2: (83 times 100) equals 8300.

Exam Tip: Breaking a number into different place value groups gives you different button-press combinations. Fewer groups often means fewer total presses.

Question 18. (b) 40629
Answer: Way 1: (4 times 10000) plus (6 times 100) plus (2 times 10) plus (9 times 1) equals 40000 plus 600 plus 20 plus 9 equals 40629. Way 2: (40 times 1000) plus (62 times 10) plus (9 times 1) equals 40000 plus 620 plus 9 equals 40629.

Exam Tip: For larger numbers, try grouping in different ways - sometimes using bigger button values reduces the total number of presses.

Question 18. (c) 56354
Answer: Way 1: (5 times 10000) plus (6 times 1000) plus (3 times 100) plus (5 times 10) plus (4 times 1) equals 50000 plus 6000 plus 300 plus 50 plus 4 equals 56354. Way 2: (56 times 1000) plus (35 times 10) plus (4 times 1) equals 56000 plus 350 plus 4 equals 56354.

Exam Tip: Mixing larger and smaller button values can help reduce the total number of presses compared to using the smallest button repeatedly.

Question 18. (d) 66666
Answer: Way 1: (6 times 10000) plus (6 times 1000) plus (6 times 100) plus (6 times 10) plus (6 times 1) equals 60000 plus 6000 plus 600 plus 60 plus 6 equals 66666. Way 2: (66 times 1000) plus (66 times 10) plus (6 times 1) equals 66000 plus 660 plus 6 equals 66666.

Exam Tip: When all digits are the same, you can see the pattern clearly - each place value group has the same digit multiplied by different powers of 10.

Question 18. (e) 367813
Answer: Way 1: (3 times 100000) plus (6 times 10000) plus (7 times 1000) plus (8 times 100) plus (1 times 10) plus (3 times 1) equals 300000 plus 60000 plus 7000 plus 800 plus 10 plus 3 equals 367813. Way 2: (367 times 1000) plus (81 times 10) plus (3 times 1) equals 367000 plus 810 plus 3 equals 367813.

Exam Tip: For very large numbers, grouping multiple digits together at once (like 367 times 1000) often reduces the total number of presses needed.

Question 19. You have to make exactly 30 button presses. What is the largest 3-digit number you can make? What is the smallest 3-digit number you can make?
Answer: To find the largest 3-digit number with exactly 30 presses, we want to use larger buttons as much as possible. Try (9 times 100) plus (8 times 10) plus (13 times 1): this equals 900 plus 80 plus 13 equals 993, using 9 plus 8 plus 13 equals 30 presses. This is the largest possible 3-digit number. For the smallest 3-digit number, we must have at least 100, so we use (1 times 100) plus (29 times 1) to get 100 plus 29 equals 129, using 1 plus 29 equals 30 presses. The largest 3-digit number is 993. The smallest 3-digit number is 129.
In simple words: For the largest number, put as much as you can into the hundreds place. For the smallest number, use the minimum in hundreds and fill the rest with ones.

Exam Tip: When constrained by a fixed number of presses, think about which buttons to prioritize - larger buttons for maximum results, or smallest buttons to keep a number low.

Question 20. 997 can be made using 25 clicks. Can you make 997 with a different number of clicks?
Answer: Yes, 997 can be made with different numbers of clicks. The efficient way (using Systematic Sippy's logic) is (9 times 100) plus (9 times 10) plus (7 times 1), taking 9 plus 9 plus 7 equals 25 clicks. But there are other ways: Example 1 - (8 times 100) plus (19 times 10) plus (7 times 1) equals 800 plus 190 plus 7 equals 997, taking 8 plus 19 plus 7 equals 34 clicks. Example 2 - (99 times 10) plus (7 times 1) equals 990 plus 7 equals 997, taking 99 plus 7 equals 106 clicks. So yes, 997 can be made with more than 25 clicks using less efficient groupings.
In simple words: You can make 997 with 25 clicks the smart way, or with 34 or 106 clicks if you group the digits differently.

Exam Tip: There are many ways to reach the same number, but some require far fewer button presses than others. The most efficient way usually matches the standard place value breakdown.

Question 21. 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. How can we get the numbers (a) 5072, (b) 8300 using as few button clicks as possible?
Answer: To use Systematic Sippy efficiently, press each button a number of times equal to the digit in that place value. (a) 5072 - Use the standard place value breakdown: (5 times 1000) plus (0 times 100) plus (7 times 10) plus (2 times 1). The minimum number of clicks is 5 plus 0 plus 7 plus 2 equals 14 clicks. (b) 8300 - Use the standard breakdown: (8 times 1000) plus (3 times 100) plus (0 times 10) plus (0 times 1). The minimum number of clicks is 8 plus 3 plus 0 plus 0 equals 11 clicks.
In simple words: Press each button once for every digit in that place. So for 5072, press +1000 five times, +100 zero times, +10 seven times, and +1 two times.

Exam Tip: The most efficient way always uses the place value digits directly. Count the digits in each position and press that button that many times.

Question 22. Is there another way to get 5072 using less than 23 button clicks? Write the expression for the same.
Answer: Yes, the minimal way uses fewer clicks. The expression is: (5 times 1000) plus (7 times 10) plus (2 times 1). The number of clicks is 5 plus 7 plus 2 equals 14 clicks, which is less than 23. This works because we skip the 100 place (since it has a digit of 0) and go directly from thousands to tens.
In simple words: Use only the buttons for digits that are not zero. So press +1000 five times, skip the +100 button, press +10 seven times, and press +1 two times.

Exam Tip: When a place value has 0, do not press that button at all - just skip it. This saves clicks.

Question 23. 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.
Answer: Using the standard place value method, here are the most efficient ways:
- 8300: (8 times 1000) plus (3 times 100). Clicks: 8 plus 3 equals 11.
- 40629: (4 times 10000) plus (6 times 100) plus (2 times 10) plus (9 times 1). Clicks: 4 plus 6 plus 2 plus 9 equals 21.
- 56354: (5 times 10000) plus (6 times 1000) plus (3 times 100) plus (5 times 10) plus (4 times 1). Clicks: 5 plus 6 plus 3 plus 5 plus 4 equals 23.
- 66666: (6 times 10000) plus (6 times 1000) plus (6 times 100) plus (6 times 10) plus (6 times 1). Clicks: 6 plus 6 plus 6 plus 6 plus 6 equals 30.
- 367813: (3 times 100000) plus (6 times 10000) plus (7 times 1000) plus (8 times 100) plus (1 times 10) plus (3 times 1). Clicks: 3 plus 6 plus 7 plus 8 plus 1 plus 3 equals 28.
In simple words: For each number, multiply each digit by its place value and add them up. Count the digits (skipping zeros) to get the minimum clicks.

Exam Tip: The minimum clicks equals the sum of all the digits in the number. This is a quick way to find the answer without working out the full breakdown.

Question 24. Do you see any connection between each number and the corresponding smallest number of button clicks?
Answer: Yes, there is a clear connection. The smallest number of button clicks needed to make a number using Systematic Sippy's method equals the sum of the digits of that number when written in standard form. For example, 8300 has digits 8, 3, 0, 0 - the sum is 8 plus 3 plus 0 plus 0 equals 11 clicks. Similarly, 56354 has digits 5, 6, 3, 5, 4 - the sum is 5 plus 6 plus 3 plus 5 plus 4 equals 23 clicks. This pattern holds for all numbers.
In simple words: Add up all the digits in a number and you get the minimum number of clicks needed.

Exam Tip: Once you spot this pattern, you can find the minimum clicks instantly - just add the digits. This is much faster than working out the full place value breakdown.

Question 25. 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.
Answer: The statement is slightly imprecise - the expressions for the least button clicks represent the standard base-10 place value expansion (used worldwide), not exclusively the Indian grouping system. This method works because the calculator buttons match powers of 10 (1, 10, 100, 1000, etc.). The standard way we write numbers is shorthand for expressing the number as a sum of its digits times these powers of 10. To minimize clicks, we press each place value button a number of times equal to the digit in that place - and that perfectly matches how base-10 numbers are written. This is why the efficient expressions reveal the place value structure of every number.
In simple words: The way we normally write numbers already shows the place value breakdown. Each digit tells us how many times to press that place value button.

Exam Tip: Place value and the decimal system are built into how we write numbers. Understanding this connection helps you see why the efficient method works the way it does.

Question 26. What if we press the +10,00,000 button ten times? What number will come up?
Answer: To find the number, we multiply 10 by 10,00,000. Calculate: 10 times 10,00,000 equals 1,00,00,000. The number that will come up is 1,00,00,000.

Exam Tip: Multiplying by 10 is the same as pressing the button 10 times. Use this to find the result quickly.

Question 27. How many zeroes will it have?
Answer: The number 1,00,00,000 has 7 zeroes.

Exam Tip: When counting zeros, look at the full number written out and count every zero position.

Question 28. What should we call it?
Answer: The number 1,00,00,000 (which is 1 followed by 7 zeroes) is called One Crore in the Indian system or Ten Million in the American or International system.

Exam Tip: Different countries use different place value names. Know both Indian and American terms for large numbers.

Question 29. How many zeros does a thousand lakh have?
Answer: First, we calculate the value of a thousand lakh. Calculate: 1000 times 1,00,000 equals 10,00,00,000. This number is 10 crore in the Indian system or 100 million in the American system. The number 10,00,00,000 has 8 zeroes.

Exam Tip: When multiplying place value units, work out the value first, then count the zeros.

Question 30. How many zeros does a hundred thousand have?
Answer: A hundred thousand is written as 100,000. It has 5 zeroes.

Exam Tip: Count zeros carefully - write the number out fully if needed to avoid mistakes.

Question 31. Read the following numbers in Indian place value notation and write their number names in both the Indian and American systems: (a) 4050678
Answer: Indian Notation: 40,50,678
Indian Name: Forty lakh fifty thousand six hundred seventy-eight.
American Name: Four million fifty thousand six hundred seventy-eight.

Exam Tip: Use commas to break the number into Indian place groups (crore, lakh, thousand, ones). Then read each group and name it.

Question 31. (b) 48121620
Answer: Indian 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.

Exam Tip: Break large numbers into clear groups. American system groups by threes from the right (ones, thousands, millions), while Indian groups are (ones, thousands, lakhs, crores).

Question 31. (c) 20022002
Answer: Indian Notation: 2,00,22,002
Indian Name: Two crore twenty-two thousand two.
American Name: Twenty million twenty-two thousand two.

Exam Tip: When intermediate groups are zeros, skip them in the name - you do not say "zero lakh."

Question 31. (d) 246813579
Answer: Indian 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.

Exam Tip: For very large numbers, break them carefully and read each group separately - then combine them into the full name.

Question 31. (e) 345000543
Answer: Indian 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.

Exam Tip: When zero groups appear in the middle (like "00,000" for thousands), skip them in the name.

Question 31. (f) 1020304050
Answer: Indian 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.

Exam Tip: "Arab" is the Indian term for 10 crore or 100 million. Some textbooks use "hundred crore" instead.

Question 32. Write the following numbers in Indian place value notation: (a) One crore one lakh one thousand ten
Answer: 1,01,01,010

Exam Tip: Place zeros in the positions where there are no values named in the words.

Question 32. (b) One billion one million one thousand one
Answer: American notation: 1,001,001,001
Equivalent Indian value: 1 arab 10 lakh 1 thousand 1 (since 1 billion equals 1 arab equals 100 crore; 1 million equals 10 lakh).
Indian Notation: 1,00,10,01,001

Exam Tip: When converting from American to Indian place values, first convert the American terms to their Indian equivalents, then write in Indian notation.

Question 32. (c) Ten crore twenty lakh thirty thousand forty
Answer: 10,20,30,040

Exam Tip: Write each group in order: crore, lakh, thousand, ones. Use zeros to fill empty places.

Question 32. (d) Nine billion eighty million seven hundred thousand six hundred
Answer: American notation: 9,080,700,600
Equivalent Indian value: 9 arab 8 crore 7 lakh 6 hundred.
Indian Notation: 9,08,07,00,600

Exam Tip: Convert American place names to Indian ones first: billion to arab, million to ten-lakh, etc.

Question 33. Compare and write '<', '>' or '=': (a) 30 thousand ___ 3 lakhs
Answer: 30 thousand equals 30,000. 3 lakhs equals 3,00,000. Since 30,000 is less than 3,00,000, we have: 30 thousand < 3 lakhs.

Exam Tip: Convert both numbers to standard form with the same units, then compare them.

Question 33. (b) 500 lakhs ___ 5 million
Answer: 500 lakhs equals 500 times 1,00,000 equals 5,00,00,000. 5 million equals 5,000,000. Since 5,00,00,000 is greater than 5,000,000, we have: 500 lakhs > 5 million.

Exam Tip: When using different naming systems, convert to a common base (like standard digits) before comparing.

Question 33. (c) 800 thousand ___ 8 million
Answer: 800 thousand equals 800,000. 8 million equals 8,000,000. Since 800,000 is less than 8,000,000, we have: 800 thousand < 8 million.

Exam Tip: One million is ten times one hundred thousand, so millions are always larger when the leading digits are the same.

Question 33. (d) 640 crore ___ 60 billion
Answer: 640 crore equals 640 times 1,00,00,000 equals 6,40,00,00,000. 60 billion equals 60 times 1,00,00,00,000 (American billion) equals 60,00,00,00,000. Since 6,40,00,00,000 is less than 60,00,00,00,000, we have: 640 crore < 60 billion.

Exam Tip: Even when numbers look similar in form, always multiply out to see the actual magnitude. A billion is much larger than a crore.

Question 34. What do you think of this conversation? Have you read or heard such headlines or statements?
Answer: This conversation highlights the difference between exact numbers and rounded approximations. Headlines and news reports often use rounded numbers (like "1 lakh visitors") because they are easier for people to understand and the exact count is not always needed to get the main idea across. It is common practice in news reporting and daily life to use approximate values when precision is not critical. The boy's comment about "99,999 visitors" if he hadn't attended is humorous but technically incorrect. The rounded figure "1 lakh" probably represents a number somewhere around 100,000, not exactly 100,000. Such rounding is normal and acceptable in most everyday situations.
In simple words: News reports round big numbers to make them easier to understand. Saying "about 1 lakh" is simpler than giving the exact count.

Exam Tip: Understand the difference between exact numbers (needed for precise work like banking or science) and approximate numbers (fine for casual discussion or headlines).

Question 35. 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.
Answer: (a) Round Up: When ordering supplies for an event (like the school ordering 750 sweets for 732 people), round up to make sure there is enough. When estimating how long a project will take, round up to give a buffer so you finish on time. When calculating building materials needed to avoid shortages. (b) Round Down: A shopkeeper might estimate a cost slightly lower (e.g., saying Rs 450 for a Rs 470 item) to make it sound better to the customer. When estimating remaining fuel in a car, round down to be cautious and avoid running out. (c) Either is Okay: In casual conversation about large numbers where exact precision does not matter (e.g., "The city has about 10 lakh people"). Rough estimates of distance or travel time when planning a trip. (d) Exact Numbers Needed: For financial transactions like bank balances, salaries, and tax calculations. Scientific measurements and laboratory experiments. Engineering specifications and construction plans. Emergency contact numbers like 139 for Railway Enquiry - these cannot be rounded. Recipes that need exact amounts of ingredients to turn out correctly.
In simple words: Round up when you need safety or plenty. Round down when you want to be careful. Round either way for rough estimates. Use exact numbers for money, science, and safety.

Exam Tip: Always think about the context before deciding whether to round and in which direction. Different situations need different levels of precision.

Question 36. Write the five nearest neighbours for these numbers: (a) 3,87,69,957
Answer: The five nearest neighbours are found by rounding to different place values.
- Nearest thousand: 3,87,70,000 (since 957 is greater than or equal to 500, round up the thousands digit)
- Nearest ten thousand: 3,87,70,000 (since 69,957 is greater than or equal to 5,000, round up the ten thousands digit)
- Nearest lakh: 3,88,00,000 (since 69,957 is greater than or equal to 50,000, round up the lakhs digit)
- Nearest ten lakh: 3,90,00,000 (since 87,69,957 is greater than or equal to 5,00,000, round up the ten lakhs digit)
- Nearest crore: 4,00,00,000 (since 87,69,957 is greater than or equal to 50,00,000, round up the crores digit)

Exam Tip: When rounding, look at the digit immediately to the right of the place you are rounding to. If it is 5 or more, round up; if it is less than 5, round down.

Question 36. (b) 29,05,32,481
Answer: The five nearest neighbours are:
- Nearest thousand: 29,05,32,000 (since 481 is less than 500, round down)
- Nearest ten thousand: 29,05,30,000 (since 2,481 is less than 5,000, round down)
- Nearest lakh: 29,05,00,000 (since 32,481 is less than 50,000, round down)
- Nearest ten lakh: 29,10,00,000 (since 5,32,481 is greater than or equal to 5,00,000, round up)
- Nearest crore: 29,00,00,000 (since 05,32,481 is less than 50,00,000, round down)

Exam Tip: For each place value, check the digits to the right. This tells you whether to round up or down.

Question 37. 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?
Answer: For all five roundings to give 5,00,00,000, the number must fall within ranges that satisfy each rounding rule. Let us work through each condition: The nearest thousand is 5,00,00,000 when the number is in the range [4,99,99,500 to 5,00,00,499]. The nearest ten thousand is 5,00,00,000 when the number is in the range [4,99,95,000 to 5,00,04,999]. The nearest lakh is 5,00,00,000 when the number is in the range [4,99,50,000 to 5,00,49,999]. The nearest ten lakh is 5,00,00,000 when the number is in the range [4,95,00,000 to 5,04,99,999]. The nearest crore is 5,00,00,000 when the number is in the range [4,50,00,000 to 5,49,99,999]. All these conditions must be true at the same time, so we need the overlap of all these ranges. The most restrictive (narrowest) range is the one for the nearest thousand: [4,99,99,500 to 5,00,00,499]. Any number within this range will satisfy all five conditions. To count the integers, we calculate: 5,00,00,499 minus 4,99,99,500 plus 1 equals 1000 integers. Therefore, there are 1000 such numbers. Examples include 4,99,99,500; 5,00,00,000; and 5,00,00,499.
In simple words: A number rounds to 5,00,00,000 from all five place values only if it sits in a narrow range between 4,99,99,500 and 5,00,00,499. There are exactly 1000 such numbers.

Exam Tip: When a number must satisfy multiple rounding conditions, find the overlap of all the ranges. The smallest range sets the final answer.

Question 38. Roxie and Estu are estimating the values of simple expressions. 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?
Answer: The exact sum equals 463,128 plus 419,682 equals 882,810. Roxie's estimate (near 8L, greater than 8L) is directionally correct. The distance between her estimate and the actual sum is 882,810 minus 800,000 equals 82,810. Estu's estimate (near 9L, less than 9L) is also directionally correct. The distance between his estimate and the actual sum is 900,000 minus 882,810 equals 17,190. Both estimates are correct in their direction. However, Estu's estimate of 9,00,000 is much closer to the actual sum of 8,82,810 because the distance is smaller (17,190 compared to 82,810).
In simple words: Both guesses are in the right direction. Estu's guess is closer to the real answer.

Exam Tip: To compare two estimates, find the distance from each to the actual answer. The smaller distance means the estimate is better.

Question 38. (b) Will the sum be greater than 8,50,000 or less than 8,50,000? Why do you think so?
Answer: The sum (8,82,810) is greater than 8,50,000. Here is the reasoning: 4,63,128 is greater than 4,50,000, and 4,19,682 is greater than 4,00,000. If we add just these lower bounds: 4.5 lakhs plus 4 lakhs equals 8.5 lakhs. Since both actual numbers are larger than their bounds (especially 463k versus 450k), the sum will definitely be more than 8,50,000. Another way: 460k plus 410k equals 870k, which is already greater than 850k.
In simple words: Both numbers are bigger than 4,50,000 and 4,00,000 respectively. Adding them always gives more than 8,50,000.

Exam Tip: When you want to know if a sum is above or below a target, find lower bounds for each number and add them. If the lower bound sum exceeds the target, the actual sum definitely will too.

Question 38. (c) Will the sum be greater than 8,83,128 or less than 8,83,128? Why do you think so?
Answer: The sum (8,82,810) is less than 8,83,128. Here is the reasoning: The target is 8,83,128. If we subtract the first number from the target: 8,83,128 minus 4,63,128 equals 4,20,000. So we need the second number to be at least 4,20,000 to reach our target. However, the second number is 4,19,682, which is less than 4,20,000. Since we are adding a number that is smaller than what is needed, the final sum will be less than 8,83,128.
In simple words: To get to 8,83,128, we would need to add 4,20,000 to the first number. But the second number is only 4,19,682, which is not quite enough.

Exam Tip: When checking if a sum will reach a target, ask "How much more is needed?" and compare it to the number being added.

Question 38. (d) Exact value of 4,63,128 + 4,19,682 = ?
Answer: 8,82,810

Exam Tip: After making estimates, always verify by computing the exact answer to check your reasoning.

Question 39. 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?
Answer: The exact difference equals 1,463,128 minus 490,020 equals 973,108. Roxie's estimate (near 10L, less than 10L) is directionally correct. The distance from her estimate to the actual difference is 1,000,000 minus 973,108 equals 26,892. Estu's estimate (near 9L, more than 9L) is also directionally correct. The distance from his estimate to the actual difference is 973,108 minus 900,000 equals 73,108. Both estimates are correct in direction. However, Roxie's estimate of 10,00,000 is closer to the actual difference of 9,73,108 because the distance is smaller (26,892 compared to 73,108).
In simple words: Both guesses go the right direction. Roxie's guess is closer to the true answer.

Exam Tip: Compare estimates by measuring how far each is from the true answer. The smaller gap wins.

Question 39. (b) Will the difference be greater than 9,50,000 or less than 9,50,000? Why do you think so?
Answer: The difference (9,73,108) is greater than 9,50,000. Here is the reasoning: Rough estimate: 14.6 lakhs minus 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 minus 5, which is 9.6 lakhs. And 9.6 lakhs (or 9,60,000) is greater than 9,50,000.
In simple words: We are removing less than 5 lakhs from 14.6 lakhs, so we keep more than 9.6 lakhs, which is more than 9.5 lakhs.

Exam Tip: Estimate by replacing numbers with nearby round values, then do quick mental arithmetic to find the approximate result.

Question 39. (c) Will the difference be greater than 9,63,128 or less than 9,63,128? Why do you think so?
Answer: The exact difference is 1,463,128 minus 490,020 equals 973,108. This is greater than 9,63,128. Here is the reasoning: Compare the number being subtracted (4,90,020) with the difference between the first number and our target (14,63,128 minus 9,63,128 equals 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, our result will be greater than 9,63,128.
In simple words: To get down to 9,63,128, we would need to subtract 5,00,000. But we only subtract 4,90,020. Since we subtract less, we end up with more than 9,63,128.

Exam Tip: When checking if a difference will exceed a target, ask "How much would we need to subtract?" and compare to the actual subtracted amount.

Question 39. (d) Exact value of 14,63,128 - 4,90,020 = ?
Answer: 9,73,108

Exam Tip: Always confirm estimates with exact calculations. This helps you learn whether your reasoning was sound.

Question 40. What is your general observation about this data?
Answer: A 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, however, varies between different 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. Overall, the data shows strong urban growth across major Indian cities during this decade.
In simple words: Almost all big cities grew a lot from 2001 to 2011. Some grew faster than others.

Exam Tip: When observing data, note overall patterns (like growth), variations between items, and any changes in ranking or order.

Question 41. What is an appropriate title for the above table?
Answer: An appropriate title could be "Population of Selected Major Indian Cities (2001 & 2011 Census)" or "Comparison of City Populations in India: 2001 vs 2011". Either of these titles clearly describes what data the table contains and the time period covered.

Exam Tip: A good table title should describe the data shown and any relevant time period or classification.

Question 42. How much is the population of Pune in 2011? Approximately, by how much has it increased compared to 2001?
Answer: Pune's population in 2011 was 31,15,431. Its population in 2001 was 25,38,473. To find the increase, we subtract: 3,115,431 minus 2,538,473 equals 576,958. Approximately, the population increased by about 5.8 lakhs, or roughly 6 lakhs.
In simple words: Pune had about 25.4 lakh people in 2001 and 31.2 lakh in 2011. That is a gain of roughly 6 lakh people.

Exam Tip: When comparing census data, state both the starting and ending populations, then calculate the difference. Rounding the difference helps communicate the approximate growth.

Question 43. Which city's population increased the most between 2001 and 2011?
Answer: To find the city with the largest increase, we calculate the difference for the main cities: Mumbai: 12,442,373 minus 11,978,450 equals 463,923. New Delhi: 11,007,835 minus 9,879,172 equals 1,128,663. Bengaluru: 8,425,970 minus 4,301,326 equals 4,124,644. Hyderabad: 6,809,970 minus 3,637,483 equals 3,172,487. Surat: 4,467,797 minus 2,433,835 equals 2,033,962. Comparing these increases, Bengaluru had the largest absolute population increase between 2001 and 2011.
In simple words: Bengaluru grew by over 41 lakh people, which is more than any other city shown.

Exam Tip: To find which city grew most, calculate the 2001-to-2011 difference for each city and compare the size of these differences.

Question 44. Are there cities whose population has almost doubled? Which are they?
Answer: "Almost doubled" means the 2011 population is close to or greater than twice the 2001 population. Let us check: Bengaluru: 2011 pop is 84.3 lakhs; twice the 2001 pop is 2 times 43.0 lakhs equals 86.0 lakhs. Bengaluru's growth is close but slightly less than double. Hyderabad: 2011 pop is 68.1 lakhs; twice the 2001 pop is 2 times 36.4 lakhs equals 72.8 lakhs. Hyderabad's growth is less than double. Surat: 2011 pop is 44.7 lakhs; twice the 2001 pop is 2 times 24.3 lakhs equals 48.6 lakhs. Surat's growth is less than double. Vadodara: 2011 pop is 35.5 lakhs; twice the 2001 pop is 2 times 16.9 lakhs equals 33.8 lakhs. Vadodara's population actually more than doubled. Considering "almost doubled," Bengaluru is very close. Vadodara actually more than doubled.
In simple words: Bengaluru almost doubled but fell just short. Vadodara actually doubled and then some.

Exam Tip: To check if a population doubled, multiply the 2001 figure by 2 and compare it to the 2011 figure.

Question 45. By what number should we multiply Patna's population to get a number/population close to that of Mumbai?
Answer: Patna's 2011 population was approximately 16.8 lakhs (1,684,222). Mumbai's 2011 population was approximately 124.4 lakhs (1,24,42,373). To find the multiplier, we divide: 124.4 lakhs divided by 16.8 lakhs approximately equals 7.4. Therefore, we should multiply Patna's population by approximately 7 to get close to Mumbai's population.
In simple words: Mumbai is about 7 times as big as Patna.

Exam Tip: To find how many times bigger one number is than another, divide the larger by the smaller.

Question 46. 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?
Answer: Multiplying by 5 can be thought of as multiplying by the fraction 10 divided by 2. Therefore, any number times 5 equals that same number times (10 divided by 2). This means: multiply the number by 10 first, then divide the result by 2. The final answer is the same because these two operations together are equivalent to a single multiplication by 5. For example: 8 times 5 equals 40. Using the alternate method: (8 times 10) divided by 2 equals 80 divided by 2 equals 40. The answer is identical.
In simple words: Since 5 equals 10 divided by 2, multiplying by 5 is the same as multiplying by 10 and then dividing by 2.

Exam Tip: Breaking a number into a fraction (like 5 as 10/2) is a useful way to rewrite multiplication in different forms.

Page 15

Question 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?
Answer: Yes, a clear relationship exists. When you multiply an M-digit number by an N-digit number, the result will always have either (M + N - 1) or (M + N) digits.
The smallest possible product occurs when the numbers are relatively small - for instance, 10 × 10 = 100 (where M = 2, N = 2, and M + N - 1 = 3 digits).
The largest possible product happens when the numbers are relatively large - for example, 99 × 99 = 9801 (where M = 2, N = 2, and M + N = 4 digits).
In simple words: Multiply an M-digit number by an N-digit number, and you get either (M + N - 1) or (M + N) digits in the answer - that's the pattern.

Exam Tip: Use the extreme cases (smallest and largest numbers with the given digits) to find the minimum and maximum number of digits in a product - this technique is much faster than checking many examples.

Question 2. Roxie says that the product of two 2-digit numbers can only be a 3- or a 4-digit number. Is she correct?
Answer: Yes, Roxie's statement is correct.
The smallest product using two 2-digit numbers is 10 × 10 = 100, which has 3 digits.
The largest product using two 2-digit numbers is 99 × 99 = 9801, which has 4 digits.
Since all products of two 2-digit numbers fall between these extremes, they must have either 3 or 4 digits - never more or fewer.
In simple words: The smallest 2-digit × 2-digit product is 100 (3 digits), and the largest is 9801 (4 digits). So all products have 3 or 4 digits.

Exam Tip: Always check the smallest and largest cases to establish the possible range of digit counts - this proves the claim without checking every combination.

Question 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?
Answer: No, there is no need to test all possible multiplications. A much smarter approach is to examine just the extreme cases - calculate the product of the smallest possible 2-digit numbers (10 × 10) and the product of the largest possible 2-digit numbers (99 × 99). By determining the minimum and maximum results, you can establish the full range of possible digit counts without checking any other pairs.
In simple words: Test only the smallest and largest pairs. This tells you the full range without trying everything.

Exam Tip: This extreme-case method is a powerful problem-solving tool - it saves time and proves claims efficiently in exams.

Question 4. Can multiplying a 3-digit number with another 3-digit number give a 4-digit number?
Answer: No, this is not possible.
The smallest product of two 3-digit numbers is 100 × 100 = 10,000, which already has 5 digits.
The largest product of two 3-digit numbers is 999 × 999 = 998,001, which has 6 digits.
Since even the smallest product has 5 digits, the result of multiplying any two 3-digit numbers will always have either 5 or 6 digits - never 4.
In simple words: The smallest 3-digit × 3-digit product is 10,000, which has 5 digits. So you can never get only 4 digits.

Exam Tip: Apply the digit-counting formula M + N - 1 and M + N to check if a given result is possible before doing lengthy calculations.

Question 5. Can multiplying a 4-digit number with a 2-digit number give a 5-digit number?
Answer: Yes, this is possible.
The smallest product of a 4-digit and a 2-digit number is 1000 × 10 = 10,000, which has 5 digits.
The largest product is 9999 × 99 = 989,901, which has 6 digits.
Because the minimum result has 5 digits and the maximum has 6 digits, a product with exactly 5 digits is definitely achievable when multiplying a 4-digit number by a 2-digit number.
In simple words: A 4-digit × 2-digit product can be 5 or 6 digits. The smallest 4-digit × 2-digit product is 10,000, which is 5 digits.

Exam Tip: Remember that when multiplying numbers, the product can have either (M + N - 1) or (M + N) digits - both are valid outcomes.

Question 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)
Answer: Yes, a clear pattern emerges when you apply the digit rule systematically.
1-digit × 1-digit = 1-digit or 2-digit
2-digit × 1-digit = 2-digit or 3-digit
2-digit × 2-digit = 3-digit or 4-digit
3-digit × 3-digit = 5-digit or 6-digit
5-digit × 5-digit = 9-digit or 10-digit (Min digits = 5 + 5 - 1 = 9; Max digits = 5 + 5 = 10)
8-digit × 3-digit = 10-digit or 11-digit (Min digits = 8 + 3 - 1 = 10; Max digits = 8 + 3 = 11)
12-digit × 13-digit = 24-digit or 25-digit (Min digits = 12 + 13 - 1 = 24; Max digits = 12 + 13 = 25)
This pattern holds universally: the product of an M-digit number and an N-digit number always has either (M + N - 1) or (M + N) digits, regardless of the actual values.
In simple words: Once you know how many digits each number has, you can predict how many digits the product will have - it follows the same rule every time.

Exam Tip: Use the formula immediately when asked about digit counts; you don't need to calculate the actual product to answer the question.

Fascinating Facts about Large Numbers

Question 1. Calculate: 1250 × 380 = ? (This represents the number of kirtanas composed by Purandaradāsa)
Answer: 1250 × 380 = 475,000 kirtanas

Working:
1250 × 380 = 125 × 10 × 38 × 10 = (125 × 38) × 100
= (125 × (40 - 2)) × 100
= ((125 × 40) - (125 × 2)) × 100
= (5000 - 250) × 100
= 4750 × 100
= 475,000

Result: 475,000 kirtanas, also expressed as Four lakh seventy-five thousand kirtanas.
In simple words: Purandaradāsa composed 475,000 songs - that's almost half a million musical pieces.

Exam Tip: Break larger numbers into simpler factors (like 1250 = 125 × 10) before multiplying - this makes the arithmetic much quicker and reduces errors.

Question 2. If Purandaradāsa composed 4,75,000 songs, how many songs per year did he have to compose?
Answer: The source material does not give us the number of years Purandaradāsa lived or spent composing. Without knowing the time span across which these 475,000 songs were created, we are unable to calculate the yearly rate of composition. To find songs per year, we would need to divide the total song count by the number of years, but that second number is missing from the information provided.
In simple words: We know he wrote 475,000 songs, but we don't know how many years that took, so we can't find out how many songs per year.

Exam Tip: When information is missing, acknowledge it clearly rather than guessing - this shows careful reading and logical thinking.

Question 3. Calculate: 2100 × 70,000 = ? (This represents the approximate distance between Earth and Sun in km)
Answer: 2100 × 70,000 = 147,000,000 km

Working:
2100 × 70,000 = (21 × 100) × (7 × 10,000)
= (21 × 7) × (100 × 10,000)
= 147 × 1,000,000
= 147,000,000

Result: 147,000,000 km, also written as One hundred forty-seven million km or Fourteen crore seventy lakh km.
In simple words: The Sun is about 147 million kilometers away from Earth - an enormous distance that light takes about 8 minutes to travel.

Exam Tip: When multiplying numbers with many zeros, separate the non-zero parts from the zeros, multiply the non-zero parts, then add all the zeros back at the end.

Page 17

Question 1. How did scientists measure the distance between the Earth and the Sun?
Answer: The source material does not explain the methods used to measure the Earth-Sun distance. Historically, astronomers have used techniques such as parallax (observing the apparent shift of nearby stars), and modern methods rely on radar measurements bouncing signals off planets and spacecraft, along with Kepler's laws of planetary motion. However, the specific measurement approach is not covered in the provided text.
In simple words: The text doesn't explain how they measured it. Scientists today use radar and other tools, but this chapter doesn't go into those details.

Exam Tip: Distinguish between what the text directly states and what falls outside its scope - this careful reading approach earns full marks.

Question 2. Calculate: 6400 × 62,500 = ? (This represents the average water discharge rate of the Amazon river in litres per second)
Answer: 6400 × 62,500 = 400,000,000 litres/sec

Working:
6400 × 62,500 = (64 × 100) × (625 × 100)
= (64 × 625) × (100 × 100)

To calculate 64 × 625:
64 × 625 = 64 × (600 + 25)
= (64 × 600) + (64 × 25)
= 38,400 + 1,600
= 40,000

Result = 40,000 × 10,000 = 400,000,000

Final Answer: 400,000,000 litres/sec, also expressed as Four hundred million litres/sec or Forty crore litres/sec.
In simple words: The Amazon River releases about 400 million liters of water every single second - more water than most countries use in a whole year.

Exam Tip: When a number looks difficult to multiply, look for ways to rewrite it (like 625 = 600 + 25) to use simpler arithmetic.

Question 3. Calculate: 13,95,000 ÷ 150 = ? (This represents the distance in km of the longest single-train journey)
Answer: 13,95,000 ÷ 150 = 9,300 km

Working:
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
In simple words: The longest train journey without stopping covers 9,300 kilometers - a distance spanning multiple countries.

Exam Tip: When dividing by larger numbers, split the dividend into two manageable parts that divide evenly, then add the results together.

Page 18

Question 1. Calculate: 10,50,00,000 ÷ 700 = ? (This represents the weight in kg an adult blue whale can weigh more than)
Answer: 10,50,00,000 ÷ 700 = 150,000 kg

Working:
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
In simple words: A blue whale can weigh more than 150,000 kilograms - that's as heavy as about 30 elephants put together.

Exam Tip: Simplify the divisor first (700 to 7 by removing zeros from both numbers) before breaking the dividend into parts.

Question 2. Calculate: 52,00,00,00,000 ÷ 130 = ? (This represents the weight in tonnes of global plastic waste generated in 2021)
Answer: 52,00,00,00,000 ÷ 130 = 400,000,000 tonnes

Working:
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, also expressed as Four hundred million tonnes or Forty crore tonnes.
In simple words: In 2021 alone, the world generated about 400 million tonnes of plastic waste - a staggering amount that will take centuries to break down.

Exam Tip: Look for simple ratios in the dividend and divisor (like 52 ÷ 13 = 4) to speed up the division process.

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Section 1.6: Did You Ever Wonder…?

Question 1. Could the entire population of Mumbai fit into 1 lakh buses?
Answer: No, the entire population of Mumbai cannot fit into 1 lakh buses.

Here is the working:
- Assume each bus can hold 50 people
- Total capacity of 1 lakh (100,000) buses = 100,000 × 50 = 50,00,000 people (50 lakh)
- Mumbai's population = 1,24,42,373 (more than 1 crore 24 lakh)
- Since 50 lakh is much smaller than 1 crore 24 lakh, the buses cannot accommodate everyone
In simple words: If each bus holds 50 people, 1 lakh buses can hold only 50 lakh people. But Mumbai has more than 1 crore 24 lakh people, so it's not enough.

Exam Tip: Always state your assumptions (like bus capacity) clearly before doing the calculation - examiners value transparent reasoning.

Question 2. The RMS Titanic ship carried about 2500 passengers. Can the population of Mumbai fit into 5000 such ships?
Answer: Yes, the population of Mumbai could (just barely) fit into 5000 Titanic-sized ships.

Here is the working:
- Total capacity of 5000 ships = 5000 × 2500 = 12,500,000 passengers
- This equals 1 crore 25 lakh people
- Mumbai's population = 1,24,42,373, which is approximately 1 crore 24 lakh
- Since 1 crore 25 lakh is slightly larger than 1 crore 24 lakh, everyone would fit with just a tiny amount of space to spare
In simple words: 5000 ships can hold 1 crore 25 lakh people. Mumbai has about 1 crore 24 lakh people. So yes, but only barely.

Exam Tip: When the answer is "just barely," mention both numbers clearly to show why the result is marginal rather than definitive.

Question 3. If I could travel 100 kilometers every day, could I reach the Moon in 10 years? (Distance to Moon = 3,84,400 km)
Answer: No, you would not reach the Moon in 10 years at this speed.

Here is the working:
- Distance traveled per year = 100 km/day × 365 days/year = 36,500 km
- Distance traveled in 10 years = 36,500 km/year × 10 years = 365,000 km
- Distance to the Moon = 384,400 km
- Since 365,000 km is less than 384,400 km, you would fall short by about 19,400 km
In simple words: In 10 years at 100 km per day, you travel only 365,000 km. The Moon is 384,400 km away, so you don't quite make it.

Exam Tip: State both values in the comparison so the examiner sees exactly how close (or far) you came to the target.

Question 4. Find out if you can reach the Sun in a lifetime, if you travel 1000 kilometers every day. (Distance to Sun = 147,000,000 km)
Answer: No, you could not reach the Sun in a 100-year lifetime traveling at 1000 km per day.

Here is the working:
- Distance traveled per year = 1000 km/day × 365 days/year = 365,000 km
- Distance traveled in 100 years = 365,000 km/year × 100 years = 36,500,000 km (approximately 3 crore 65 lakh km)
- Distance to the Sun = 147,000,000 km (approximately 14.7 crore km)
- Since 36,500,000 km is much smaller than 147,000,000 km, you would cover only about one-quarter of the distance
In simple words: Even traveling 1000 km every single day for a whole lifetime, you'd cover only 3 crore 65 lakh km. The Sun is 14.7 crore km away - far too far to ever reach.

Exam Tip: Express large numbers in both standard form and in crores/lakhs for clarity - this helps readers grasp the scale difference.

Question 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?
Answer: No, you could not lift one lakh sheets of paper at once.

Here is the working:
- Total weight = 100,000 sheets × 5 grams/sheet = 500,000 grams
- Converting to kilograms = 500,000 grams ÷ 1000 g/kg = 500 kg
- Assumption: A typical person cannot lift 500 kg (this far exceeds human lifting capacity; most people can lift only 10-30 kg)
- Conclusion: It is impossible for an average person to lift one lakh sheets of paper
In simple words: One lakh sheets would weigh 500 kilograms. That's heavier than a car. You definitely cannot lift it.

Exam Tip: Clearly state your assumptions about human capability or other factors - this shows you understand what "reasonable assumption" means.

Question 5. (b) If 250 babies are born every minute across the world, will a million babies be born in a day?
Answer: No, a million babies will not be born in a day at this rate.

Here is the working:
- Babies born per hour = 250 babies/min × 60 min/hour = 15,000 babies/hour
- Babies born per day = 15,000 babies/hour × 24 hours/day = 360,000 babies/day
- Target = 1 million = 1,000,000
- Since 360,000 is less than 1,000,000, a million babies will not be born in a single day at this birth rate
In simple words: At 250 babies born per minute, about 360,000 are born in a whole day. That's less than half a million, nowhere near a million.

Exam Tip: Convert the rate step by step (minutes to hours to days) to avoid arithmetic errors with large multipliers.

Question 5. (c) Can you count 1 million coins in a day? Assume you can count 1 coin every second.
Answer: No, you cannot count 1 million coins in a day at a rate of 1 coin per second.

Here is the working:
- Number of seconds in a day = 60 sec/min × 60 min/hour × 24 hours/day = 86,400 seconds
- Coins counted in a day = 86,400 coins (at 1 coin per second)
- Target = 1 million = 1,000,000
- Since 86,400 is much smaller than 1,000,000, you would fall far short - you would need over 11 days to count a million coins
In simple words: A day has only 86,400 seconds. So you can count only 86,400 coins in a day. A million is way more than that.

Exam Tip: When the gap between your rate and the target is huge, mention how long it would actually take - this adds depth to your answer.

Figure it Out

Question 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
Answer: The largest multiple of 5 is 9,876,543,210.

Reasoning: To form the largest possible number, arrange all digits in descending order: 9, 8, 7, 6, 5, 4, 3, 2, 1, 0, giving 9876543210. For this number to be a multiple of 5, it must end in either 0 or 5. Since 0 is smaller than 5, placing 0 at the end gives us the largest number while satisfying the divisibility rule.
In simple words: Arrange the digits from biggest to smallest: 9876543210. This ends in 0, so it divides by 5, and it's the biggest number you can make.

Exam Tip: For "largest" with constraints, always arrange digits in descending order, then adjust the last digit only if the constraint requires it.

Question 1. (b) Smallest even number
Answer: The smallest even number is 1,023,456,798.

Reasoning: To form the smallest possible number, arrange the digits in ascending order while ensuring the first digit is not 0. This gives us 1023456789. However, for the number to be even, it must end in an even digit (0, 2, 4, 6, or 8). To keep the number as small as possible overall, we use the smallest available even digit at the end. The remaining digits after placing 1 first are 0, 2, 3, 4, 5, 6, 7, 8, 9. To minimize the overall value, we arrange 0, 2, 3, 4, 5, 6, 7 in the middle positions, leaving 8 and 9 for the end. Since we need an even digit at the end, we place 8 there, giving 1,023,456,798.
In simple words: Start with the smallest digits in order (but not 0 first), then end with an even digit. This gives 1,023,456,798.

Exam Tip: For "smallest" constraints with multiple conditions, arrange ascending order first, then adjust only the last digit to satisfy the even/odd or divisibility requirement.

Page 20

Question 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.
Answer: The 7-digit number with the maximum number of letters is 77,77,777, which has 66 letters when written in words.

Word form: "Seventy seven lakhs seventy seven thousand seven hundred seventy seven"

Letter count (ignoring spaces and hyphens):
Seventy(7) + seven(5) + lakhs(5) + seventy(7) + seven(5) + thousand(8) + seven(5) + hundred(7) + seventy(7) + seven(5) = 66 letters total

This number maximizes letters because the word "seventy" (7 letters) is one of the longest number-name components, and using the digit 7 repeatedly throughout ensures we use this longest word multiple times.
In simple words: Write 77,77,777 in words. The word "seventy" appears many times, and each "seventy" and "seven" adds many letters, giving 66 letters total.

Exam Tip: When asked to maximize letters, use digits whose word forms are longest (like "seventy" for 7) and repeat them as much as possible.

Question 3. Write a 9-digit number where exchanging any two digits results in a bigger number. How many such numbers exist?
Answer: There is only 1 such number: 987,654,321.

Reasoning: For any exchange of two digits to produce a larger number, the original number must already have its digits arranged in strictly decreasing order from left to right. If any digit is out of this descending sequence, swapping it with a smaller digit to its right would create a larger number, violating the condition.

Using the digits 1 through 9 each exactly once, the only arrangement that satisfies this requirement is 987,654,321 (digits in descending order: 9, 8, 7, 6, 5, 4, 3, 2, 1). Any swap of two digits here would place a smaller digit to the left of a larger one, making the number smaller, not bigger. If digits could repeat, 999,999,999 would also work, but the standard interpretation assumes distinct digits for a 9-digit number using different values.
In simple words: The digits must go from biggest to smallest: 987,654,321. Swapping any pair of digits makes it smaller, not bigger. There's only one such arrangement.

Exam Tip: Recognize that "exchange results in bigger" means the original is the absolute smallest arrangement - which is the descending order.

Question 4. Strike out 10 digits from the number 12345123451234512345 so that the remaining number is as large as possible.
Answer: The largest possible remaining number is 55555.

Working using a greedy approach:
The original 20-digit number is: 12345 12345 12345 12345
We must keep exactly 10 digits and strike 10.

Strategy: Scan from left to right and keep digits that allow us to maintain the largest possible value at each step.
- Keep the first 5 that appears (at position 5). Strike the four 1-4 digits before it (4 struck). Remaining to keep: 6
- Keep the next 5 that appears (at position 10). Strike the four 1-4 digits between (8 struck total). Remaining to keep: 5
- Keep the next 5 that appears (at position 15). Strike the four 1-4 digits between (12 struck total). This is too many; we've struck all 10.
- Backtrack: Keep exactly five of the 5's, which gives us the number 55555.
In simple words: The digit 5 appears four times in the pattern. Keep all four 5's from the first four repetitions of "12345", and you get four 5's. Then find one more 5 from later in the sequence to make five 5's total - wait, let me recalculate: if the pattern 12345 repeats four times, there are four 5's total. Striking optimally leaves us with 5555. Actually, keeping the rightmost portions: the answer requires careful greedy selection, and the maximum turns out to be a string of 5's, specifically 55555 or similar, depending on exact positioning.

Exam Tip: Use the greedy algorithm: at each step, keep the largest available digit that still allows you to keep enough remaining digits to reach your target count.

Question 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?
Answer: This is a well-known puzzle. Let's examine consecutive number pairs:

ONE/TWO share 'O'
TWO/THREE share 'T' and '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', and 'N'
NINE/TEN share 'N' and 'E'

Upon checking, it appears that all consecutive number-name pairs in standard English share at least one letter. Such a pair may not exist in the usual sequence of English number names, or the answer might involve numbers far higher than those typically checked. The puzzle demonstrates how intricately connected language and patterns can be.
In simple words: Every pair of consecutive numbers shares at least one letter. There might not be any pair that doesn't share letters, or it might be so far up that we rarely check it.

Exam Tip: When a puzzle seems unsolvable after reasonable effort, say so clearly - examiners appreciate honest reasoning over forced answers.

Question 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?
Answer: The 1000th digit is '3', which is the first digit of the number 370.

Working:
- Digits from 1-9: 9 numbers × 1 digit each = 9 digits
- Digits from 10-99: 90 numbers × 2 digits each = 180 digits
- Total digits up to 99: 9 + 180 = 189 digits
- Remaining digits needed: 1000 - 189 = 811 digits
- These 811 digits come from 3-digit numbers (100 onwards)
- Position within 3-digit numbers: 811 ÷ 3 = 270 remainder 1
- This means we need the 1st digit of the (270 + 1) = 271st three-digit number
- The 271st three-digit number is: 100 + 270 = 370
- The 1st digit of 370 is '3'
In simple words: Count the digits: single numbers give 9 digits, two-digit numbers give 180 more (189 total). We need 811 more digits from three-digit numbers. That's 270 complete three-digit numbers plus 1 more digit. The 271st three-digit number is 370, and its first digit is 3.

Exam Tip: Break the sequence into ranges (1-9, 10-99, 100-999...), calculate cumulative digit counts, then pinpoint the exact number and position.

Question 6. (b) At which number would it occur?
Answer: The 1000th digit occurs within the number 370.
In simple words: The number containing the 1000th digit is 370.

Exam Tip: When asked "at which number," identify the specific number that contains that digit, not just its position within the number.

Question 6. (c) What number would contain the millionth digit?
Answer: The number containing the millionth digit is 185,185.

Working:
- Digits from 1-9: 9 digits
- Digits from 10-99: 180 digits. Total: 189 digits
- Digits from 100-999: 900 numbers × 3 digits = 2700 digits. Total: 2889 digits
- Digits from 1000-9999: 9000 numbers × 4 digits = 36,000 digits. Total: 38,889 digits
- Digits from 10,000-99,999: 90,000 numbers × 5 digits = 450,000 digits. Total: 488,889 digits
- Remaining needed: 1,000,000 - 488,889 = 511,111 digits
- These come from 6-digit numbers (100,000 onwards)
- Position within 6-digit numbers: 511,111 ÷ 6 = 85,185 remainder 1
- This means the 1st digit of the (85,185 + 1) = 85,186th six-digit number
- The 85,186th six-digit number is: 100,000 + 85,185 = 185,185
- The millionth digit is the 1st digit of 185,185, which is '1'
- The number containing it is 185,185
In simple words: Keep counting digits by ranges. After all 1-digit, 2-digit, 3-digit, 4-digit, and 5-digit numbers, we're at 488,889 digits. We need 511,111 more from 6-digit numbers. That takes us to the number 185,185.

Exam Tip: Use a cumulative table to track digit counts by range - this prevents arithmetic errors and keeps your work organized.

Question 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
Answer: The expression is (2 × +10,000) + (8 × +100).
Total clicks = 2 + 8 = 10 clicks

Breaking down 20,800:
20,800 = 20,000 + 800
= (2 × 10,000) + (8 × 100)
= (2 clicks of +10,000 button) + (8 clicks of +100 button)
In simple words: 20,800 is made from two groups of 10,000 and eight groups of 100. So click +10,000 twice and +100 eight times - that's 10 clicks total.

Exam Tip: Break the target number into place values that match your available buttons to minimize the total clicks.

Question 7. (b) 92,100
Answer: The expression is (9 × +10,000) + (21 × +100).
Total clicks = 9 + 21 = 30 clicks

Breaking down 92,100:
92,100 = 90,000 + 2,100
= (9 × 10,000) + (21 × 100)
= (9 clicks of +10,000 button) + (21 clicks of +100 button)
In simple words: 92,100 breaks into nine 10,000s and twenty-one 100s. So that's 9 + 21 = 30 clicks.

Exam Tip: For numbers not evenly divisible by your button values, use division with remainder to find the required multiplier.

Question 7. (c) 1,20,500
Answer: The expression is (12 × +10,000) + (5 × +100).
Total clicks = 12 + 5 = 17 clicks

Breaking down 1,20,500:
1,20,500 = 120,000 + 500
= (12 × 10,000) + (5 × 100)
= (12 clicks of +10,000 button) + (5 clicks of +100 button)
In simple words: 1,20,500 is twelve 10,000s and five 100s. That's 12 + 5 = 17 clicks.

Exam Tip: Notice that the number before 00 (like 120 in 1,20,500) tells you exactly how many +10,000 clicks you need.

Question 7. (d) 65,30,000
Answer: The expression is (653 × +10,000) + (0 × +100).
Total clicks = 653 + 0 = 653 clicks

Breaking down 65,30,000:
65,30,000 = 6,530,000
= (653 × 10,000) + (0 × 100)
= (653 clicks of +10,000 button) + (0 clicks of +100 button)
In simple words: 65,30,000 is 653 groups of 10,000 and no groups of 100. So 653 clicks total.

Exam Tip: When the last two digits are 00, you need 0 clicks of the +100 button - don't overlook this.

Question 7. (e) 70,25,700
Answer: The expression is (702 × +10,000) + (57 × +100).
Total clicks = 702 + 57 = 759 clicks

Breaking down 70,25,700:
70,25,700 = 7,025,700
= (702 × 10,000) + (57 × 100)
= (702 clicks of +10,000 button) + (57 clicks of +100 button)
In simple words: 70,25,700 is 702 groups of 10,000 and 57 groups of 100. So 702 + 57 = 759 clicks.

Exam Tip: Divide the entire number by 10,000 to get the quotient and remainder, which directly gives your two button counts.

Question 8. How many lakhs make a billion?
Answer: 10,000 lakhs make one billion.

Working:
- An American billion = 1,000,000,000 (one thousand million)
- One lakh = 100,000
- Number of lakhs in a billion = 1,000,000,000 ÷ 100,000
- Simplify by removing five zeros from both numbers: 10,000 ÷ 1 = 10,000
- Therefore, 10,000 lakhs = 1 billion
In simple words: A billion is 1,000,000,000. A lakh is 100,000. Divide to get 10,000 lakhs per billion.

Exam Tip: Remove equal numbers of zeros from both dividend and divisor to simplify division of large numbers.

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Question 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.
Answer:

(a) Target: 1,10,000
One approach: 1,13,000 using 4000 × (20 + 5) + 13000 = 4000 × 25 + 13000 = 100,000 + 13,000 = 113,000
Alternative: 70000 + (4000 × 5) + 13000 + 300 + 20 = 70000 + 20000 + 13000 + 300 + 20 = 103,320
The first attempt gets us to 113,000 (only 3,000 more than the target).

(b) Target: 2,00,000
Exact match found: 150000 + 70000 - (4000 × 5) = 220,000 - 20,000 = 200,000
Cards used: 150000, 70000, 4000, 5

(c) Target: 5,80,000
Approach: 150000 × (20 ÷ 5) - 13000 - 4000 = 150000 × 4 - 13000 - 4000 = 600,000 - 13,000 - 4,000 = 583,000
Cards used: 150000, 20, 5, 13000, 4000
Result is very close (only 3,000 more).

(d) Target: 12,45,000
Approach: (70000 × 20) - 150000 - 5 = 1,400,000 - 150,000 - 5 = 1,249,995
Cards used: 70000, 20, 150000, 5
Result is very close (only 5 short of 1,250,000).

(e) Target: 20,90,800
Approach: (4000 × (150000 ÷ 300)) + 70000 + 13000 + (20 × 5) = (4000 × 500) + 70000 + 13000 + 100 = 2,000,000 + 70,000 + 13,000 + 100 = 2,083,100
Cards used: 4000, 150000, 300, 70000, 13000, 20, 5
Result is reasonably close (about 8,000 more).
In simple words: Use the given cards with addition, subtraction, multiplication, and division to get as near as possible to each target. Some targets have exact matches, while others allow only close approximations.

Exam Tip: Start by identifying the largest card(s) close to your target, then add or subtract smaller cards to fine-tune the result.

Question 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 is approximately 180 metres)
Answer: Approximately 180,000 coins are needed to match the height of the Statue of Unity.

Working:
- Statue height = 180 metres
- Convert to millimeters: 180 metres × 1000 mm/metre = 180,000 mm
- Coin thickness = 1 mm per coin
- Number of coins = Total height ÷ Coin thickness = 180,000 mm ÷ 1 mm/coin = 180,000 coins
In simple words: The statue is 180 metres tall. Convert that to 180,000 mm. Each coin is 1 mm thick. So you need 180,000 coins stacked on top of each other.

Exam Tip: Always convert units to the same scale before dividing - this prevents mistakes when working with very large or very small numbers.

Question 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?
Answer: Such a trip would take approximately 12 to 14 days.

Working:
- Minimum days = Total distance ÷ Maximum daily distance = 12,000 km ÷ 1000 km/day = 12 days
- Maximum days = Total distance ÷ Minimum daily distance = 12,000 km ÷ 900 km/day = 13.33 days, which rounds to 14 days
- Therefore, the trip would take between 12 and 14 days depending on the bird's exact daily flying distance
In simple words: If the bird flies 1000 km per day, the trip takes 12 days. If it flies only 900 km per day, it takes about 13-14 days. So somewhere between 12 and 14 days.

Exam Tip: When given a range, always calculate for both extremes to show the full possible span of outcomes.

Question 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.
Answer: The bird covered approximately 1,233 km per day and approximately 51 km per hour.

Working - Distance per day:
- Distance per day = Total distance ÷ Number of days = 13,560 km ÷ 11 days ≈ 1,232.7 km/day
- Rounded: Approximately 1,233 km per day

Working - Distance per hour:
- Distance per hour = Distance per day ÷ Hours per day = 1,232.7 km/day ÷ 24 hours/day ≈ 51.36 km/hour
- Rounded: Approximately 51 km per hour
In simple words: Divide the total 13,560 km by 11 days to get about 1,233 km each day. Then divide 1,233 by 24 hours to get about 51 km every hour.

Exam Tip: Break multi-part questions into smaller steps - first find the daily rate, then convert that to hourly by dividing by 24.

Question 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 40 m)
Answer: The heights are vastly bigger compared to Somu's 40-meter building:
- Bald eagle maximum height: 6000 m ÷ 40 m = 150 times bigger
- Mount Everest: 8850 m ÷ 40 m = 221.25 times bigger (approximately 221 times)
- Aeroplane maximum height: 12,800 m ÷ 40 m = 320 times bigger

So eagles can fly 150 times higher than Somu's building, Everest is 221 times higher, and aeroplanes can fly 320 times higher.
In simple words: Somu's building is 40 metres. Eagles fly 150 times as high, Everest is 221 times as high, and planes fly 320 times as high.

Exam Tip: When comparing large numbers to a reference, always use division to find the ratio - this makes the scale difference clear and meaningful.

What is the main focus of Class 7 Maths Ganita Prakash Chapter 1?

Class 7 Maths Ganita Prakash Chapter 1 ("Large Numbers Around Us") focuses on introducing students to very large numbers that appear in everyday life. The chapter teaches students how to read, write, and estimate large numbers properly.

It covers several key concepts:
- Understanding the Indian place value system (ones, tens, hundreds, thousands, lakhs, crores)
- Understanding the International place value system (ones, tens, hundreds, thousands, millions, billions)
- Reading and writing numbers using both systems
- Comparing and ordering large numbers
- Rounding off numbers to different place values
- Using properties like associativity and commutativity to calculate with large numbers more efficiently
- Real-world applications of large numbers (distances, populations, measurements)

The chapter uses engaging examples, thought experiments, and hands-on activities to help students build a strong foundation in handling large numbers. This skill is vital for practical knowledge and for advancing in higher mathematics and science studies.
In simple words: This chapter teaches you how to read, write, and work with very big numbers like lakhs, crores, millions, and billions. It shows you how these huge numbers appear in real life - in distances, populations, and other measurements.

Exam Tip: Become comfortable switching between the Indian system (lakhs, crores) and the International system (millions, billions) - exams often ask you to convert or compare between the two.