CBSE Class 7 Mathematics Large Numbers Around Us Worksheet Set 01

Long Answer Questions

Question 1. Large numbers are read and written using the Indian place value system.
(i) Write these numbers in words: 5,06,009; 70,40,300; 8,00,05,070.
(ii) Write these names in figures with Indian commas: Six crore four lakh thirty thousand two; Twelve lakh nine hundred.
(iii) Explain the rule by which commas are placed in the Indian system, and show how the rule works on the number 92531478.
Answer:
(i) 5,06,009 = Five lakh six thousand nine.
70,40,300 = Seventy lakh forty thousand three hundred.
8,00,05,070 = Eight crore five thousand seventy (the lakhs group is 00, so no lakhs are named).
(ii) Six crore four lakh thirty thousand two = 6,04,30,002.
Twelve lakh nine hundred = 12,00,900.
(iii) Starting from the right, the first comma comes after three digits (the hundreds group). After that a comma comes after every two digits. This gives the 3-2-2-2 pattern of thousands, lakhs and crores.
For 92531478: the last three digits are 478; then 31; then 25; the digit 9 is left. So it is 9,25,31,478, read as nine crore twenty-five lakh thirty-one thousand four hundred seventy-eight.
Each comma marks where a new number name begins.
In simple words: Group digits 3-2-2 from the right, then read each group with its name.

Teacher's Note:
1. Empty groups are not read aloud, but they must be written as zeroes.
2. Always put commas from the right.

Question 2. In the Land of Tens, Thoughtful Thousands has only a +1000 button, Handy Hundreds has only a +100 button and Tedious Tens has only a +10 button. Systematic Sippy has all the buttons +1, +10, +100, +1000, +10000, +100000 and wants to use as few clicks as possible.
(i) How many times must each of the first three calculators be pressed to show 54,000?
(ii) Handy Hundreds claims, “There are numbers that Tedious Tens and Thoughtful Thousands cannot show, but I can.” Is the claim correct? Give reasons.
(iii) Write Systematic Sippy’s expression for 54,000 and the number of clicks it uses.
Answer:
(i) Thoughtful Thousands: 54,000 ÷ 1000 = 54 presses.
Handy Hundreds: 54,000 ÷ 100 = 540 presses.
Tedious Tens: 54,000 ÷ 10 = 5400 presses. The smaller the button, the more presses are needed.
(ii) No, the claim is wrong. Every number Handy Hundreds can show is a multiple of 100, and every multiple of 100 is also a multiple of 10.
So Tedious Tens can show each of those numbers too, by pressing 10 times for every one press of Handy Hundreds. For example, 700 needs 7 presses on Handy Hundreds and 70 on Tedious Tens.
Hence there is no number that Handy Hundreds can show but Tedious Tens cannot.
(iii) 54,000 = (5 × 10,000) + (4 × 1000).
Number of clicks = 5 + 4 = 9, the sum of the digits of 54,000.
In simple words: Divide by the button value for presses; Tedious Tens can show everything Handy Hundreds can.

Teacher's Note:
1. One counter-example is enough to reject a claim, and one general reason is enough to prove it false here.
2. Least clicks = sum of the digits.

Teacher's Note:
1. Learn the four pairs in the table by heart.
2. Read a number in both systems for practice.

Question 4. The population of a town recorded in a census is 4,67,890.
(i) Round this number to the nearest hundred, nearest thousand, nearest ten thousand and nearest lakh.
(ii) Which of these four approximations gives the clearest quick idea of the town’s size? Why?
(iii) Give one situation connected with this town where only the exact figure 4,67,890 may be used, and one where an approximate figure is enough.
Answer:
(i) Nearest hundred: 4,67,900, because the last two digits, 90, are more than 50.
Nearest thousand: 4,68,000, because 890 is more than 500.
Nearest ten thousand: 4,70,000, because 7,890 is more than 5,000.
Nearest lakh: 5,00,000, because 67,890 is more than 50,000.
(ii) “About 5 lakh people” is the clearest. Lakhs are the natural unit for town populations, and one short phrase tells the listener how big the place is. The nearest hundred, 4,67,900, is almost as long as the exact figure.
(iii) Exact: preparing voter lists, ration cards or the final census record, where every person must be counted.
Approximate: a newspaper headline, a school project chart, or a speech saying nearly five lakh people live in the town.
In simple words: Round step by step; 'about 5 lakh' gives the quickest idea, but official records need the exact count.

Teacher's Note:
1. Look at the part after the rounding place each time.
2. Exact numbers are needed where every unit matters.

Question 5. Roxie found that multiplying a number by 5 is the same as halving it and then multiplying by 10.
(i) Explain why this shortcut works.
(ii) Using similar regrouping, calculate 364 × 50, 88 × 125 and 16 × 625, showing each step.
(iii) Write a shortcut of your own for multiplying a number by 250, and test it on 36.
Answer:
(i) Five is half of ten: 5 = \( \frac{10}{2} \). So multiplying by 5 is the same as multiplying by 10 and dividing by 2, and the division can be done first. Halving is easy and multiplying by 10 only adds a zero.
(ii) 364 × 50 = 364 × \( \frac{100}{2} \) = \( \frac{364}{2} \) × 100 = 182 × 100 = 18,200.
88 × 125 = 88 × \( \frac{1000}{8} \) = \( \frac{88}{8} \) × 1000 = 11 × 1000 = 11,000.
16 × 625 = 16 × \( \frac{10{,}000}{16} \) = \( \frac{16}{16} \) × 10,000 = 10,000.
(iii) 250 = \( \frac{1000}{4} \), so to multiply by 250, divide by 4 and then multiply by 1000.
Test on 36: 36 ÷ 4 = 9 and 9 × 1000 = 9000. Check: 36 × 250 = 9000, so the shortcut works.
In simple words: Write the multiplier as a round number divided by a small number, then divide first.

Teacher's Note:
1. 5 = 10 ÷ 2, 50 = 100 ÷ 2, 125 = 1000 ÷ 8, 625 = 10,000 ÷ 16.
2. Always check a new shortcut by ordinary multiplication.

Question 6. Multiplications of special numbers often hide a pattern.
(i) Evaluate 11 × 11, 111 × 111 and 1111 × 1111. Describe the pattern and predict 11111 × 11111.
(ii) Evaluate 3 × 5, 33 × 35 and 333 × 335. Describe the pattern and predict 3333 × 3335.
(iii) Evaluate 66 × 61 and 666 × 661, then predict 6666 × 6661.
Answer:
(i) 11 × 11 = 121, 111 × 111 = 12321, 1111 × 1111 = 1234321.
The digits climb from 1 up to the number of 1s used and then come down again (a palindrome). So 11111 × 11111 = 123454321.
(ii) 3 × 5 = 15, 33 × 35 = 1155, 333 × 335 = 111555.
Each time one more 1 and one more 5 appear. So 3333 × 3335 = 11115555, that is 1,11,15,555.
(iii) 66 × 61 = 4026 and 666 × 661 = 440226.
Each step adds one more 4 at the front and one more 2 near the end, with a 0 in the middle. So 6666 × 6661 = 44402226, that is 4,44,02,226.
Counting the digits of the factors also tells how long each product will be, which is a quick check.
In simple words: Each pattern grows by one digit of each kind as the factors get longer.

Teacher's Note:
1. Verify at least one prediction by multiplying.
2. Check the length: two 4-digit numbers give 7 or 8 digits.

Question 7. Somu is 1 m tall. Each floor of his building is about four times his height, and the building has 10 floors. The Statue of Unity is about 180 m tall and the Kunchikal waterfall drops from about 450 m.
(i) Find the height of Somu’s building.
(ii) How much taller than the building is the statue? About how many times the building’s height is it?
(iii) How many floors would the building need to match the waterfall?
(iv) If a coin is 1 mm thick, how many coins must be stacked to match the height of the Statue of Unity?
Answer:
(i) Each floor ≈ 4 × 1 = 4 m, so the building ≈ 10 × 4 = 40 m.
(ii) 180 − 40 = 140 m taller. Also 180 ÷ 40 = 4.5, so the statue is about 4½ times as tall.
(iii) 450 ÷ 4 = 112.5, so the building would need about 113 floors. (The waterfall is 450 − 40 = 410 m taller than the present building.)
(iv) 180 m = 180 × 100 cm = 18,000 cm = 1,80,000 mm.
Each coin is 1 mm thick, so 1,80,000 coins are needed.
Comparing a huge measure with something familiar, like our height or a coin, gives a feel for large numbers.
In simple words: Compare the big heights with Somu’s 40 m building and with 1 mm coins.

Teacher's Note:
1. 'How much taller' means subtract; 'how many times' means divide.
2. 1 m = 100 cm = 1000 mm.

Question 8. The population of Hyderabad in 2011 was 68,09,970. A bus can carry 50 people and a large ship can carry 2,500 people.
(i) How many people can 1 lakh buses carry? Can the whole city fit into 1 lakh buses?
(ii) How many buses would actually be needed to carry everyone?
(iii) How many such ships would be needed instead?
(iv) Write the population of Hyderabad in words in the American system.
Answer:
(i) 1,00,000 × 50 = 50,00,000 (50 lakh) people. The population 68,09,970 is more than 50 lakh, so no, the whole city cannot fit.
(ii) 68,09,970 ÷ 50 = 1,36,199.4. Part of a bus is no use, so round up: 1,36,200 buses. Check: 1,36,200 × 50 = 68,10,000, just enough.
(iii) 68,09,970 ÷ 2500 ≈ 2723.99, so about 2,724 ships. Check: 2724 × 2500 = 68,10,000.
(iv) 68,09,970 = 6,809,970, read as six million eight hundred nine thousand nine hundred seventy.
In simple words: 1 lakh buses hold only 50 lakh people; Hyderabad needs 1,36,200 buses or 2,724 ships.

Teacher's Note:
1. When dividing people into vehicles, always round up.
2. Check by multiplying back.

Question 9. A calculator shows a number only by adding with its buttons.
(i) On Systematic Sippy (buttons +1, +10, +100, +1000, +10000, +100000), the smallest possible number of clicks for a number always turns out to be the sum of its digits. Explain why this happens.
(ii) Another calculator has only the buttons +10,000 and +100. Write an expression for 85,40,300 and find the total number of clicks.
(iii) Give one 5-digit number that this second calculator can never show, and explain why.
Answer:
(i) Each button adds one unit to exactly one place: +1 to the ones, +10 to the tens, and so on. To build a digit d in a place, that place’s button is pressed d times.
Using a smaller button instead costs more: one thousand needs 1 click of +1000 but 10 clicks of +100 or 100 clicks of +10.
So the cheapest way is to use each place’s own button as many times as its digit, and the total is the sum of the digits. This least-click expression is exactly the place value expansion.
(ii) 85,40,300 = (854 × 10,000) + (3 × 100).
Check: 854 × 10,000 = 85,40,000 and 85,40,000 + 300 = 85,40,300.
Total clicks = 854 + 3 = 857.
(iii) For example, 45,678. Both buttons add multiples of 100, so every total is a multiple of 100. 45,678 is not a multiple of 100, so it can never be shown.
In simple words: Each place’s own button is cheapest; the second calculator can only reach multiples of 100.

Teacher's Note:
1. Any 5-digit number not ending in 00 is a correct answer to (iii).
2. Always check an expression by adding it back.

Question 10. Estu makes three claims about large quantities. Test each one by calculation and state whether it is true or false.
(i) “Counting 1 million coins is possible in a single day, if I count one coin every second.”
(ii) “If 300 babies are born every minute across the world, then 5 lakh babies are born in a day.”
(iii) “I can lift 50,000 sheets of paper at one time, if each sheet weighs 4 g.”
Finally, explain in one or two lines why breaking such problems into stages makes large-number calculations easier.
Answer:
(i) A day has 60 × 60 × 24 = 86,400 seconds, so only 86,400 coins can be counted. This is far less than 10,00,000, so the claim is false. (10,00,000 ÷ 86,400 ≈ 11.6, so it would take nearly 12 days.)
(ii) Babies in one hour = 300 × 60 = 18,000. Babies in one day = 18,000 × 24 = 4,32,000. This is less than 5,00,000, so the claim is false, though not far off.
(iii) Total weight = 50,000 × 4 g = 2,00,000 g = 200 kg. Nobody can lift 200 kg at once, so the claim is false.
Why stages help: one frightening calculation becomes two or three simple ones (per minute, per hour, per day). Each stage can be checked, so mistakes are caught early.
In simple words: All three claims are false; working in small stages makes big calculations easy to check.

Teacher's Note:
1. Turn grams into kilograms by dividing by 1000.
2. Compare each result with the number in the claim.

Case Study Based Questions

Question 11. A village seed bank: Lakshmamma’s village runs a seed bank. Old farmers in the village say that India once grew about one lakh varieties of rice, but today most farmers buy only a handful of varieties from the market. To save the old seeds, the village seed bank has slowly collected different kinds of rice seed from nearby districts. Its register shows that 78,500 packets of indigenous seed have been collected and stored so far. The members plan to keep adding new varieties every year, and they often talk about the day their collection will reach one lakh packets.
(i) How much less than one lakh is 78,500?
(ii) Write 1,06,000 in words in the Indian system.
(iii) Write the smallest 6-digit number in figures and in words.
(iv) If the seed bank adds 100 new varieties every year, how many years will it take to collect one lakh varieties?
Answer:
(i) 1,00,000 − 78,500 = 21,500. The collection is 21,500 packets short of one lakh.
(ii) 1,06,000 = One lakh six thousand.
(iii) 1,00,000, read as one lakh.
(iv) 1,00,000 ÷ 100 = 1000 years, which shows how large one lakh really is.
In simple words: The bank is 21,500 short of a lakh, and at 100 a year a lakh would take 1000 years.

Teacher's Note:
1. Subtract for 'how much less', divide for 'how many years'.
2. 1,06,000 has 06 in the thousands group.

Teacher's Note:
1. Add two numbers at a time to avoid mistakes.
2. The printed value is smaller, so it is rounding down.

Question 13. The calculator shop in the Land of Tens: A shop in the Land of Tens sells strange calculators. Thoughtful Thousands has only a +1000 button. Handy Hundreds has only a +100 button. Tedious Tens has only a +10 button. Systematic Sippy has the buttons +1, +10, +100, +1000, +10000 and +100000, and it always tries to reach a number using as few clicks as it can. A customer wants each of these calculators to display the number 36,000 on its screen, and then wants to compare how each calculator had to work.
(i) How many times must the +1000 button of Thoughtful Thousands be pressed?
(ii) How many times must the +100 button of Handy Hundreds be pressed?
(iii) How many times must the +10 button be pressed to show one lakh?
(iv) Write Systematic Sippy’s expression for 36,000 and the number of clicks it needs.
Answer:
(i) 36,000 ÷ 1000 = 36 presses.
(ii) 36,000 ÷ 100 = 360 presses.
(iii) 1,00,000 ÷ 10 = 10,000 presses.
(iv) 36,000 = (3 × 10,000) + (6 × 1000), which needs 3 + 6 = 9 clicks, the sum of the digits.
In simple words: Divide by the button value; Sippy needs only the sum of the digits, 9 clicks.

Teacher's Note:
1. The bigger the button, the fewer the presses.
2. Least clicks = sum of the digits.

Question 14. A district census report: A district office prepared a report after the census. On the cover it printed the total population of the district as 7234567, without any commas. The officer explained to the students visiting the office that the same number is grouped in two different ways around the world: in the Indian system the digits are grouped as thousands, lakhs and crores, while in the American or International system they are grouped as thousands, millions and billions. The students were asked to rewrite the figure in both systems and read it aloud.
(i) Write the number with commas in the Indian system.
(ii) Write its number name in the Indian system.
(iii) Write the number with commas in the American system and read it in that system.
(iv) How many zeroes does 1 crore have, and how many lakhs make 1 crore?
Answer:
(i) 72,34,567.
(ii) Seventy-two lakh thirty-four thousand five hundred sixty-seven.
(iii) 7,234,567, read as seven million two hundred thirty-four thousand five hundred sixty-seven.
(iv) 1 crore has 7 zeroes (1,00,00,000), and 100 lakhs make 1 crore.
In simple words: 72,34,567 in Indian commas is 7,234,567 in American commas.

Teacher's Note:
1. Commas always start from the right.
2. Same number, different names.

Teacher's Note:
1. Note the 2011 column comes first in this table.
2. Round to lakhs for quick comparison: 30 ÷ 17 ≈ 1.8.

Question 16. Shopping for a school function: A school is preparing for its annual day. The store keeper has to buy 125 notebooks costing ₹24 each as prizes, 48 chairs costing ₹250 each, and two sets of 50 decorative lamps with each lamp costing ₹1250. Instead of doing long multiplications, the school clerk uses the regrouping shortcuts learnt in class: he changes 125 into \( \frac{1000}{8} \) and 250 into \( \frac{1000}{4} \) so that the multiplication becomes a simple division followed by multiplication by 1000.
(i) Find the cost of the notebooks using the shortcut.
(ii) Find the cost of the chairs using the shortcut.
(iii) Find the value of 2 × 1250 × 50, the cost of the lamps.
(iv) Find the total of the three amounts.
Answer:
(i) Notebooks: 125 × 24 = \( \frac{1000}{8} \) × 24 = 3 × 1000 = ₹3,000.
(ii) Chairs: 48 × 250 = \( \frac{48}{4} \) × 1000 = 12 × 1000 = ₹12,000.
(iii) Lamps: 2 × 1250 × 50 = 1250 × (2 × 50) = 1250 × 100 = ₹1,25,000.
(iv) Total = 3000 + 12,000 + 1,25,000 = ₹1,40,000, that is one lakh forty thousand rupees.
In simple words: Use 125 = 1000 ÷ 8 and 250 = 1000 ÷ 4 to get ₹3,000, ₹12,000, ₹1,25,000; total ₹1,40,000.

Teacher's Note:
1. Divide first, then multiply by 1000.
2. Pair 2 with 50 to make 100.

Question 17. The longest train rides: The longest single-train journey in the world runs in Russia between Moscow and Vladivostok. The train covers about 9,300 km and the journey takes about 7 days. The longest train route in India runs from Dibrugarh in Assam to Kanyakumari in Tamil Nadu; it covers 4,219 km in about 76 hours. Travellers often describe these distances only in round figures, because the exact kilometre count changes a little with every change in the track.
(i) About how many kilometres does the Russian train cover each day?
(ii) Round the Indian route’s distance to the nearest hundred kilometres.
(iii) How much longer is the Russian route than the Indian route?
(iv) About how many times the Indian route is the Russian route?
Answer:
(i) 9300 ÷ 7 ≈ 1328.6, so about 1,330 km a day (roughly 1,300 km).
(ii) 4,219 rounds to 4,200 km, because 19 is less than 50.
(iii) 9300 − 4219 = 5081 km longer.
(iv) 9300 ÷ 4219 ≈ 2.2, so the Russian route is a little more than 2 times the Indian route.
In simple words: The Russian train does about 1,330 km a day and its route is about 2 times the Indian one.

Teacher's Note:
1. Divide total distance by days for distance per day.
2. For nearest hundred, look at the last two digits.

Question 18. The blue whale is the heaviest animal alive. The weight of a large adult blue whale in kilograms can be found by working out 10,50,00,000 ÷ 700. A newborn blue whale already weighs about 2,700 kg, which is about the weight of an adult hippopotamus, and the heart of a blue whale has been recorded at nearly 700 kg. A blue whale can eat up to 3,500 kg of krill in a single day. The largest known land animal, Argentinosaurus, is estimated to have weighed about 90,000 kg.
(i) Find the weight of the adult blue whale in kilograms.
(ii) How much heavier is the adult blue whale than Argentinosaurus?
(iii) How much krill can a blue whale eat in 30 days?
(iv) Write the whale’s weight in words in the Indian system.
Answer:
(i) 10,50,00,000 ÷ 700 = 1,50,000 kg. (Check: 1,50,000 × 700 = 10,50,00,000.)
(ii) 1,50,000 − 90,000 = 60,000 kg heavier.
(iii) 3500 × 30 = 1,05,000 kg of krill.
(iv) 1,50,000 is one lakh fifty thousand kilograms.
In simple words: The whale weighs 1,50,000 kg, 60,000 kg more than Argentinosaurus.

Teacher's Note:
1. Cancel the two zeroes of 700 first: 1,05,00,000 ÷ 7.
2. Check divisions by multiplying back.

Question 19. How far is the Sun? The approximate distance in kilometres between the Earth and the Sun can be found by evaluating 2100 × 70,000. This distance keeps changing a little through the year; at its farthest the Sun is about 152 million kilometres away. Numbers of this size are hard to picture, so scientists often compare them with journeys we know. Roxie imagines a spacecraft that can travel exactly 1,000 km every single day without a break, and asks whether such a craft could reach the Sun within one human lifetime. Take 1 year = 365 days.
(i) Find the Earth–Sun distance by regrouping the factors.
(ii) Write this distance in words in both the Indian and the American systems.
(iii) Round the distance to the nearest crore kilometres.
(iv) How many years would Roxie’s spacecraft take? Could it reach the Sun in a lifetime?
Answer:
(i) 2100 × 70,000 = (21 × 7) × (100 × 10,000) = 147 × 10,00,000 = 14,70,00,000 km.
(ii) Indian: 14,70,00,000, fourteen crore seventy lakh. American: 147,000,000, one hundred forty-seven million.
(iii) It lies between 14 crore and 15 crore, and 70,00,000 is more than 50,00,000, so it rounds to 15,00,00,000 km (15 crore km).
(iv) Days = 14,70,00,000 ÷ 1000 = 1,47,000 days. Years = 1,47,000 ÷ 365 ≈ 402.7, about 403 years.
No one lives 403 years, so the spacecraft could not reach the Sun within a lifetime.
In simple words: The Sun is about 14 crore 70 lakh km away; at 1,000 km a day it would take about 403 years.

Teacher's Note:
1. Multiply the non-zero parts, then add all the zeroes.
2. Work in stages: days first, then years.

Teacher's Note:
1. More digits always beats bigger digits for size.
2. List all positions before choosing the largest.