Exploring Large Numbers — Questions and Answers
Welcome to HSLC Guru. This lesson gives full, worked solutions to every “Work it Out” (1.1–1.10) box and the end-of-chapter Exercise 1 of Chapter 1 Exploring Large Numbers from ASSEB Class 7 New Mathematics, along with extra practice questions and a Key Terms table.
Summary
In this chapter we learn to read and write numbers with more than five digits using commas. In the Indian place-value system commas are placed in the 3-2-2-2 pattern (thousand, lakh, crore); in the International system they follow the 3-3-3-3 pattern (thousand, million, billion). One hundred lakh make one crore.
Any number can be written in expanded form as the sum of the place values of its digits. To handle large numbers easily we often use an approximate (estimated) value instead of the exact value. Rounding off gives the nearest value; round up gives the upper nearest value; round down gives the lower nearest value.
The chapter ends with short methods of multiplication — multiplying by 11, 12, 25 and 125, the Vedic one-line method, and multiplication by a series of 9s using factorisation and re-grouping. The number of digits in a product equals the total number of digits of the two numbers, or one less.
Summary: This ASSEB Class 7 New Mathematics Chapter 1 “Exploring Large Numbers” solution explains reading and writing numbers with more than five digits, expanded form, the Indian (3-2-2-2) and International (3-3-3-3) place-value systems, exact value versus approximate value (rounding off, round up, round down), and short methods of multiplication including the Vedic method. Every Work it Out (1.1–1.10) box and the end-of-chapter Exercise 1 are solved step by step for HSLC Guru.
Textbook Questions and Answers
In-lesson Tasks
1. The Narendra Modi Stadium has a total of 1,32,000 seats. How many more is that than one lakh?
Answer: 1,32,000 − 1,00,000 = 32,000. So the stadium has 32,000 more seats than one lakh.
2. Fill in the blanks in the picture (99998 … 100000 … 100003) with the correct numbers.
Answer: Adding 1 each time: 99998 → 99999 → 100000 → 100001 → 100002 → 100003. So the blanks are 99999, 100001, 100002.
3. Card game (Raju=1, Amina=10, Joseph=100, Tora=1000, Mijing=10000, Sujata=100000, Bijit=1000000):
- 5 cards from Amina’s hand: 5 × 10 = 50.
- To get 6000 from Tora: 6000 ÷ 1000 = 6 cards.
- If Joseph gives 7 cards: 7 × 100 = 700.
- To form Bijit’s number (1000000) from Sujata: 1000000 ÷ 100000 = 10 cards.
4. Write the following numbers in expanded form.
Answer:
23501 = 2 × 10000 + 3 × 1000 + 5 × 100 + 0 × 10 + 1 × 1
60521 = 6 × 10000 + 0 × 1000 + 5 × 100 + 2 × 10 + 1 × 1
123462 = 1 × 100000 + 2 × 10000 + 3 × 1000 + 4 × 100 + 6 × 10 + 2 × 1
786543 = 7 × 100000 + 8 × 10000 + 6 × 1000 + 5 × 100 + 4 × 10 + 3 × 1
903210 = 9 × 100000 + 0 × 10000 + 3 × 1000 + 2 × 100 + 1 × 10 + 0 × 1
5. Number-of-digits questions from the multiplication patterns:
- The product of two two-digit numbers is always a three- or four-digit number (10 × 10 = 100, 99 × 99 = 9801).
- The product of two three-digit numbers has 5 or 6 digits.
- A five-digit number × a five-digit number gives a product with 9 or 10 digits.
- A three-digit number × a two-digit number gives a product with 4 or 5 digits.
Work it Out 1.1
1. Write the following numbers in words.
Answer:
(a) 12,43,700 = twelve lakh forty-three thousand seven hundred.
(b) 95,12,550 = ninety-five lakh twelve thousand five hundred fifty.
(c) 7,20,205 = seven lakh twenty thousand two hundred five.
(d) 55,80,710 = fifty-five lakh eighty thousand seven hundred ten.
2. Write the following in numbers.
Answer:
(a) Five lakh twenty thousand three hundred twelve = 5,20,312.
(b) Thirty lakh fourteen thousand one hundred five = 30,14,105.
(c) Seventy-seven lakh fifty-five thousand six hundred sixty-six = 77,55,666.
Work it Out 1.2
1. Using 20 cards, form the largest and smallest four-digit numbers. What are those two numbers?
Answer: With the fewest cards (each digit 0–9) the number of cards equals the sum of the digits, so the digits must add up to 20. Largest four-digit number = 9920 (9+9+2+0 = 20); smallest four-digit number = 1199 (1+1+9+9 = 20).
2. To obtain 245619, how can the cards be selected to minimise the number of cards?
Answer: 245619 = 2 × 100000 + 4 × 10000 + 5 × 1000 + 6 × 100 + 1 × 10 + 9 × 1. Take 2 from Sujata, 4 from Mijing, 5 from Tora, 6 from Joseph, 1 from Amina and 9 from Raju — a total of 27 cards.
3. To obtain 5309 with the fewest cards, how can the cards be selected?
Answer: 5309 = 5 × 1000 + 3 × 100 + 0 × 10 + 9 × 1. Take 5 from Tora, 3 from Joseph and 9 from Raju — a total of 17 cards.
4. To obtain 4572, 93107 and 875432, how can the cards be selected? Write the expanded form for each.
Answer:
4572 = 4 × 1000 + 5 × 100 + 7 × 10 + 2 × 1 (Tora 4, Joseph 5, Amina 7, Raju 2 = 18 cards).
93107 = 9 × 10000 + 3 × 1000 + 1 × 100 + 0 × 10 + 7 × 1 (Mijing 9, Tora 3, Joseph 1, Raju 7 = 20 cards).
875432 = 8 × 100000 + 7 × 10000 + 5 × 1000 + 4 × 100 + 3 × 10 + 2 × 1 (Sujata 8, Mijing 7, Tora 5, Joseph 4, Amina 3, Raju 2 = 29 cards).
Work it Out 1.3
1. Insert commas and write in words in both the Indian and International systems.
Answer:
(a) 98020120 — Indian: 9,80,20,120 (nine crore eighty lakh twenty thousand one hundred twenty); International: 98,020,120 (ninety-eight million twenty thousand one hundred twenty).
(b) 50553624 — Indian: 5,05,53,624 (five crore five lakh fifty-three thousand six hundred twenty-four); International: 50,553,624 (fifty million five hundred fifty-three thousand six hundred twenty-four).
(c) 430265327 — Indian: 43,02,65,327 (forty-three crore two lakh sixty-five thousand three hundred twenty-seven); International: 430,265,327 (four hundred thirty million two hundred sixty-five thousand three hundred twenty-seven).
(d) 4654489890 — Indian: 4,65,44,89,890 (four arab sixty-five crore forty-four lakh eighty-nine thousand eight hundred ninety); International: 4,654,489,890 (four billion six hundred fifty-four million four hundred eighty-nine thousand eight hundred ninety).
2. Write the following in numbers using both the Indian and International systems.
Answer:
(a) Nine crore eleven lakh twenty-one thousand two hundred two = Indian 9,11,21,202; International 91,121,202.
(b) Fifty million twenty-six thousand five hundred seven = International 50,026,507; Indian 5,00,26,507.
(c) Twenty-two billion thirty-five million seven thousand seven hundred forty = International 22,035,007,740; Indian 22,03,50,07,740.
3. Fill in the blanks with >, = or <.
Answer:
(a) 2,78,56,145 > 2,78,40,561.
(b) 120,002,000 < 120,020,000.
(c) 35 million = 3,50,00,000 (35 million = 3,50,00,000).
(d) 212 billion > 2,120 crore (212 billion = 2,12,000 crore).
Work it Out 1.4
1. Find the nearest approximate value of the following numbers to the thousands, lakhs and crores place.
Answer:
(a) 5,63,59,957 — thousands: 5,63,60,000; lakhs: 5,64,00,000; crores: 6,00,00,000.
(b) 49,05,62,481 — thousands: 49,05,62,000; lakhs: 49,06,00,000; crores: 49,00,00,000.
2. For each number, mentioning five different places, write the nearest approximate value.
Answer:
(a) 2,45,06,853 — thousand: 2,45,07,000; ten-thousand: 2,45,10,000; lakh: 2,45,00,000; ten-lakh: 2,50,00,000; crore: 2,00,00,000.
(b) 84,75,56,432 — thousand: 84,75,56,000; ten-thousand: 84,75,60,000; lakh: 84,76,00,000; ten-lakh: 84,80,00,000; crore: 85,00,00,000.
3. Write five numbers whose nearest approximate value to different place values will be 75,00,000.
Answer: (many answers are possible) — 74,50,000 (to the lakh), 75,48,000 (to the lakh), 74,96,000 (to the ten-thousand), 75,03,000 (to the ten-thousand), 74,99,500 (to the thousand). Each of these rounds to 75,00,000.
Work it Out 1.5
1. A Jorhat farmer’s rice production in 2023 was 9,856 kg per hectare and in 2024 it rose to 12,347 kg per hectare. (a) Find the difference; (b) estimate the difference by rounding each year to the nearest thousand; (c) how does this help the farmer plan?
Answer: (a) Exact difference = 12,347 − 9,856 = 2,491 kg. (b) Rounding to the nearest thousand: 9,856 → 10,000 and 12,347 → 12,000; estimated difference = 12,000 − 10,000 = 2,000 kg. (c) “About 2,000 kg more” is an easy figure that helps the farmer plan seeds, fertiliser, storage and expected income for next year without exact numbers.
2. Earth–Sun average distance is 149,597,870 km and Mars–Sun is 227,943,824 km. (a) Find the exact difference; (b) estimate it by rounding to the nearest 10 million.
Answer: (a) Exact difference = 227,943,824 − 149,597,870 = 78,345,954 km. (b) Rounding to the nearest 10 million: 149,597,870 → 150,000,000 and 227,943,824 → 230,000,000; estimated difference = 230,000,000 − 150,000,000 = 80,000,000 km.
Work it Out 1.6
1. Find the product — (a) 327 × 25 (b) 425 × 25 (c) 123 × 125 (d) 440 × 125 (e) 926 × 125.
Answer: To multiply by 25, multiply by 100 and divide by 4; to multiply by 125, multiply by 1000 and divide by 8.
(a) $327 \times 25 = \frac{327 \times 100}{4} = \frac{32700}{4} = 8175$
(b) $425 \times 25 = \frac{42500}{4} = 10625$
(c) $123 \times 125 = \frac{123 \times 1000}{8} = \frac{123000}{8} = 15375$
(d) $440 \times 125 = \frac{440000}{8} = 55000$
(e) $926 \times 125 = \frac{926000}{8} = 115750$
Work it Out 1.7
1. Find the product using the Vedic method — (a) 45 × 36 (b) 24 × 13 (c) 87 × 16 (d) 37 × 54.
Answer: Multiply the units digits, then add the cross (diagonal) products, then multiply the tens digits, carrying over to the next step where needed.
(a) 45 × 36 → 6|(24+15)|30 = 6|39|30 = 1620.
(b) 24 × 13 → 2|(6+4)|12 = 2|10|12 = 312.
(c) 87 × 16 → 8|(48+7)|42 = 8|55|42 = 1392.
(d) 37 × 54 → 15|(12+35)|28 = 15|47|28 = 1998.
Work it Out 1.8
1. Multiply using the Vedic method — (a) 154 × 624 (b) 312 × 126 (c) 412 × 125.
Answer: Three-digit Vedic multiplication uses five steps (units, one cross pair, three cross pairs, one cross pair, hundreds).
(a) 154 × 624 = 96096.
(b) 312 × 126 = 39312.
(c) 412 × 125 = 51500 (by 125: 412000 ÷ 8 = 51500).
Work it Out 1.9
1. Solve using the steps shown above — (a) 1254 × 3294 (b) 1026 × 3150 (c) 4444 × 3333.
Answer: Four-digit Vedic multiplication uses seven steps; adding all the diagonal cross-products (with carrying) gives the same result as ordinary multiplication.
(a) 1254 × 3294 = 4130676.
(b) 1026 × 3150 = 3231900.
(c) 4444 × 3333 = 14811852.
Work it Out 1.10
1. Solve using the methods shown above — (a) 187 × 9999 (b) 5687 × 99999. (Note: place a 0 in front of 187 and 5687.)
Answer: For a series of 9s, multiply the number by 10, 100, 1000 … and subtract the number.
(a) 187 × 9999 = 187 × 10000 − 187 = 1870000 − 187 = 1869813.
(b) 5687 × 99999 = 5687 × 100000 − 5687 = 568700000 − 5687 = 568694313.
2. (a) 2431 × 99 (b) 62543 × 9999.
Answer:
(a) 2431 × 99 = 2431 × 100 − 2431 = 243100 − 2431 = 240669.
(b) 62543 × 9999 = 62543 × 10000 − 62543 = 625430000 − 62543 = 625367457.
Exercise 1
1. Which number is equal to 50 million? (A) 5 lakhs (B) 50 lakhs (C) 5 crores (D) 50 crores
Answer: (C) 5 crores. 50 million = 50 × 10,00,000 = 5,00,00,000 = 5 crore.
2. Which number is one million fifty thousand three hundred and seven? (A) 1,050,307 (B) 10,500,307 (C) 1,500,307 (D) 150,307
Answer: (A) 1,050,307.
3. Which of the following is the difference of the place values of 8 and 6 in the number 7981654? (A) 74940 (B) 79400 (C) 84900 (D) 89400
Answer: (B) 79400. Place value of 8 = 80000, place value of 6 = 600; difference = 80000 − 600 = 79400.
4. The difference between the largest and the smallest number whose approximate value to the nearest 10 is 520? (A) 7 (B) 8 (C) 9 (D) 10
Answer: (C) 9. Numbers rounding to 520 range from 515 to 524; largest 524, smallest 515, difference = 9.
5. How many hundreds are there in the number 35472? (A) 354 (B) 400 (C) 5472 (D) 35400
Answer: (A) 354. 35472 ÷ 100 = 354.72, so there are 354 complete hundreds.
6. An organisation bought 240 chart papers for 232 students. Which method was used? (A) Exact value (B) Round off to the nearest ten (C) Upper nearest approximate value (D) Lower nearest approximate value
Answer: (C) Upper nearest approximate value. 232 was rounded up to 240 so that every student gets a chart paper.
7. The total prize money of the 2024 T-20 World Cup was about ₹937462500. (A) Express it with commas in both systems and write in words; (B) write it in expanded form; (C) approximate it to the nearest thousands, lakhs and crores.
Answer: (A) Indian: 93,74,62,500 — ninety-three crore seventy-four lakh sixty-two thousand five hundred. International: 937,462,500 — nine hundred thirty-seven million four hundred sixty-two thousand five hundred.
(B) Expanded form = 9 × 100000000 + 3 × 10000000 + 7 × 1000000 + 4 × 100000 + 6 × 10000 + 2 × 1000 + 5 × 100 + 0 × 10 + 0 × 1.
(C) To the thousand: 93,74,63,000; to the lakh: 93,75,00,000; to the crore: 94,00,00,000.
8. Using all the digits 1, 5, 6 and 8 (without repeating) form a number whose approximate value to the nearest hundred is 6500 and to the nearest thousand is 7000.
Answer: 6518. For the nearest-thousand value to be 7000 the number must lie in 6500–7499, and for the nearest-hundred value to be 6500 it must lie in 6450–6549; so it must be between 6500 and 6549. 6518 rounds to 6500 (hundred) and 7000 (thousand).
9. Using all the digits 2, 3, 5, 7, 8 and 9 (without repeating) form a number whose approximate value to the nearest thousand is 895000 and to the nearest ten thousand is 900000.
Answer: 895237 (895273, 895327, 895372 also work). The number must lie in 895000–895499; 895237 rounds to 895000 (thousand) and 900000 (ten-thousand).
10. Assertion (P): The expanded form of 230147 in the Indian system is 2×100000 + 3×10000 + 0×1000 + 1×100 + 4×10 + 7×1. Reason (Q): In the Indian expanded form the digits multiplied by 1, 10, 100, 1000 … must be from 0 to 9.
Answer: (a) Both P and Q are true and Q is the correct explanation of P.
11. Observe the patterns — (P) 236891, 237891, 238891, 239891 (Q) 143656, 143756, 143856, 143956 (R) 736541, 746541, 756541, 766541 (S) 913567, 913577, 913587, 913597. Choose the option that arranges their increasing steps in ascending order.
Answer: (c) S, Q, P, R. The common difference is S = 10, Q = 100, P = 1000, R = 10000; in ascending order that is S, Q, P, R.
12. Assertion (P): The nearest approximate value of 865492 to its highest place value is 9,00,000. Reason (Q): If the digit to the right of the given place value is 5 or more, that digit is increased by 1 and the digits to the right are made 0.
Answer: (a) Both P and Q are true and Q is the correct explanation of P. The highest place value of 865492 is the lakh place (8); the digit to its right is 6 (≥5), so 8 becomes 9, giving 9,00,000.
Rounding on a Number Line
On the number line below, 72 is nearer to 70, 77 is nearer to 80, and 75 lies exactly in the middle. When a number is exactly in the middle, the larger value (80) is chosen.
Additional Questions and Answers
Multiple Choice Questions
1. 1 crore = how many lakhs? (A) 10 (B) 100 (C) 1000 (D) 10000
Answer: (B) 100.
2. In the Indian system, in which pattern are commas placed? (A) 3-3-3-3 (B) 2-2-2-3 (C) 3-2-2-2 (D) 2-3-2-3
Answer: (C) 3-2-2-2.
3. 1 MB = how many bytes? (A) 1024 (B) 1000 (C) 1048576 (D) 1073741824
Answer: (C) 1048576 (1024 × 1024).
4. The approximate value of 8572 to the nearest hundred is? (A) 8500 (B) 8600 (C) 8000 (D) 9000
Answer: (B) 8600.
5. In the International system, 1 million = ? (A) 1 lakh (B) 10 lakh (C) 1 crore (D) 10 crore
Answer: (B) 10 lakh.
6. To multiply a number by 25, we multiply it by 100 and divide by which number? (A) 2 (B) 4 (C) 8 (D) 5
Answer: (B) 4.
7. 999 × 999 = ? (A) 998001 (B) 999999 (C) 980001 (D) 998010
Answer: (A) 998001.
8. How many seconds are there in a year (365 days)? (A) 3,15,36,000 (B) 8,64,00,000 (C) 3,60,00,000 (D) 1,00,00,000
Answer: (A) 3,15,36,000.
9. The product of two three-digit numbers has how many digits? (A) 4 or 5 (B) 5 or 6 (C) 6 or 7 (D) 3 or 4
Answer: (B) 5 or 6.
10. 50 million = how many crores? (A) 5 (B) 50 (C) 500 (D) half
Answer: (A) 5.
Fill in the Blanks
1. One hundred lakh is called ____.
Answer: one crore.
2. In the Indian system, commas are placed in the ____ pattern.
Answer: 3-2-2-2.
3. 1 GB = ____ MB.
Answer: 1024.
4. If the digit to the right of the given place value is 5 or more, the digit is increased by ____.
Answer: 1.
5. Writing a number as the sum of the place values of its digits is called its ____ form.
Answer: expanded.
True or False
1. There is a largest number.
Answer: False (numbers are infinite; a bigger number can always be found).
2. The approximate value of 75 to the nearest ten is 80.
Answer: True.
3. In the International system, commas are placed in the 3-3-3-3 pattern.
Answer: True.
4. 1 kilobyte = 1000 bytes.
Answer: False (1 kilobyte = 1024 bytes).
5. The number of digits in a product equals the total number of digits of the two numbers, or one less.
Answer: True.
Short Answer Questions
1. What is the difference between rounding off, round up and round down?
Answer: Rounding off takes the nearest place value; round up (upper nearest approximate value) takes a value greater than the actual value; round down (lower nearest approximate value) takes a value smaller than the actual value.
2. Explain the short method of multiplying by 11 with the example 3452 × 11.
Answer: Put a 0 in front of the number (03452) and add each digit to the digit on its right: (0+3), (3+4), (4+5), (5+2), (2+0) = 3, 7, 9, 7, 2. So 3452 × 11 = 37972.
3. What is the main difference between the Indian and International number systems?
Answer: The Indian system places commas in the 3-2-2-2 pattern and uses lakh and crore; the International system places commas in the 3-3-3-3 pattern and uses million and billion.
4. How is 4956 × 9999 found by taking 10000 as the base?
Answer: 4956 × 9999 = 4956 × 10000 − 4956 = 49560000 − 4956 = 49555044. That is, multiply the number by 10000 and subtract the number itself.
Key Terms
| Term | Meaning |
|---|---|
| Place value | The value of a digit according to its position in the number |
| Expanded form | A number written as the sum of the place values of its digits |
| Indian system | Comma pattern 3-2-2-2, using lakh and crore |
| International system | Comma pattern 3-3-3-3, using million and billion |
| Crore | One hundred lakh (1,00,00,000) |
| Approximate value / Estimation | A value close to the exact value that is easy to state |
| Rounding off | Approximating to the nearest place value |
| Round up | Upper nearest approximate value (greater than the actual value) |
| Round down | Lower nearest approximate value (less than the actual value) |
| Factor | A number used in multiplication |
| Vedic multiplication | A short one-line method of multiplication |