Jaka EduTech
Jaka EduTech
Parent Sign In

© 2026 Jaka EduTech

Jaka EduTech

📖 Mathematics

Std 7
Books Book Guide
1 2 3 4 5 6 7 8
Chapters
1 Large Numbers Around Us 2 Arithmetic Expressions 3 A Peek Beyond the Point 4 Expressions Using Letter-Numbers 5 Parallel and Intersecting Lines 6 Number Play 7 A Tale of Three Intersecting Lines 8 Working with Fractions
1
Chapter 1
Skill: 30%

Large Numbers Around Us

Large Numbers Around Us

Chapter at a Glance

This chapter introduces large numbers up to crores and billions, exploring how they are represented, read, written, and compared. Using real-world contexts like the history of rice seed varieties, population growth, waterfall heights, and migratory bird routes, the chapter highlights how place value functions. It contrasts the Indian place-value system with the American (International) system and dives into the practical difference between exact and approximate values. It also introduces mental estimation techniques and multi-digit multiplication properties (such as calculating the length of products).

Key Definitions & Terminology

  • Lakh: The smallest 6-digit number, represented as $1,00,000$ (1 followed by 5 zeros) in the Indian system. It is equal to a hundred thousand in the American system.
  • Crore: Represented as $1,00,00,000$ (1 followed by 7 zeros) in the Indian system. It is equal to 100 lakhs, or ten million in the American system.
  • Arab: Represented as $1,00,00,00,00,000$ (1 followed by 9 zeros) in the Indian system. It is equal to 100 crores, or one billion in the American system.
  • Indian Place Value System: A system of grouping digits using a 3-2-2-2... pattern from right to left (Thousands, Lakhs, Crores, Arabs).
  • American (International) Place Value System: A system of grouping digits uniformly in a 3-3-3-3... pattern from right to left (Thousands, Millions, Billions).
  • Rounding Up: Approximating a number to a larger convenient value when safety margins or extra quantity is preferred (e.g. buying slightly more food than exactly needed for a crowd).
  • Rounding Down: Approximating a number to a smaller convenient value when simplifying costs or underestimating is appropriate (e.g. a shopkeeper quoting ₹450 instead of ₹470).

Formulas, Rules & Properties

  • Digit Length of Products: The product of an $n$-digit number and an $m$-digit number will have:
  • Minimum digits: $n + m - 1$
  • Maximum digits: $n + m$
  • Multiplication Shortcuts:
  • Multiplying by 5: Halve the number, then multiply by 10: $(x \div 2) \times 10$.
  • Multiplying by 25: Divide the number by 4, then multiply by 100: $(x \div 4) \times 100$.
  • Multiplying by 125: Divide the number by 8, then multiply by 1000: $(x \div 8) \times 1000$.
  • System Conversion Rules:
  • $10\text{ Thousands} = 1\text{ Ten Thousand}$
  • $100\text{ Thousands} = 1\text{ Lakh}$
  • $100\text{ Lakhs} = 1\text{ Crore}$
  • $100\text{ Crores} = 1\text{ Arab}$
  • $1\text{ Million} = 10\text{ Lakhs}$
  • $10\text{ Million} = 1\text{ Crore}$
  • $100\text{ Million} = 10\text{ Crores}$
  • $1\text{ Billion} = 1\text{ Arab} = 100\text{ Crores} = 10,000\text{ Lakhs}$

Core Concepts & Topics

  • Place Value Systems & Commas:
  • Commas act as punctuation marks to make large numbers readable.
  • In the Indian system, the first comma is placed after 3 digits from the right, and subsequent commas are placed after every 2 digits.
  • In the American system, commas are placed after every 3 digits from the right.
  • Calculator Simulations (Place-Value Decompositions):
  • Thoughtful Thousands (+1000 button), Tedious Tens (+10 button), and Handy Hundreds (+100 button) explore place value increments.
  • Creative Chitti (+1, +10, +100, +1000, +10000, +100000, +1000000 buttons) represents numbers in multiple ways (e.g. $5072 = 50 \times 100 + 7 \times 10 + 2 \times 1$, or $3 \times 1000 + 20 \times 100 + 72 \times 1$).
  • Systematic Sippy clicked buttons as few times as possible. The absolute minimum clicks needed to represent any number is equal to the sum of the digits in its standard place value form.
  • Exact vs. Approximate Values:
  • Approximations (e.g., stating a town's population of 76,068 is about 75,000) are useful when exact figures do not change the general context.
  • Exact numbers are critical in banking, phone dialing, and precise inventory control.
  • Nearest Neighbors (Rounding Limits):
  • Rounding to the nearest thousand, ten thousand, lakh, ten lakh, and crore requires identifying the closest multiple based on midpoint boundaries. E.g. for $6,72,85,183$:
    • Nearest thousand: $6,72,85,000$ (midpoint boundary is $6,72,85,500$)
    • Nearest ten thousand: $6,72,90,000$
    • Nearest lakh: $6,73,00,000$
    • Nearest ten lakh: $6,70,00,000$
    • Nearest crore: $7,00,00,000$
  • Product Patterns:
  • Symmetric products: $11 \times 11 = 121$; $111 \times 111 = 12,321$; $1,111 \times 1,111 = 1,234,321$.
  • Linear expansion sums: $66 \times 61 = 4,026$; $666 \times 661 = 4,40,226$; $6,666 \times 6,661 = 4,44,02,226$.
  • Consecutive ending in 5: $3 \times 5 = 15$; $33 \times 35 = 1,155$; $333 \times 335 = 1,11,555$.
  • Squares of 100s: $101 \times 101 = 10,201$; $102 \times 102 = 10,404$; $103 \times 103 = 10,609$.

Worked Examples

  • Population Comparisons:
  • Question: The 2011 Census population of Chintamani was about 75,000, and the estimated 2024 population is 1,06,000. How much less/more than a lakh are they, and what is the increase?
  • Solution:
    • $1,00,000 - 75,000 = 25,000$ less than a lakh.
    • $1,06,000 - 1,00,000 = 6,000$ more than a lakh.
    • Population increase $= 1,06,000 - 75,000 = 31,000$.
  • Statue of Unity vs. Somu's Building:
  • Question: Somu is 1 m tall. Each floor of his building is 4 times his height. If the building is 10 floors high, what is its height? How does it compare to the Statue of Unity (180 m) and Kunchikal waterfall (450 m)?
  • Solution:
    • Building height $= 10\text{ floors} \times 4\text{ m/floor} = 40\text{ m}$.
    • Height difference with Statue of Unity $= 180\text{ m} - 40\text{ m} = 140\text{ m}$ (The Statue is 140 m taller).
    • Height difference with Kunchikal waterfall $= 450\text{ m} - 40\text{ m} = 410\text{ m}$ (The waterfall is 410 m higher).
    • Floors needed to match the waterfall $= 450\text{ m} \div 4\text{ m/floor} = 112.5$ (approx 113 floors).
  • Number Names to Figures (Indian System):
  • Fifty lakhs five thousand and fifty: $50,05,050$.
  • Ten lakhs two hundred and thirty-five: $10,00,235$.
  • Calculator Clicks Riddle (Creative Chitti):
  • Question: Find the largest and smallest 3-digit numbers you can make using exactly 30 clicks on Chitti's calculator (+100, +10, +1 buttons).
  • Solution:
    • Largest 3-digit number: To maximize $N < 1000$, we maximize the hundreds digit ($a=9$) and tens digit ($b=8$), leaving the remaining 13 clicks for ones ($c=13$). $N = 9 \times 100 + 8 \times 10 + 13 \times 1 = 993$.
    • Smallest 3-digit number: To minimize $N \ge 100$, we click tens ($b=8$) and ones ($c=22$) using 30 clicks. $N = 8 \times 10 + 22 \times 1 = 102$.
  • Digit Striking Riddle:
  • Question: Strike out 10 digits from $12345123451234512345$ so that the remaining number is as large as possible.
  • Solution: To maximize the leading digits of the remaining 10-digit number, we target the highest digit '5'.
    • Strike first four digits (1234) $\rightarrow$ first remaining digit is 5.
    • Strike next four digits (1234) $\rightarrow$ second remaining digit is 5.
    • We have struck 8 digits. The remaining sequence is 1234512345.
    • Strike the next two smallest digits (1 and 2) $\rightarrow$ remaining sequence is 34512345.
    • The largest 10-digit number is $5534512345$.
  • Consecutive Numbers without Common English Letters:
  • Question: Find the first two consecutive numbers that do not share any common English letters in their names.
  • Solution: By counting: 1 (O-N-E) and 2 (T-W-O) share 'o'. 2 and 3 (T-H-R-E-E) share 't'. Continuing this, the first pair is 3 and 4 (T-H-R-E-E and F-O-U-R) which share no letters. So the first consecutive pair is 3 and 4.
  • 1000th Written Digit:
  • Question: If you write numbers sequentially (1, 2, 3...), what is the 1000th digit written?
  • Solution:
    • Digits in 1-digit numbers (1 to 9) $= 9$ digits.
    • Digits in 2-digit numbers (10 to 99) $= 90 \times 2 = 180$ digits.
    • Digits up to 99 $= 189$ digits.
    • Remaining digits needed $= 1000 - 189 = 811$ digits.
    • $811 \div 3 = 270$ full 3-digit numbers with 1 digit left over.
    • The 270th 3-digit number is $100 + 270 - 1 = 369$.
    • The next number is 370. The 1st digit of 370 is 3, which is the 1000th digit.
  • Population Estimation (Buses/Ships):
  • Question: Can Mumbai's population (approx 1.24 crores) fit into 1 lakh buses (capacity 50 each)? Can it fit in 5,00,000 Titanic-sized ships (capacity 2,500 each)?
  • Solution:
    • 1 lakh buses capacity $= 1,00,000 \times 50 = 50,00,000$ (50 lakhs). Since 1.24 crore (124 lakhs) $> 50$ lakhs, they cannot fit.
    • 5,000 ships capacity $= 5,000 \times 2,500 = 1,25,00,000$ (1.25 crore). Since 1.24 crore $< 1.25$ crore, they can fit.
  • Statue of Unity Coin Stack:
  • Question: How many 1 mm thick coins must be stacked to match the height of the Statue of Unity (180 m)?
  • Solution: $180\text{ m} = 1,80,000\text{ mm}$. Since each coin is $1\text{ mm}$ thick, $1,80,000 \div 1 = 1,80,000$ coins.
  • Albatross Migration Days:
  • Question: How many days does it take for an albatross flying 900–1000 km/day to complete a 12,000 km journey?
  • Solution:
    • At $900\text{ km/day}$: $12,000 \div 900 \approx 13.3$ days.
    • At $1000\text{ km/day}$: $12,000 \div 1000 = 12$ days.
    • The trip takes approximately 12 to 14 days.

Practical Activities & Experiments

  • Rice Variety Challenge: Exploring how many varieties of rice a person can eat in a lifetime by calculating days in 100 years ($365 \times 100 = 36,500$ days) to compare it to 1 lakh ($1,00,000$) rice varieties.
  • Height Audits: Using Somu (1 m tall) to estimate the height of a building, comparing it to landmark heights visually.
  • Thoughtful calculators game: Pressing $+1000, +100, +10$ buttons specific times to form numbers like $53,000$ or $1,00,000$.
  • Matchstick Digits puzzle: Representing numbers with matchstick lines (e.g. $42,019$ takes 23 sticks). Rearranging exactly four sticks in $63,890$ to form a larger number ($88,078$). Placing the digit '1' to obtain the largest possible number.
  • Goa Cycle Rally & Godwit flight calculations: Speed, time, and distance estimations of natural movements using division and multiplication.
Chapter 1 of 8 Next