Large Numbers Around Us | CBSE Class 7 Maths Notes

This note covers lakhs and crores, Indian and International number names, place value, calculator button patterns, exact and approximate values, population comparisons, multiplication shortcuts, digits in products, large quantities, thought experiments and number puzzles.

How big is one lakh?

A lakh is one hundred thousand, written as 1,00,000 in the Indian system. It is the smallest six-digit number. A digit is one of the symbols 0 to 9 used to write numbers. The number 1,00,000 contains a 1 followed by five zeroes.

Counting across a boundary helps explain the size of a lakh. After the largest three-digit number, 999, comes 1,000. After 9,999 comes 10,000. After the largest five-digit number, 99,999, comes 1,00,000. Adding one can therefore increase the number of digits.

How can days help us imagine a lakh?

Roxie and Estu consider tasting different varieties of rice over a lifetime of 100 years. For this calculation, take a year as 365 days and ignore leap years. Here, × means multiplication, + means addition, and = means that two quantities have equal values.

Worked example 1. Could someone taste one lakh different rice varieties in 100 years at one, two or three new varieties each day? Use 365 days per year, ignoring leap years.

Answer: 365 × 100 = 36,500 days. One daily variety gives 36,500 varieties; two give 73,000; three give 1,09,500. One or two a day are insufficient. Three a day would allow the target to be reached within 100 years.

If y represents the number of years, the number of days under the same assumption is 365 × y. The letter stands for a quantity whose value can be chosen; it is not a separate measurement unit.

Whether a lakh feels large depends on the comparison. The Ahmedabad cricket stadium has a seating capacity of more than one lakh. Most humans have 80,000 to 1,20,000 hairs on their heads. The same number can describe a very different amount of space or time.

How do we read and write numbers in the Indian system?

Place value is the value contributed by a digit because of its position in a number. Reading from the right, the places begin with ones, tens, hundreds, thousands, ten thousands, lakhs and ten lakhs. Moving a place to the left makes the value of one unit ten times larger.

Commas make these places easier to read. Group the last three digits together, then group the remaining digits in pairs, working towards the left. The Indian grouping pattern is therefore three digits, two digits, two digits, and so on.

How do comma groups guide the number name?

In 12,78,830, read the groups as twelve lakh, seventy-eight thousand and eight hundred thirty. In 15,75,000, the final group is zero, so the name is fifteen lakh seventy-five thousand. Zeroes still preserve places even when no corresponding group is spoken.

Worked example 2. Write 5,04,085 in words, then write “four lakh seven thousand seven hundred and four” in figures.

Answer: 5,04,085 is five lakh four thousand eighty-five. The second number is 4,07,704. In each case, retain the zeroes needed to keep the thousands and final three-digit groups in their correct positions.

To reverse the process, first identify the named groups. Write the lakh group, then the two-digit thousands group, then the three-digit group for hundreds, tens and ones. Check that every spoken part has been represented and that missing places contain zeroes.

For example, fifty lakhs five thousand and fifty becomes 50,05,050. Ten lakhs two hundred and thirty-five becomes 10,00,235. Omitting a zero would change the place occupied by another digit and therefore change the number, rather than merely shorten its written form.

What do calculator buttons teach us about place value?

Imagine calculators starting at zero. A button labelled +1000 adds one thousand on each press. Thoughtful Thousands has this button; Tedious Tens has +10; Handy Hundreds has +100. The number displayed depends on the amount added each time and the number of presses.

The symbol ÷ means division. To find how many equal additions are needed, divide the target number by the button value. Conversely, multiply the number of presses by the button value to find the display. This connects repeated addition with multiplication and division.

Worked example 3. How many presses make one lakh using +1000, +100 and +10 separately, starting at zero?

Answer: 1,00,000 ÷ 1,000 = 100 presses; 1,00,000 ÷ 100 = 1,000 presses; 1,00,000 ÷ 10 = 10,000 presses. Smaller additions require more presses to reach the same target.

Result: The fewest clicks follow place value

Systematic Sippy uses +1, +10, +100, +1000, +10000 and +100000. For numbers within these places, the minimum, meaning the smallest possible, number of presses is the sum of the digits. Ten presses of a smaller button can be replaced by one press of the next larger available button.

Worked example 4. Make 5,072 using the fewest presses of Sippy’s buttons, starting at zero.

Answer: 5,072 = (5 × 1,000) + (7 × 10) + (2 × 1). Brackets group parts of the calculation. Press +1000 five times, +10 seven times and +1 twice: 14 clicks altogether. No hundreds button is needed.

Creative Chitti can represent the same number differently. Fifty hundreds, seven tens and two ones make 5,072. Three thousands, twenty hundreds and seventy-two ones also make 5,072. An expression combines numbers and operation signs to describe a calculation; different expressions may have the same value.

How are crores, millions and billions related?

A crore is one hundred lakhs, written as 1,00,00,000. An arab is one hundred crores. The International system, also called the American system here, names large numbers using millions and billions. A million is ten lakhs, and a billion is one arab.

Result: Successive large Indian units are linked by hundreds

One lakh is one hundred thousands; one crore is one hundred lakhs; one arab is one hundred crores. These relationships connect the named units. They do not mean that adjacent digit places differ by a hundred: adjacent places still differ by a factor of ten.

The International system groups digits in threes throughout, beginning at the right. Changing the comma pattern changes the reading system, not the quantity. Thus 9,87,65,01,234 and 9,876,501,234 contain the same digits in the same order and represent the same number.

Worked example 5. Compare 500 lakhs and 5 million.

Answer: One million is ten lakhs, so 5 million is 50 lakhs. Hence 500 lakhs is greater than 5 million. The symbol > means “greater than”, < means “less than”, and = means “equal to”: 500 lakhs > 5 million.

Before comparing quantities with different number names, express them in a common unit. Comparing 500 with 5 alone would ignore what is being counted. The same method shows that 10,000 lakhs make one billion.

When should we use exact values and approximations?

An exact value gives the precise number being used. An approximation gives a nearby, simpler value sufficient for a particular purpose. Very often, exact numbers are not required and just an approximation is sufficient. The purpose of the number determines the useful amount of detail.

Chintamani’s 2011 census population is 76,068. Saying that it is about 75,000 gives an idea of its size. The approximate statement should retain “about”; it does not claim that the exact count was 75,000.

Cartoon: Book fair (NCERT Class 7, page 9, unnumbered). A child holding a newspaper says, “1 lakh people visited the book fair.” Another child responds, “Ohh, so many people!” A third child then says, “If I had not gone, they would have written ‘99,999 people visited the book fair last week’.” The joke treats a rounded report as an exact count.

How does the purpose affect rounding?

Rounding up replaces a number with a larger approximate value. Rounding down replaces it with a smaller one. A principal arranging sweets for 732 people might order 750 instead of 700. Having enough sweets matters more here than choosing the smaller convenient number.

For an item costing ₹470, a shopkeeper may describe the cost as around ₹450 instead of around ₹500. The symbol ₹ means Indian rupees. This describes a choice of approximation; it does not change the actual price of the item.

Cartoon: Rounded telephone number (NCERT Class 7, page 10, unnumbered). An adult says, “Bappi, call and check if Toofan Express is on time.” Bappi uses a telephone beside a directory listing railway enquiry as 139 and police as 100 or 112. The doorbell rings and a police officer appears. The adult asks, “What number did you call?” Bappi replies, “I called the rounded off number, of course! My teacher told us that efficient people deal with rounded off numbers.” A telephone number needs its exact digits.

A useful approximation therefore depends on context. A rounded population can communicate size, while rounding a telephone number can prevent the intended call. Do not assume that every long number is suitable for rounding simply because its exact form looks inconvenient.

How do nearest values and estimates help us calculate?

A nearest thousand is the closest multiple of one thousand; a multiple is a number obtained by multiplying a given number by a whole number. Nearest ten thousand, lakh, ten lakh and crore use the same idea at different sizes of interval.

For 6,72,85,183, the five nearest values are as follows. Each row answers a different rounding question. The original number stays the same, but the chosen size of interval changes how much detail the approximate answer retains.

How can an estimate be checked?

An estimate is an approximate result judged from the available numbers. The sum is the result of addition; the difference is the result of subtraction. The sign − means subtraction. Check both the closeness of an estimate and whether it lies above or below the exact answer.

Worked example 6. Estimate and then calculate 4,63,128 + 4,19,682. Compare estimates of 8,00,000 and 9,00,000.

Answer: The exact sum is 8,82,810. It is above 8,00,000 and below 9,00,000. Its distance from 8,00,000 is 82,810; its distance from 9,00,000 is 17,190. Therefore 9,00,000 is the closer estimate.

Reasoning can also narrow an answer before the full calculation. Since 4,19,682 is less than 4,20,000, the sum is less than 8,83,128. This comparison uses the same first number and a slightly larger second number, so the resulting total provides an upper comparison.

Similarly, 14,63,128 − 4,90,020 = 9,73,108. The difference lies below 10,00,000 and above 9,00,000, with 10,00,000 closer. Estimation supports checking; an approximate result should remain distinguishable from an exact calculation.

What can population tables tell us?

Population means the number of people in a place. A table comparing different years allows us to examine both the size of populations and changes over time. Read the year headings carefully before subtracting: the later population and earlier population play different roles.

The following figures compare some Indian cities in 2011 and 2001. They are historical values for those years, rather than present-day population estimates. Each comma helps identify the size of the number, but the comparison must use the entire value.

How are size and increase different?

Most cities in this table increased in population. Kolkata is an exception: its 2011 figure is smaller than its 2001 figure. A statement about “most cities” must not become a claim about every city. Mumbai has the largest listed 2011 population, but Bengaluru has the largest increase.

Worked example 7. Pune’s populations were 31,15,431 in 2011 and 25,38,473 in 2001. Find the increase and describe it approximately.

Answer: 31,15,431 − 25,38,473 = 5,76,958. The increase is approximately 6 lakh people. Keep the exact subtraction separate from the rounded description of the change.

Bengaluru’s increase is 41,24,644. Bengaluru, Hyderabad, Surat and Vadodara have populations that roughly doubled (Vadodara's slightly more than doubled). “Roughly doubled” is a comparison of scale; it does not mean that multiplying every earlier figure by two gives its later figure exactly.

How can regrouping make multiplication quicker?

A product is the result of multiplication. The numbers being multiplied are its factors. Factorising means expressing a number as a product of factors; regrouping means collecting suitable factors together to simplify a calculation without changing its value.

Multiplication by 10, 100 or 1,000 is particularly useful. Look for factors which combine to make these numbers. Another approach is to express a multiplier as a convenient number divided by another number, then divide first when the calculation is straightforward.

How do the shortcuts for 5, 25 and 125 work?

Since 5 = 10 ÷ 2, multiplying by five can be done by halving and multiplying by ten. Similarly, 25 = 100 ÷ 4, so multiplication by twenty-five can be done by dividing by four and multiplying by one hundred.

Worked example 8. Calculate 116 × 5 and 824 × 25 using convenient division.

Answer: 116 ÷ 2 = 58, then 58 × 10 = 580. For the second product, 824 ÷ 4 = 206, then 206 × 100 = 20,600. These operations preserve the original multipliers, 5 and 25.

For multiplication by 125, use 125 = 1,000 ÷ 8. Thus 72 × 125 becomes 72 ÷ 8 × 1,000, giving 9,000. The shortcut works because the combined effect of the division and multiplication equals multiplication by the original factor.

Worked example 9. Calculate 125 × 40 × 8 × 25 by regrouping.

Answer: Group 125 with 8 and 40 with 25. Then 125 × 8 = 1,000 and 40 × 25 = 1,000. The complete product is 1,000 × 1,000 = 10,00,000, or one million.

Check that every original factor appears exactly once after regrouping. A quick method should simplify the work while preserving the calculation. For 2 × 1,768 × 50, grouping 2 and 50 makes 100, leaving 1,768 × 100 = 1,76,800.

How many digits can a product have?

Products reveal patterns, but a few examples alone do not explain why a pattern holds. One useful method is to consider the smallest possible factors and a number larger than every possible factor. These give limits within which the product must fall.

Result: Two two-digit factors give three or four digits

The smallest two-digit number is 10, so the smallest product of two two-digit numbers is 10 × 10 = 100. Each two-digit factor is less than 100, so their product is less than 100 × 100 = 10,000.

The product is therefore at least 100 but below 10,000. This range contains three-digit and four-digit numbers. The conclusion follows from the limits on the factors; it does not require checking every possible pair of two-digit numbers.

Worked example 10. Can multiplying two three-digit numbers produce a four-digit number?

Answer: No. The smallest three-digit number is 100, and 100 × 100 = 10,000 already has five digits. Larger three-digit factors cannot give a smaller product. Such products have five or six digits.

What is the wider pattern?

For two positive whole-number factors, add their digit counts. The product has either that many digits or one fewer. A positive whole number is a counting number beginning with 1. Keeping this condition excludes zero from the digit-count rule.

Two five-digit factors produce a nine-digit or ten-digit result. An eight-digit factor and a three-digit factor produce ten or eleven digits. A twelve-digit factor and a thirteen-digit factor produce twenty-four or twenty-five digits.

Calculations such as 11 × 11, 111 × 111 and 1111 × 1111 invite pattern spotting. Use patterns to make predictions, then verify them by calculation or reasoning. Recognising a repeated appearance and explaining why a result must hold are related but different tasks.

How do large quantities become easier to imagine?

Large measurements become more meaningful when compared with a familiar quantity. A metre, abbreviated m, is a unit of length; a kilometre, abbreviated km, equals 1,000 metres. The comparison must use compatible units before asking how many times larger one height or distance is.

Somu is 1 metre tall, and each floor of his building is about four times his height. This gives an approximate height of four metres per floor. The Statue of Unity is about 180 metres high. Kunchikal waterfall is said to drop from about 450 metres.

An illustrated tall building has Somu near its top. Beside it, the statue and waterfall appear against a vertical scale labelled from 0 to 500 metres. The illustration supports comparing heights on a common scale.

Reference: NCERT Class 7, page 3, unnumbered

What do calculations reveal about very large measurements?

The calculation 2,100 × 70,000 gives 14,70,00,000 kilometres, or 147 million kilometres, as an approximate Earth-to-Sun distance. This distance keeps varying throughout the year. The farthest distance is about 152 million kilometres, so the first value should not be treated as an unchanging exact distance.

For water flow, 6,400 × 62,500 = 40,00,00,000. This is the average number of litres of water the Amazon discharges into the Atlantic Ocean every second. A litre is a unit used to measure liquid volume, meaning the amount of space the liquid occupies.

For a journey, 13,95,000 ÷ 150 = 9,300 gives the distance in kilometres of the longest single-train journey in the world, which runs between Moscow and Vladivostok in Russia. Its duration is about seven days. Reading the calculated number in both naming systems links arithmetic to the size of the quantity being described.

How can we solve large-number thought experiments?

A thought experiment explores an imagined situation by making assumptions and calculating their consequences. An assumption is a condition accepted for the calculation. State it clearly, particularly when it concerns capacity, speed or time, because the conclusion depends on it.

Worked example 11. Could 1 lakh buses accommodate Mumbai’s listed population of 1,24,42,373, assuming each bus holds 50 people?

Answer: 1,00,000 × 50 = 50,00,000 places. This is fewer than 1,24,42,373 people. Therefore one lakh buses are insufficient under the stated assumption of fifty people per bus.

Why should a calculation be broken into stages?

Roxie imagines travelling 100 kilometres each day towards the Moon. Use a distance of 3,84,400 kilometres and 365 days per year, ignoring leap years. Calculate one year first, then ten years, and finally compare the resulting distance with the target.

1. Find the distance covered in one year: 100 × 365 = 36,500 kilometres.
2. Find the distance covered in ten years: 36,500 × 10 = 3,65,000 kilometres.
3. Compare this with the required 3,84,400 kilometres: the ten-year distance is smaller.
4. Subtract to find the shortfall: 3,84,400 − 3,65,000 = 19,400 kilometres.

This is a numerical comparison based on the imagined daily travel, rather than a plan for an actual journey. Keeping the time assumption visible prevents an approximate model from being mistaken for a fully exact description of travel.

Counting coins gives another scale comparison. Assuming one coin every second without a break, a day of 24 hours, with 60 minutes per hour and 60 seconds per minute, permits 86,400 coins. That is far below one million, which is 10,00,000.

For each such problem, record the target, the amount per unit of time or per container, and the number of those units. Multiply or divide as needed, then state what the comparison means in the original situation.

How can place value help solve number puzzles?

Number puzzles combine size comparisons with restrictions. Read the restriction before constructing a number: a digit may have to be used exactly once, a first digit may not be zero, or only particular calculator buttons may be available. A large result is valid only if it satisfies the stated conditions.

How do digit order and available buttons matter?

When all digits from 0 to 9 must be used exactly once, the largest multiple of 5 is 9,87,65,43,210. A multiple of five ends in 0 or 5. Placing the remaining larger digits towards the left gives the larger place values priority.

An even number is divisible by two without a remainder, meaning nothing is left over after equal division. Under the same digit restriction, the smallest even number is 1,02,34,56,798. The first digit cannot be zero; the ending must be even.

A calculator restricted to +10,000 and +100 cannot use an unavailable +1,000 button. Starting at zero, it can make 92,100 through nine presses of +10,000 and twenty-one presses of +100. This totals thirty presses and respects the available buttons.

How do toothpick digits create a different challenge?

Digits 0 to 9 are displayed as red straight segments on a black strip. Their shapes resemble a digital display. The digit 7 uses three sticks; changing a number requires changing the displayed segments.

Reference: NCERT Class 7, page 23, unnumbered

The displayed number 42,019 requires twenty-three sticks. Adding two sticks can produce the larger number 42,078. Starting instead with 63,890, rearranging exactly four sticks can produce 88,078. Adding sticks and rearranging sticks impose different restrictions.

For an insertion puzzle, placing another digit 1 among the digits of 42,019 changes the place values. For a stick puzzle, the drawn shapes also matter. In both cases, check the construction against every condition before accepting the resulting number.

Glossary

• Digit — One of the symbols from zero to nine used to write numbers.
• Place value — The value a digit contributes because of its position within a number.
• Lakh — One hundred thousand, written with five zeroes as 1,00,000 in Indian notation.
• Crore — One hundred lakhs, written with seven zeroes as 1,00,00,000 in Indian notation.
• Arab — One hundred crores, equal to one billion in the International naming system.
• Million — One thousand thousands, equal to ten lakhs in the Indian naming system.
• Billion — One thousand million, equal to one arab or one hundred crores.
• Approximation — A nearby, simpler value used when the exact number is unnecessary.
• Rounding up — Replacing a number with an approximate value greater than its actual value.
• Rounding down — Replacing a number with an approximate value less than its actual value.
• Product — The result obtained when two or more numbers are multiplied together.
• Factor — A number multiplied with another number to obtain a particular product.
• Estimate — An approximate result judged from the available quantities and the calculation required.
• Thought experiment — An exploration of an imagined situation using stated assumptions and logical calculations.

Common errors and misconceptions

• Misconception: A lakh and a million are equal. Correct: One million equals ten lakhs. Convert both quantities to the same naming unit before comparing them.
• Misconception: Both systems group all digits in threes. Correct: Indian notation uses three digits at the right, then pairs; International notation continues grouping in threes.
• Misconception: Zeroes inside a number can be omitted. Correct: They preserve place values. In 5,04,085, removing them would change the value represented by the other digits.
• Misconception: An approximate population is an exact count. Correct: About 75,000 communicates size, while 76,068 is the exact Chintamani census figure used for 2011.
• Misconception: Every number should be rounded for convenience. Correct: A telephone number needs exact digits. The purpose determines whether rounding is suitable.
• Misconception: The biggest population means the biggest increase. Correct: Compare the two years by subtraction. Mumbai has the largest listed population, while Bengaluru has the largest increase.
• Misconception: Two two-digit factors always give four digits. Correct: Their product can have three or four digits; the smallest such product is 100.

Exam-style questions with model answers

Q1. Write 5,04,085 in words and state how many lakhs make one million. [2 marks]

1. 5,04,085 is five lakh four thousand eighty-five. Its zeroes preserve the places of the other digits.
2. One million equals ten lakhs, written as 10,00,000 in Indian notation and 1,000,000 in International notation.

Q2. A calculator starts at zero and has +1, +10, +100, +1000, +10000 and +100000 buttons. Show how to make 5,072 with the fewest presses and explain why this is minimal. [3 marks]

1. Use place value to write 5,072 = (5 × 1,000) + (7 × 10) + (2 × 1), with no hundreds contribution.
2. Press +1000 five times, +10 seven times and +1 twice. The total number of presses is 5 + 7 + 2 = 14.
3. Ten smaller-unit presses can be replaced by one next-larger-unit press. The place-value form already uses fewer than ten presses at each place, giving the minimum.

Q3. Find 4,63,128 + 4,19,682 and decide which estimate, 8,00,000 or 9,00,000, is closer. Show the distances from both estimates. [4 marks]

1. The exact sum is 4,63,128 + 4,19,682 = 8,82,810. Keep this exact answer distinct from either rounded estimate.
2. The distance above 8,00,000 is 8,82,810 − 8,00,000 = 82,810, so that estimate is below the sum.
3. The distance below 9,00,000 is 9,00,000 − 8,82,810 = 17,190, so that estimate is above the sum.
4. Since 17,190 is smaller than 82,810, the estimate 9,00,000 is closer to the exact sum.

Q4. Pune’s population was 25,38,473 in 2001 and 31,15,431 in 2011. Find the increase, round it to the nearest lakh and distinguish the exact result from the estimate. [3 marks]

1. Subtract the earlier figure from the later figure: 31,15,431 − 25,38,473 = 5,76,958. This is the exact increase using the supplied populations.
2. To the nearest lakh, 5,76,958 is approximately 6,00,000. The population therefore increased by about six lakh people.
3. The exact difference retains the full numerical detail. The rounded difference communicates the size of the change, so it should be described as approximate.

Q5. Calculate 824 × 25 and 125 × 40 × 8 × 25 using convenient division or regrouping. Explain both methods. [4 marks]

1. Use 25 = 100 ÷ 4, so multiplication by twenty-five can be replaced by division by four followed by multiplication by one hundred.
2. Calculate 824 ÷ 4 = 206, then 206 × 100 = 20,600. Thus the first product is 20,600.
3. In the second expression, pair 125 with 8 and 40 with 25. Each pair has a product of 1,000.
4. Multiply these pair products: 1,000 × 1,000 = 10,00,000. Every original factor has been used exactly once.

Q6. Roxie imagines travelling 100 km each day. The Earth-to-Moon distance to use is 3,84,400 km. Take 365 days per year and ignore leap years. Can she cover this distance in 10 years? Calculate the annual distance, ten-year distance and shortfall. [4 marks]

1. The daily distance is 100 kilometres, and the calculation assumes 365 days in every year. Leap years are ignored, as required by the question.
2. In one year she would cover 100 × 365 = 36,500 kilometres. This first stage converts the daily distance to a yearly distance.
3. In ten years she would cover 36,500 × 10 = 3,65,000 kilometres, using the same daily travel assumption throughout the period.
4. Compare the total with 3,84,400 kilometres. Since 3,65,000 is smaller, she would not cover the required distance within ten years.
5. The shortfall is 3,84,400 − 3,65,000 = 19,400 kilometres. This is the distance still remaining under the stated assumptions.

Q7. Prove that the product of two positive two-digit whole numbers must have either three or four digits. Use the smallest two-digit number and a bound from the smallest three-digit number. [5 marks]

1. The smallest two-digit number is 10. Therefore each of the two positive whole-number factors is at least 10.
2. The smallest possible product is 10 × 10 = 100. Every allowed product is therefore at least 100 and has at least three digits.
3. The smallest three-digit number is 100. Every two-digit factor is less than 100, which supplies an upper comparison for the multiplication.
4. Multiplying the upper comparison values gives 100 × 100 = 10,000. The product of the actual two-digit factors must be less than 10,000.
5. A whole number at least 100 and less than 10,000 has three or four digits. Therefore the conclusion holds for every allowed pair of factors.

Q8. Mumbai’s population in the supplied data is 1,24,42,373. Assume a bus accommodates 50 people. Would 1,00,000 buses accommodate everyone? Calculate their total capacity and compare it with the population. [3 marks]

1. Each bus holds fifty people under the stated assumption. For one lakh buses, the total capacity is 1,00,000 × 50 = 50,00,000 people.
2. The supplied Mumbai population is 1,24,42,373. This is greater than the total capacity of 50,00,000 places in the buses.
3. Therefore one lakh buses would not accommodate everyone. This conclusion depends on the assumed capacity of fifty people in each bus.

Key takeaways

• One lakh is one hundred thousand; one crore is one hundred lakhs; one arab is one hundred crores.
• Indian and International comma patterns differ, but regrouping the same digits does not change the number they represent.
• One million equals ten lakhs, while one billion equals one hundred crores or one arab.
• Place-value calculator buttons connect repeated addition with multiplication and reveal why efficient representations use the digits of the number.
• Keep exact counts separate from approximations, and choose whether to round according to the purpose of the calculation.
• Population size and population increase answer different questions; compare the years carefully before calculating a change.
• Regrouping convenient factors can simplify multiplication, while the smallest and largest possible factors help explain product digit counts.
• Break large-number thought experiments into stages, state assumptions clearly and compare the calculated result with the required target.

Test yourself

What comes immediately after 99,999?

1,00,000 comes next. It is one lakh and the smallest six-digit number.

How is 12,78,830 read in the Indian system?

It is twelve lakh seventy-eight thousand eight hundred thirty, read group by group.

How many presses of +100 make 1,00,000, starting at zero?

It takes 1,000 presses because 1,00,000 divided by 100 equals 1,000.

Why is rounding a telephone number unsuitable?

Its exact digits identify the intended connection; rounding changes the number being called.

What is 6,72,85,183 rounded to the nearest lakh?

The nearest lakh is 6,73,00,000, the closer of the neighbouring whole-lakh values.

How can 72 × 125 be calculated quickly?

Use 125 = 1,000 ÷ 8. Divide 72 by 8, then multiply 9 by 1,000 to get 9,000.

Can two three-digit numbers have a four-digit product?

No. Their smallest possible product is 100 × 100 = 10,000, which already has five digits.

At one coin per second without breaks, can you count a million coins in a day? Use 24 hours, 60 minutes per hour and 60 seconds per minute.

No. The total is 24 × 60 × 60 = 86,400 coins, fewer than the 10,00,000 coins required.