Class 7 Mathematics Chapter 1: Large Numbers Around Us

A number that is easy to say and hard to picture

You can say "one lakh" in a moment. Picturing it is much harder.

Write it down and it looks calm enough: 1,00,000. That is a 1 with five zeros after it. But the gap between saying a number and feeling its size is where most mistakes with large numbers begin.

Climbing to a lakh one digit at a time

Start with the largest 3-digit number, 999. Add 1 and you get 1000, the smallest 4-digit number.

Do it again. The largest 4-digit number is 9999, and adding 1 gives 10,000. The largest 5-digit number is 99,999, and adding 1 gives 1,00,000.

So a lakh is simply the smallest 6-digit number. Each new digit arrives the same way: the old digits run out of room.

Where the commas go

In the Indian system we write a lakh as 1,00,000, not 100,000.

The commas are not decoration. Reading from the right, the first comma marks off three digits, and after that they come every two. That grouping is what lets you read 1,00,000 straight off as "one lakh".

Four ways to count to a lakh

Imagine four keys that each add a fixed amount.

A +1 key needs 1,00,000 presses. A +10 key needs 10,000. A +100 key needs 1,000. A +1000 key needs only 100.

Each time the key grows ten times bigger, the number of presses shrinks ten times. The total never changes.

The same amount, counted differently

Those four answers are four names for one quantity.

One lakh is 1,00,000 ones. It is also 10,000 tens, or 1,000 hundreds, or 100 thousands. Nothing has been added or removed. Only the size of the bundle changed.

Reading a lakh backwards

The trick works in reverse too, and that is often the useful direction.

Suppose you press a +100 key 1,000 times. You do not need to add anything up. Each press is one hundred, so 1,000 presses give 1,000 hundreds, which is one lakh.

A seed library

A seed library keeps one small packet of each variety it collects.

Say it has 2,000 packets on its shelves. To reach a lakh of packets it would need 50 times as many, because 2,000 multiplied by 50 is 1,00,000. Fifty whole libraries of that size.

One lakh people

A large school hall seats 500 people.

To seat a lakh of people you would need 200 such halls, since 500 multiplied by 200 is 1,00,000. Two hundred halls, all full at once.

One lakh seconds

Here is the fact that surprises most people.

One lakh seconds is not a few minutes. An hour holds 3,600 seconds, and 1,00,000 divided by 3,600 is about 27.8. So one lakh seconds runs for nearly 28 hours. That is longer than a whole day.

One lakh days

Now try days instead of seconds.

A year holds about 365 days, and 1,00,000 divided by 365 is about 274. One lakh days is roughly 274 years. Nobody has ever lived that long.

Could you taste a lakh of them?

Suppose a cook tries one new dish every day for a hundred years.

A hundred years is about 36,500 days, because 365 multiplied by 100 is 36,500. That is nowhere near a lakh. At two dishes a day it is 73,000, still short. Only at three a day does the total pass a lakh, at 1,09,500.

Why estimate before you calculate

Notice what the dish question did not need.

It did not need an exact answer. It needed to know whether the total lands above or below a lakh. A rough comparison settled it, and rough comparisons are usually faster and safer than exact arithmetic.

Zeros are worth counting carefully

Most errors with large numbers are zero errors.

Ten thousand is 10,000, with four zeros. One lakh is 1,00,000, with five. They look similar on a page and differ by a factor of ten. Counting the zeros is worth the two seconds it takes.

What a lakh is not

A lakh is not "a very big number". It is an exact amount.

It is ten times ten thousand, and one hundredth of one crore. Treating it as vague is how a ten-times error slips through unnoticed.

Carrying this forward

Everything here repeats one step higher up.

Ten lakhs make 10,00,000, and a hundred lakhs make one crore. The pattern of digits running out of room never changes. Only the names do.

Scenario:

A seed library in a small town keeps one packet of every rice variety it can find. Volunteers bring seeds back from villages and label each packet by hand. The shelves currently hold two thousand packets. A visiting teacher asks the volunteers whether the library could ever hold one lakh packets. Nobody has thought about it in those terms before. They know the collection grows by about fifty packets a month, because that is roughly what the volunteers can label and store. The teacher suggests they work out the answer rather than guess at it.

Analysis:

The question turns on two numbers: how far the library has to go, and how fast it moves. The gap is 98,000 packets. The rate is 600 a year, because fifty a month is 600 a year. Dividing the gap by the rate gives about 163 years. The arithmetic is simple; the surprise is how large a lakh turns out to be against a rate that feels steady.

Activity:

Count how many separate items are stored in one cupboard at home. Then work out how many such cupboards would hold one lakh items.

Title:

Guess, then divide

Safety:

No safety concerns. This activity needs only paper and a timer.

What You Notice:

Most guesses for one lakh seconds come out far too short. Dividing a large number by a familiar unit turns a number you cannot picture into one you can.

One lakh seconds lasts nearly 28 hours, so it is longer than a whole day.

One lakh days is about 274 years, far longer than anyone has lived.

Reading one book every week, a lakh of books would take about 1,923 years.

Five zeros for a lakh, seven for a crore.

Three from the right, then two by two.

Two countries, two sets of commas

A long number can be written two ways, and both are correct.

India groups the digits one way. Most of the English-speaking world groups them another. The digits never change. Only the commas move, and with them the names you read out.

The Indian grouping

Start from the right and mark off three digits. After that, group in twos.

So a lakh is 1,00,000 and a crore is 1,00,00,000. The pattern of commas is three, then two, then two, for as long as the number runs.

The International grouping

Start from the right and mark off three digits. Then keep going in threes.

So the same two numbers become 100,000 and 10,000,000. The grouping is even all the way along, which is why the names change every three digits instead of every two.

The names that go with each

Each grouping carries its own ladder of names.

The Indian ladder runs thousand, lakh, crore, arab. The International ladder runs thousand, million, billion. Both describe the same quantities.

The one bridge worth memorising

If you remember a single conversion, remember this one.

One crore is ten million. In figures, 1,00,00,000 is the same as 10,000,000. Count the zeros on each side and you find seven both times.

And the bridge the other way

The reverse is just as useful.

One million is ten lakh. In figures, 1,000,000 is the same as 10,00,000. Again there are six zeros on each side.

Climbing the Indian ladder

Each step up is a jump of either ten or a hundred.

A thousand is 1,000. Ten thousand is 10,000. A lakh is 1,00,000. Ten lakh is 10,00,000. A crore is 1,00,00,000. Each name here is ten times the one before.

Climbing the International ladder

The International ladder climbs in bigger strides.

A thousand is 1,000. A million is 1,000,000, a thousand times larger. A billion is 1,000,000,000, a thousand times larger again. Three zeros at every step.

Reading a number you have never seen

Suppose you meet 2,15,43,000.

The commas are Indian, so read from the left. The leading 2 sits in the crore group, the 15 in the lakh group, the 43 in the thousand group. It reads as two crore fifteen lakh forty-three thousand.

The same number, regrouped

Now rewrite 2,15,43,000 with International commas.

The digits are 21543000, which is eight digits. Grouping in threes from the right gives 21,543,000. That reads as twenty-one million five hundred forty-three thousand.

A billion is a hundred crore

This is the step most people have to stop and think about.

One billion is 1,000,000,000, which has nine zeros. In the Indian system that same number is 1,00,00,00,000. It is one hundred crore, and it also has a name of its own, one arab.

Which grouping should you use?

Use whichever the question uses, and say which one you used.

An answer is not wrong for being in millions, but it is unclear if the reader cannot tell which system you meant. Writing the commas properly is how you make that clear without a sentence of explanation.

A crore of seconds

Dividing by something familiar still works at this scale.

An hour holds 3,600 seconds, and a day holds 86,400. Divide 1,00,00,000 by 86,400 and you get about 116. So a crore of seconds runs for about 116 days, which is near enough four months.

Sharing out a crore

Suppose a crore of rupees is shared equally among a thousand people.

Each person receives 1,00,00,000 divided by 1,000, which is 10,000. So a crore split a thousand ways gives ten thousand rupees each. Large totals shrink fast when they are divided.

Zeros are the quickest check

Commas can be miscounted. Zeros are harder to argue with.

A lakh carries five zeros, a crore seven, and a billion nine. If the name you have written does not match the zero count, one of the two is wrong. The check takes a second and needs no conversion at all.

Why both systems survive

It would be tidier if the world agreed on one grouping. It has not.

Indian newspapers, budgets and census tables use lakhs and crores. International reports and most scientific writing use millions and billions. Anyone reading both needs to move between them, which is why the two bridges are worth knowing by heart.

Where the mistakes come from

Almost every error here is a grouping error, not an arithmetic one.

Writing 1,000,000 and calling it a crore is the common one. It is a million, which is only ten lakh. Checking the count of zeros against the name you have written catches it immediately.

Scenario:

A school magazine prints two short reports on the same page. The first says that a nearby state has a population of 6,80,00,000 people. The second, copied from an international news site, says a neighbouring country has 90 million. A reader writes in to complain that the magazine has mixed up its units and that the two figures cannot be compared. The editor is not sure whether the reader is right. Both numbers were copied correctly from their sources, and neither report is wrong on its own terms.

Analysis:

The reader is half right. Both figures are correct, but they are written in different systems, so no comparison is possible until one is converted. Converting is straightforward: one million is ten lakh, so 90 million is 900 lakh, which is 9 crore. The state figure is 6 crore 80 lakh. Once both sit in crores the comparison is easy, and the state turns out to have fewer people than the neighbouring country.

Activity:

Find two news reports that give large figures in different systems. Convert both to crores and write the comparison in one sentence.

Title:

Regroup a long number both ways

Safety:

No safety concerns. Paper and pencil only.

What You Notice:

The digits never move, yet the name changes completely. This is why writing the commas properly matters more than it looks.

A crore of seconds runs for about 116 days, which is close to four months.

One hundred crore has its own Indian name, one arab, and equals a billion.

A crore of rupees shared among a thousand people gives ten thousand each.

Crore has seven zeros, million has six.

India groups three then twos, the world groups threes.

A headline with a round number

A newspaper reports that one lakh people visited the book fair. Nobody counted them one by one.

The true figure might have been 99,412, or 1,02,878. The headline is not lying. It is approximating.

Exact or approximate

Some numbers must be exact. The marks on your report card, the price on a bill, the number of seats in a bus.

Others need only be close. A town's population, the crowd at a match, the distance to a city. Knowing which kind you need is part of the question.

When close is enough

The census gives the population of a town as 76,068. Saying about 75,000 tells you nearly as much.

It also tells you something the exact figure hides, which is the size of the thing. Seventy-five thousand is easier to hold in your head.

Rounding up

Sometimes you must not fall short. A caterer expecting 847 guests orders 900 meals, not 800.

Running out of food is a real failure. Having a few meals spare is not. So the number is rounded up.

Rounding down

Sometimes you must not overstate. A lift that can carry 680 kilograms is marked as 650 kilograms.

Claiming 700 would be dangerous. Here the safe direction is downwards, so the number is rounded down.

Which way to round

Notice that no rule about digits decided either case. What decided it was what the number is FOR.

Ask which mistake would cost more, falling short or going over. Then round away from that mistake.

The nearest neighbours

With a large number it helps to know the nearest thousand, lakh and crore. These are its nearest neighbours.

Take 4,78,36,215. Its neighbours are worth working out one at a time.

Working them out

The nearest thousand is 4,78,36,000, since the 215 left over is below 500. The nearest ten thousand is 4,78,40,000, because 6,215 is more than half of ten thousand.

The nearest lakh is 4,78,00,000. The nearest ten lakh is 4,80,00,000. And the nearest crore is 5,00,00,000, since 78 lakh is past the halfway mark.

The neighbours get further apart

Read that list again and something stands out. The nearest thousand is only 215 away, and the nearest crore is more than 21 lakh away.

The bigger the unit you round to, the more you throw away. That is the price of a simpler number.

Estimating a sum

Estimates are rounding put to work. Take 3,62,475 + 2,91,318.

Round each to the nearest lakh and you get 4,00,000 and 3,00,000, suggesting about 7,00,000. Round to the nearest ten thousand instead and you get 3,60,000 and 2,90,000, suggesting 6,50,000.

Which estimate to trust

The second is far better, and the exact answer is 6,53,793. Rounding to a smaller unit threw less away.

That is the rule in general. A coarse estimate is quicker, and a finer one is closer, and you choose by how much accuracy the job needs.

Estimating a difference

The same works for subtraction. For 8,25,640 - 3,97,210, round to the nearest ten thousand to get 8,30,000 less 4,00,000, or about 4,30,000.

The exact answer is 4,28,430, so the estimate was within 1,570. Doing the estimate first means a wildly wrong answer cannot slip past you.

A shortcut for multiplying by five

Now a different use for round numbers. To work out 248 times 5, halve the 248 and multiply by 10.

That gives 124 times 10, which is 1,240. Halving and then multiplying by ten is easier than multiplying by five.

Why that shortcut works

It works because 5 is half of 10. Multiplying by 10 and then halving gives the same answer as multiplying by five.

Doing the halving first just keeps the numbers small. Nothing has been approximated here, so the answer is exact.

The same trick with 25, 50 and 125

Twenty-five is a quarter of 100, so 376 times 25 is 94 times 100, which is 9,400.

Fifty is half of 100, so 164 times 50 is 82 times 100, or 8,200. And 125 is an eighth of 1000, so 72 times 125 is 9 times 1000, which is 9,000.

How long is a product?

Here is a harder question. Multiply any two 2-digit numbers. How many digits can the answer have?

You could try them all, and there are more than two thousand pairs. There is a much better way.

Trapping it between two products

The smallest 2-digit number is 10, so the smallest such product is 10 times 10, which is 100. Every product is therefore at least 100, so at least three digits.

The largest is 99 times 99. Rather than work it out, notice it must be below 100 times 100, which is 10,000. So every product has fewer than five digits. Three digits or four, and nothing else is possible.

What carries forward

Some numbers must be exact and others need only be close, and the question tells you which. Round up when falling short would cost more, and round down when overstating would, because the use decides the direction and not the digits.

The nearest neighbours of a large number are its nearest thousand, lakh and crore, and rounding to a smaller unit throws less away. Multiplying by 5, 25, 50 or 125 is easier through 10, 100 or 1000. And the length of a product can be settled by trapping it between two easy multiplications instead of trying every case.

Scenario:

A school is planning its annual day. The hall has 680 seats and the head teacher expects 847 guests. The caterer asks how many meals to prepare. The treasurer adds two grants of 3,62,475 rupees and 2,91,318 rupees and wants a figure to announce from the stage before the exact total is audited. A teacher ordering 248 badges at 5 rupees each starts reaching for a calculator. A pupil asks whether the product of two 2-digit numbers could ever have five digits.

Analysis:

Four numbers, and each one needs a different kind of answer.

Activity:

Collect five numbers from a newspaper and sort them into those printed exactly and those printed approximately.

Title:

Round it, and say why

Safety:

No hazards here.

What You Notice:

Figures given to the last digit are usually exact; round figures are usually approximations. The same number rounds in opposite directions depending on the use you invented, which is the point of the topic.

Two numbers as close as 847 and 680 can round in opposite directions, because the use and not the digits decides.

Rounding 4,78,36,215 to the nearest thousand moves it 215, and to the nearest crore moves it more than 21 lakh.

The length of a product is settled by bounding, which is quicker and surer than trying all 8,100 pairs of 2-digit numbers.

Round away from the costlier mistake.

Five is half of ten, so halve and times ten.