Hari's Random Thoughts by Hariharan Ramamurthy: Mathematics basics from NCERT

Sunday, January 29, 2017

Mathematics basics from NCERT

Mathematics

1.1 introduction

Counting things is easy for us now.
We can count objects in large numbers,for example, the number of students in the school, and represent them through numerals.
We can also communicate large numbers using suitable number names.

It is not as if we always knew how to convey large quantities in conversation or through symbols. Many thousands years ago, people knew only small numbers. Gradually, they learnt how to handle larger numbers. They also learnt how to express large numbers in symbols.
All this came through collective efforts of human beings. Their path was not easy, they struggled all along the

way. In fact, the development of whole of Mathematics can be understood

this way. As human beings progressed, there was greater need for development

of Mathematics and as a result Mathematics grew further and faster.

We use numbers and know many things about them. Numbers help us

count concrete objects. They help us to say which collection of objects

is bigger and arrange them in order e.g., first, second, etc. Numbers are

used in many different contexts and in many ways. Think about various

situations where we use numbers. List five distinct situations in which

numbers are used.

We enjoyed working with numbers in our previous classes. We have added,

subtracted, multiplied and divided them. We also looked for patterns in number

sequences and done many other interesting things with numbers. In this chapter,

we shall move forward on such interesting things with a bit of review and

revision as well.

Chapter 1

Knowing our Numbers

1.2 Comparing Numbers

As we have done quite a lot of this earlier, let us see if we remember which is

the greatest among these :

(i) 92, 392, 4456, 89742

(ii) 1902, 1920, 9201, 9021, 9210

So, we know the answers.

Discuss with your friends, how you find the number that is the greatest.

Can you instantly find the greatest and the smallest numbers in each row?

1. 382, 4972, 18, 59785, 750.

Ans.

59785 is the greatest and

18 is the smallest.

2. 1473, 89423, 100, 5000, 310.

Ans.

____________________

3. 1834, 75284, 111, 2333, 450 .

Ans.

____________________

4. 2853, 7691, 9999, 12002, 124.

Ans.

____________________

Was that easy? Why was it easy?

We just looked at the number of digits and found the answer.

The greatest number has the most thousands and the smallest is

only in hundreds or in tens.

Make five more problems of this kind and give to your friends

to solve.

Now, how do we compare 4875 and 3542?

This is also not very difficult.These two numbers have the

same number of digits. They are both in thousands. But the digit

at the thousands place in 4875 is greater than that in 3542.

Therefore, 4875 is greater than 3542.

Next tell which is greater, 4875 or

4542? Here too the numbers have the

same number of digits. Further, the digits

at the thousands place are same in both.

What do we do then? We move to the

next digit, that is to the digit at the

hundreds place. The digit at the hundreds

place is greater in 4875 than in 4542.

Therefore, 4875 is greater than 4542.

Find the greatest and the smallest

numbers.

(a) 4536, 4892, 4370, 4452.

(b) 15623, 15073, 15189, 15800.

(d) 6895, 23787, 24569, 24659.

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867

027

If the digits at hundreds place are also same in the two numbers, then what

do we do?

Compare 4875 and 4889 ; Also compare 4875 and 4879.

1.2.1 How many numbers can you make?

Suppose, we have four digits 7, 8, 3, 5. Using these digits we want to make

different 4-digit numbers in such a way that no digit is repeated in them. Thus,

7835 is allowed, but 7735 is not. Make as many 4-digit numbers as you can.

Which is the greatest number you can get? Which is the smallest number?

The greatest number is 8753 and the smallest is 3578.

Think about the arrangement of the digits in both. Can you say how the largest

number is formed? Write down your procedure.

1. Use the given digits without repetition and make the greatest and smallest 4-digit

numbers.

(a) 2, 8, 7, 4 (b) 9, 7, 4, 1 (c) 4, 7, 5, 0

(d) 1, 7, 6, 2 (e) 5, 4, 0, 3

(

Hint :

0754 is a 3-digit number.)

2. Now make the greatest and the smallest 4-digit numbers by using any one

digit twice.

(a) 3, 8, 7 (b) 9, 0, 5 (c) 0, 4, 9 (d) 8, 5, 1

(

Hin

t :

Think in each case which digit will you use twice.)

3. Make the greatest and the smallest 4-digit numbers using any four different

digits with conditions as given.

(a) Digit 7 is always at Greatest

ones place

Smallest

(Note, the number cannot begin with the digit 0. Why?)

(b) Digit 4 is always Greatest

at tens place

Smallest

hundreds place

Smallest

(d) Digit 1 is always at Greatest

thousands place

Smallest

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How much was the increase in population

during 1991-2001? Try to find out.

Do you know what is India’s population

today? Try to find this too.

1. Read these numbers. Write them using placement boxes and then write their

expanded forms.

(i) 475320

(ii)

9847215

(iii)

97645310

(iv)

30458094

(a) Which is the smallest number?

(b) Which is the greatest number?

2. Read these numbers.

(i) 527864

(ii)

95432

(iii)

18950049

(iv)

70002509

(a) Write these numbers using placement boxes and then using commas in Indian

as well as International System of Numeration..

(b) Arrange these in ascending and descending order.

3. Take three more groups of large numbers and do the exercise given above.

Do you know?

India’s population increased by

about

27 million during 1921-1931;

37 million during 1931-1941;

44 million during 1941-1951;

78 million during 1951-1961

Can you help me write the numeral?

To write the numeral for a number you can follow the boxes again.

(a) Forty two lakh seventy thousand eight.

(b) Two crore ninety lakh fifty five thousand eight hundred.

1. You have the following digits 4, 5, 6, 0, 7 and 8. Using them, make five numbers

each with 6 digits.

(a) Put commas for easy reading.

(b) Arrange them in ascending and descending order.

2. Take the digits 4, 5, 6, 7, 8 and 9. Make any three numbers each with 8 digits.

Put commas for easy reading.

3. From the digits 3, 0 and 4, make five numbers each with 6 digits. Use commas.

ATHEMATICS

1. How many

centimetres make a

kilometre?

2. Name five large cities

in India. Find their

population. Also, find

the distance in

kilometres between

each pair of these cities.

1.3 Large Numbers in Practice

In earlier classes, we have learnt that we use centimetre (cm) as a unit of length.

For measuring the length of a pencil, the width of a book or

notebooks etc., we use centimetres. Our ruler has marks on each centimetre.

For measuring the thickness of a pencil, however, we find centimetre too big.

We use millimetre (mm) to show the thickness of a pencil.

(a) 10 millimetres = 1 centimetre

To measure the length of the classroom or

the school building, we shall find

centimetre too small. We use metre for the

purpose.

(b) 1 metre = 100 centimetres

= 1000 millimetres

Even metre is too small, when we have to

state distances between cities, say, Delhi

and Mumbai, or Chennai and Kolkata. For

this we need kilometres (km).

EXERCISE 1.1

1. Fill in the blanks:

(a) 1 lakh = _______ ten thousand.

(b) 1 million = _______ hundred thousand.

(d) 1 crore = _______ million.

(e) 1 million = _______ lakh.

2. Place commas correctly and write the numerals:

(a) Seventy three lakh seventy five thousand three hundred seven.

(b) Nine crore five lakh forty one.

(d) Fifty eight million four hundred twenty three thousand two hundred two.

(e) Twenty three lakh thirty thousand ten.

3. Insert commas suitably and write the names according to Indian System of

Numeration :

(a) 87595762 (b) 8546283 (c) 99900046 (d) 98432701

4. Insert commas suitably and write the names according to International System

of Numeration :

(a) 78921092 (b) 7452283 (c) 99985102 (d) 48049831

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How many millimetres make 1 kilometre?

Since 1 m = 1000 mm

1 km = 1000 m = 1000 × 1000 mm = 10,00,000 mm

We go to the market to buy rice or wheat; we buy it in

kilograms (kg). But items like ginger or chillies which

we do not need in large quantities, we buy in grams (g).

We know 1 kilogram = 1000 grams.

Have you noticed the weight of the medicine tablets

given to the sick? It is very small. It is in milligrams

(mg).

1 gram = 1000 milligrams.

What is the capacity of a bucket for holding water? It

is usually 20 litres (

). Capacity is given in litres. But

sometimes we need a smaller unit, the millilitres.

A bottle of hair oil, a cleaning liquid or a soft drink

have labels which give the quantity of liquid inside in

millilitres (ml).

1 litre = 1000 millilitres.

Note that in all these units we have some words

common like kilo, milli and centi. You should remember

that among these

kilo

is the greatest and

milli

is the

smallest; kilo shows 1000 times greater, milli shows

1000 times smaller, i.e. 1 kilogram = 1000 grams,

1 gram = 1000 milligrams.

Similarly, centi shows 100 times smaller, i.e. 1 metre

100 centimetres.

1. A bus started its journey and reached different places with a speed of

60 km/hour. The journey is shown below.

(i) Find the total distance covered by the bus from A to D.

(ii) Find the total distance covered by the bus from D to G.

(iii) Find the total distance covered by the bus, if it starts

from A and returns back to A.

(iv) Can you find the difference of distances from C to D and D to E?

1. How many

milligrams

make one

kilogram?

2. A box contains

2,00,000

medicine tablets

each weighing

20 mg. What is

the total weight

of all the

tablets in the

box in grams

and in

kilograms?

ATHEMATICS

The sales during the last year

Apples 2457 kg

Oranges 3004 kg

Combs 22760

Tooth brushes 25367

Pencils 38530

Note books 40002

Soap cakes 20005

(v) Find out the time taken by the bus to

reach

(a) A to B (b) C to D

Raman’s shop

Things Price

Apples

40 per kg

Oranges

30 per kg

Combs

3 for one

Tooth brushes

10 for one

Pencils

1 for one

Note books

6 for one

Soap cakes

8 for one

(a) Can you find the total weight of apples and oranges Raman sold last year?

Weight of apples = __________ kg

Weight of oranges = _________ kg

Therefore, total weight = _____ kg + _____ kg = _____ kg

Answer – The total weight of oranges and apples = _________ kg.

(b) Can you find the total money Raman got by selling apples?

together?

(d) Make a table showing how much money Raman received from selling

each item. Arrange the entries of amount of money received in

descending order. Find the item which brought him the highest amount.

How much is this amount?

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We have done a lot of problems that have addition, subtraction, multiplication

and division. We will try solving some more here. Before starting, look at these

examples and follow the methods used.

Example 1 :

Population of Sundarnagar was 2,35,471 in the year 1991. In the

year 2001 it was found to be increased by 72,958. What was the population of

the city in 2001?

Solution :

Population of the city in 2001

= Population of the city in 1991 + Increase in population

= 2,35,471 + 72,958

Now, 235471

+ 72958

308429

Salma added them by writing 235471 as 200000 + 35000 + 471 and

72958 as 72000 + 958. She got the addition as 200000 + 107000 + 1429 = 308429.

Mary added it as 200000 + 35000 + 400 + 71 + 72000 + 900 + 58 = 308429

Answer : Population of the city in 2001 was 3,08,429.

All three methods are correct.

Example 2 :

In one state, the number of bicycles sold in the year 2002-2003

was 7,43,000. In the year 2003-2004, the number of bicycles sold was 8,00,100.

In which year were more bicycles sold? and how many more?

Solution :

Clearly, 8,00,100 is more than 7,43,000. So, in that state, more

bicycles were sold in the year 2003-2004 than in 2002-2003.

Now, 800100

– 743000

057100

Can you think of alternative ways of solving this problem?

Answer : 57,100 more bicycles were sold in the year 2003-2004.

Example 3 :

The town newspaper is published every day. One copy has

12 pages. Everyday 11,980 copies are printed. How many total pages are

printed everyday?

Check the answer by adding

743000

+ 57100

800100 (the answer is right)

ATHEMATICS

Solution :

Each copy has 12 pages. Hence, 11,980 copies will have

12 × 11,980 pages. What would this number be? More than 1,00,000 or lesser.

Try to estimate.

Now, 11980

× 12

23960

+ 119800

143760

Answer:Everyday 1,43,760 pages are printed.

Example 4 :

The number of sheets of paper available for making notebooks

is 75,000. Each sheet makes 8 pages of a notebook. Each notebook contains

200 pages. How many notebooks can be made from the paper available?

Solution :

Each sheet makes 8 pages.

Hence, 75,000 sheets make 8 × 75,000 pages,

Now, 75000

× 8

600000

Thus, 6,00,000 pages are available for making notebooks.

Now, 200 pages make 1 notebook.

Hence, 6,00,000 pages make 6,00,000 ÷ 200 notebooks.

3000

Now, 200 600000

– 600

0000 The answer is 3,000 notebooks.

EXERCISE 1.2

1. A book exhibition was held for four days in a school. The number of tickets sold

at the counter on the first, second, third and final day was respectively 1094,

1812, 2050 and 2751. Find the total number of tickets sold on all the four days.

2. Shekhar is a famous cricket player. He has so far scored 6980 runs in test matches.

He wishes to complete 10,000 runs. How many more runs does he need?

3. In an election, the successful candidate registered 5,77,500 votes and his nearest

rival secured 3,48,700 votes. By what margin did the successful candidate win

the election?

4. Kirti bookstore sold books worth Rs 2,85,891 in the first week of June and books

worth Rs 4,00,768 in the second week of the month. How much was the sale for

)

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the two weeks together? In which week was the sale greater and by how much?

5. Find the difference between the greatest and the least number that can be written using

the digits 6, 2, 7, 4, 3 each only once.

6. A machine, on an average, manufactures 2,825 screws a day. How many screws did

it produce in the month of January 2006?

7. A merchant had Rs 78,592 with her. She placed an order for purchasing 40 radio sets

at Rs 1200 each. How much money will remain with her after the purchase?

8. A student multiplied 7236 by 65 instead of multiplying by 56. By how much was his

answer greater than the correct answer? (Hint: Do you need to do both the

multiplications?)

9. To stitch a shirt, 2 m 15 cm cloth is needed. Out of 40 m cloth, how many shirts can be

stitched and how much cloth will remain?

(

Hint:

convert data in cm.)

10. Medicine is packed in boxes, each weighing 4 kg 500g. How many such boxes can be

loaded in a van which cannot carry beyond 800 kg?

11. The distance between the school and the house of a student’s house is 1 km 875 m.

Everyday she walks both ways. Find the total distance covered by her in six days.

12. A vessel has 4 litres and 500 ml of curd. In how many glasses, each of 25 ml

capacity, can it be filled?

1.3.1 Estimation

News

1. India drew with Pakistan in a hockey match watched by 51,000 spectators

in the stadium and 40 million television viewers world wide.

2. Approximately, 2000 people were killed and more than 50000 injured in a

cyclonic storm in coastal areas of India and Bangladesh.

3. Over 13 million passengers are carried over 63,000 kilometre route of

railway track every day.

Can we say that there were exactly as many people as the numbers quoted

in these news items? For example,

In (1), were there exactly 51,000 spectators in the stadium? or did exactly

40 million viewers watched the match on television?

Obviously, not. The word

approximately

itself

shows that the number of people were near about these

numbers. Clearly, 51,000 could be 50,800 or 51,300

but not 70,000. Similarly, 40 million implies much

more than 39 million but quite less than 41 million

but certainly not 50 million.

ATHEMATICS

The quantities given in the examples above are not exact counts, but are

estimates to give an idea of the quantity.

Discuss what each of these can suggest.

Where do we approximate?

Imagine a big celebration at your home. The first

thing you do is to find out roughly how many guests may visit you. Can you get an

idea of the exact number of visitors? It is practically impossible.

The finance minister of the country presents a budget annually. The minister provides

for certain amount under the head ‘Education’. Can the amount be absolutely accurate?

It can only be a reasonably good estimate of the expenditure the country needs for

education during the year.

Think about the situations where we need to have the exact numbers and compare

them with situations where you can do with only an approximately estimated number.

Give three examples of each of such situations.

1.3.2 Estimating to the nearest tens by rounding off

Look at the following :

(a) Find which flags are closer to 260.

(b) Find the flags which are closer to 270.

Locate the numbers 10,17 and 20 on your ruler. Is 17 nearer to 10 or 20? The

gap between 17 and 20 is smaller when compared to the gap between 17 and 10.

So, we round off 17 as 20, correct to the nearest tens.

Now consider 12, which also lies between 10 and 20. However, 12 is

closer to 10 than to 20. So, we round off 12 to 10, correct to the nearest tens.

How would you

round off 76 to the nearest tens? Is it not 80?

We see that the numbers 1,2,3 and 4 are nearer to 0 than to 10. So, we

round off 1, 2, 3 and 4 as 0. Number 6, 7, 8, 9 are nearer to 10, so, we round

them off as 10. Number 5 is equidistant from both 0 and 10; it is a common

practice to round it off as 10.

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1.3.3 Estimating to the nearest hundreds by rounding off

Is 410 nearer to 400 or to 500?

410 is closer to 400, so it is rounded off to 400, correct to the nearest

hundred.

889 lies between 800 and 900.

It is nearer to 900, so it is rounded off as 900 correct to nearest hundred.

Numbers 1 to 49 are closer to 0 than to 100, and so are rounded off to 0.

Numbers 51 to 99 are closer to 100 than to 0, and so are rounded off to 100.

Number 50 is equidistant from 0 and 100 both. It is a common practice to round it off

as 100.

Check if the following rounding off is correct or not :

841

800; 9537

9500; 49730

49700;

2546

2500; 286

200; 5750

5800;

168

200; 149

100; 9870

9800.

Correct those which are wrong.

1.3.4 Estimating to the nearest thousands by rounding off

We know that numbers 1 to 499 are nearer to 0 than to 1000, so these numbers are

rounded off as 0.

The numbers 501 to 999 are nearer to 1000 than 0 so they are rounded off as

1000.

Number 500 is also rounded off as 1000.

Check if the following rounding off is correct or not :

2573

3000; 53552

53000;

6404

6000; 65437

65000;

7805

7000; 3499

4000.

Correct those which are wrong.

Round these numbers to the nearest tens.

28 32 52 41 39 48

64 59 99 215 1453 2936

ATHEMATICS

Round off the given numbers to the nearest tens, hundreds and thousands.

Given Number Approximate to

Nearest Rounded Form

75847 Tens ________________

75847

Hundreds

________________

75847

Thousands

________________

75847 Ten

thousands

________________

1.3.5 Estimating outcomes of number situations

How do we add numbers? We add numbers by following the algorithm (i.e. the

given method) systematically. We write the numbers taking care that the digits in the

same place (ones, tens, hundreds etc.) are in the same column. For example,

3946 + 6579 + 2050 is written as —

Th H T O

3946

6579

+ 2050

We add the column of ones and if necessary carry forward the appropriate

number to the tens place as would be in this case. We then add the tens

column and this goes on. Complete the rest of the sum yourself. This

procedure takes time.

There are many situations where we need to find answers more quickly.

For example, when you go to a fair or the market, you find a variety of attractive

things which you want to buy. You need to quickly decide what you can buy.

So, you need to estimate the amount you need. It is the sum of the prices of

things you want to buy.

A trader is to receive money from two sources. The money he is to receive

is Rs 13,569 from one source and Rs 26,785 from another. He has to pay

Rs 37,000 to someone else by the evening. He rounds off the numbers to their

nearest thousands and quickly works out the rough answer. He is happy that

he has enough money.

Do you think he would have enough money? Can you tell without doing

the exact addition/subtraction?

Sheila and Mohan have to plan their monthly expenditure. They know

their monthly expenses on transport, on school requirements, on groceries,

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on milk, and on clothes and also on other regular

expenses. This month they have to go for visiting

and buying gifts. They estimate the amount they

would spend on all this and then add to see, if what

they have, would be enough.

Would they round off to thousands as the

trader did?

Think and discuss five more situations where we have to estimate sums or

remainders.

Did we use rounding off to the same place in all these?

There are no rigid rules when you want to estimate the outcomes of numbers.

The procedure depends on the degree of accuracy required and how quickly

the estimate is needed. The most important thing is, how sensible the guessed

answer would be.

1.3.6 To estimate sum or difference

As we have seen above we can round off a number to any place. The trader

rounded off the amounts to the nearest thousands and was satisfied that he had

enough. So, when you estimate any sum or difference, you should have an idea

of why you need to round off and therefore the place to which you would round

off. Look at the following examples.

Example 5 :

Estimate: 5,290 + 17,986.

Solution :

You find 17,986 > 5,290.

Round off to thousands.

17,986 is rounds off to

18,000

+5,290 is rounds off to + 5,000

Estimated sum =

23,000

Does the method work? You may attempt to find the actual answer and

verify if the estimate is reasonable.

Example 6 :

Estimate: 5,673 – 436.

Solution :

To begin with we round off to thousands. (Why?)

5,673 rounds off to

6,000

– 436 rounds off to – 0

Estimated difference =

6,000

This is not a reasonable estimate. Why is this not reasonable?

ATHEMATICS

To get a closer estimate, let us try rounding each number to hundreds.

5,673 rounds off to

5,700

– 436 rounds off to – 400

Estimated difference =

5,300

This is a better and more meaningful estimate.

1.3.7 To estimate products

How do we estimate a product?

What is the estimate for 19 × 78?

It is obvious that the product is less than 2000. Why?

If we approximate 19 to the nearest tens, we get 20 and then approximate 78

to nearest tens, we get 80 and 20 × 80 = 1600

Look at 63 × 182

If we approximate both to the nearest hundreds we

get 100 × 200 = 20,000. This is much larger than the

actual product. So, what do we do? To get a more

reasonable estimate, we try rounding off 63 to the

nearest 10, i.e. 60, and also 182 to the nearest ten, i.e.

180. We get 60 × 180 or 10,800. This is a good

estimate, but is not quick enough.

If we now try approximating 63 to 60 and 182 to

the nearest hundred, i.e. 200, we get 60 × 200, and this

number 12,000 is a quick as well as good estimate of

the product.

The general rule that we can make is, therefore,

Round off each factor to its

greatest place, then multiply the rounded off factors

. Thus, in the above

example, we rounded off 63 to tens and 182 to hundreds.

Now, estimate 81 × 479 using this rule :

479 is rounded off to 500 (rounding off to hundreds),

and 81 is rounded off to 80 (rounding off to tens).

The estimated product = 500 × 80 = 40,000

An important use of estimates for you will be to check your answers.

Suppose, you have done the multiplication 37 × 1889, but

are not sure about your answer. A quick and reasonable estimate

of the product will be 40 × 2000 i.e. 80,000. If your answer

is close to 80,000, it is probably right. On the other hand, if

it is close to 8000 or 8,00,000, something is surely wrong in

your multiplication.

Same general rule may be followed by addition and

subtraction of two or more numbers.

Estimate the

following products :

(a) 87 × 313

(b) 9 × 795

(d) 958 × 387

Make five more

such problems and

solve them.

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EXERCISE 1.3

1. Estimate each of the following using general rule:

(a) 730 + 998 (b) 796 – 314 (c) 12,904 +2,888 (d) 28,292 – 21,496

Make ten more such examples of addition, subtraction and estimation of their outcome.

2. Give a rough estimate (by rounding off to nearest hundreds) and also a closer estimate

(by rounding off to nearest tens) :

(a) 439 + 334 + 4,317 (b) 1,08,734 – 47,599 (c) 8325 – 491

(d) 4,89,348 – 48,365

Make four more such examples.

3. Estimate the following products using general rule:

(a) 578 × 161 (b) 5281 × 3491 (c) 1291 × 592 (d) 9250 × 29

Make four more such examples.

1.4 Using Brackets

Suman bought 6 notebooks from the market and the cost was Rs 10 per notebook.

Her sister Sama also bought 7 notebooks of the same type. Find the total money

they paid.

Seema calculated the Meera calculated the

amount like this amount like this

6 × 10 + 7 × 10 6 + 7 =13

= 60 + 70

= 130 and 13 × 10 = 130

Ans. Rs 130 Ans. Rs 130

1. Write the expressions for each of the following using brackets.

(a) Four multiplied by the sum of nine and two.

(b) Divide the difference of eighteen and six by four.

2. Write three different situations for (5 + 8) × 6.

(One such situation is : Sohani and Reeta work for 6 days; Sohani

works 5 hours a day and Reeta 8 hours a day. How many hours do

both of them work in a week?)

3. Write

five sit

uations for the following where brackets would be

necessary.

(a) 7(8 – 3) (b) (7 + 2) (10 – 3)

ATHEMATICS

You can see that Seema’s and Meera’s ways to get the answer are a bit different.

But both give the correct result. Why?

Seema says, what Meera has done is 7 + 6 × 10.

Appu points out that 7 + 6 × 10 = 7 + 60 = 67. Thus, this is not what Meera

had done. All the three students are confused.

To avoid confusion in such cases we may use brackets. We can pack the numbers

6 and 7 together using a bracket, indicating that the pack is to be treated as a single

number. Thus, the answer is found by (6 + 7) × 10 = 13 × 10.

This is what Meera did. She first added 6 and 7 and then multiplied the sum

by 10.

This clearly tells us :

First, turn everything inside the brackets ( ) into a

single number and then do the operation outside which in this case is to

multiply by 10.

1.4.1 Expanding brackets

Now, observe how use of brackets allows us to follow our procedure

systematically. Do you think that it will be easy to keep a track of what steps we

have to follow without using brackets?

(i) 7 × 109 = 7 × (100 + 9) = 7 × 100 + 7 × 9 = 700 + 63 = 763

(ii)

102 × 103 = (100 + 2) × (100 + 3) = (100 + 2) × 100 + (100 + 2) × 3

= 100 × 100 + 2 × 100 + 100 × 3 + 2 × 3

= 10,000 + 200 + 300 + 6 = 10,000 + 500 + 6

= 10,506

(iii) 17 × 109 = (10 + 7) × 109 = 10 × 109 + 7 × 109

= 10 × (100 + 9) + 7 × (100 + 9)

= 10 × 100 + 10 × 9 + 7 × 100 + 7 × 9

= 1000 + 90 + 700 + 63 = 1,790 + 63

= 1,853

1.5 Roman Numerals

We have been using the Hindu-Arabic numeral system so far. This

is not the only system available. One of the early systems of

writing numerals is the system of Roman numerals. This system

is still used in many places.

For example, we can see the use of Roman numerals in clocks; it

is also used for classes in the school time table etc.

Find three other examples, where Roman numerals are used.

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The Roman numerals :

I, II, III, IV, V, VI, VII, VIII, IX, X

denote 1,2,3,4,5,6,7,8,9 and 10 respectively. This is followed by XI for 11, XII

for 12,... till XX for 20. Some more Roman numerals are :

IVXL C D M

1 5 10 50 100 500 1000

The rules for the system are :

(a) If a symbol is repeated, its value is added as many times as it occurs:

i.e. II is equal 2, XX is 20 and XXX is 30.

(b) A symbol is not repeated more than three times. But the symbols V, L and D

are never repeated.

value, its value gets added to the value of greater symbol.

VI = 5 + 1 = 6, XII = 10 + 2 = 12

and LXV = 50 + 10 + 5 = 65

(d) If a symbol of smaller value is written to the left of a symbol of greater

value, its value is subtracted from the value of the greater symbol.

IV = 5 – 1 = 4, IX = 10 – 1 = 9

XL = 50 – 10 = 40, XC = 100 – 10 = 90

(e) The symbols V, L and D are never written to the left of a symbol of greater

value, i.e. V, L and D are never subtracted.

The symbol I can be subtracted from V and X only.

The symbol X can be subtracted from L, M and C only.

Following these rules we get,

1 = I 10 = X 100 = C

2= II 20= XX

3 = III 3 0 = XXX

4= IV 40= XL

5= V 50= L

6= VI 60= LX

7 = VII 7 0 = LXX

8 = VIII 8 0 =

LXXX

9= IX 90= XC

(a) Write in Roman numerals the missing numbers in the table.

(b) XXXX, VX, IC, XVV are not written. Can you tell why?

Write in

Roman

numerals.

1. 73

2. 92

ATHEMATICS

Example 7 :

Write in Roman Numerals (a) 69 (b) 98.

Solution :

(a) 69 = 60 + 9 (b) 98 = 90 + 8

= (50 + 10) + 9 = (100 – 10) + 8

= LX + IX = XC + VIII

= LX IX = XCVIII

What have we discussed?

1. Given two numbers, one with more digits is the greater number. If the number of

digits in two given numbers is the same, that number is larger, which has a greater

leftmost digit. If this digit also happens to be the same, we look at the next digit and

so on.

2. In forming numbers from given digits, we should be careful to see if the conditions

under which the numbers are to be formed are satisfied. Thus, to form the greatest

four digit number from 7, 8, 3, 5 without repeating a single digit, we need to use all

four digits, the greatest number can have only 8 as the leftmost digit.

3. The smallest four digit number is 1000 (one thousand). It follows the largest three

digit number 999. Similarly, the smallest five digit number is 10,000. It is ten thousand

and follows the largest four digit number 9999.

Further, the smallest six digit number is 100,000. It is one lakh and follows the largest

five digit number 99,999. This carries on for higher digit numbers in a similar manner.

4. Use of commas helps in reading and writing large numbers. In the Indian system of

numeration we have commas after 3 digits starting from the right and thereafter every

2 digits. The commas after 3, 5 and 7 digits separate thousand, lakh and crore

respectively. In the International system of numeration commas are placed after every

3 digits starting from the right. The commas after 3 and 6 digits separate thousand

and million respectively.

5. Large numbers are needed in many places in daily life. For example, for giving number

of students in a school, number of people in a village or town, money paid or received

in large transactions (paying and selling), in measuring large distances say betwen

various cities in a country or in the world and so on.

6. Remember kilo shows 1000 times larger, Centi shows 100 times smaller and milli

shows 1000 times smaller, thus, 1 kilometre = 1000 metres, 1 metre = 100 centimetres

or 1000 millimetres etc.

7. There are a number of situations in which we do not need the exact quantity but need

only a reasonable guess or an estimate. For example, while stating how many spectators

watched a particular international hockey match, we state the approximate number,

say 51,000, we do not need to state the exact number.

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8. Estimation involves approximating a quantity to an accuracy required. Thus, 4117

may be approximated to 4100 or to 4000, i.e. to the nearest hundred or to the

nearest thousand depending on our need.

9. In number of situations, we have to estimate the outcome of number operations. This

is done by rounding off the numbers involved and getting a quick, rough answer.

10. Estimating the outcome of number operations is useful in checking answers.

11. Use of brackets allows us to avoid confusion in the problems where we need to

carry out more than one number operation.

12. We use the Hindu-Arabic system of numerals. Another system of writing numerals is

the Roman system.

1. Give five examples where the number of things counted would be more than 6-digit number.

2. Starting from the greatest 6-digit number, write the previous five numbers in

descending order.

3. Starting from the smallest 8-digit number, write the next five numbers in

ascending order and read them.

1.2.6 Larger numbers

If we add one more to the greatest 6-digit number we get the smallest 7-digit

number. It is called

ten lakh.

Write down the greatest 6-digit number and the smallest 7-digit number.

Write the greatest 7-digit number and the smallest 8-digit number. The smallest

8-digit number is called

one crore.

Complete the pattern :

9 + 1 = 10

99 + 1 = 100

999 + 1 = _______

9,999 + 1 = _______

99,999 + 1 = _______

9,99,999 + 1 = _______

99,99,999 + 1 = 1,00,00,000

We come across large numbers in

many different situations.

For example, while the number of

children in your class would be a

2-digit number, the number of

children in your school would be

a 3 or 4-digit number.

The number of people in the nearby town would be much larger.

Is it a 5 or 6 or 7-digit number?

Do you know the number of people in your state?

How many digits would that number have?

What would be the number of grains in a sack full of wheat? A 5-digit number,

a 6-digit number or more?

Remember

1 hundred = 10 tens

1 thousand = 10 hundreds

= 100 tens

1 lakh = 100 thousands

= 1000 hundreds

1 crore = 100 lakhs

= 10,000 thousands

1. What is 10 – 1 =?

2. What is 100 – 1 =?

3. What is 10,000 – 1 =?

4. What is 1,00,000 – 1 =?

5. What is 1,00,00,000 – 1 =?

(

Hint :

Use the said pattern.)

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1.2.7 An aid in reading and writing large numbers

Try reading the following numbers :

(a) 279453 (b) 5035472

Was it difficult?

Did you find it difficult to keep track?

Sometimes it helps to use indicators to read and write large numbers.

Shagufta uses indicators which help her to read and write large numbers.

Her indicators are also useful in writing the expansion of numbers. For example,

she identifies the digits in ones place, tens place and hundreds place in 257 by

writing them under the tables O, T and H as

H T O Expansion

2 5 7 2 × 100 + 5 × 10 + 7 × 1

Similarly, for 2902,

Th H T O Expansion

2 9 0 2 2 × 1000 + 9 × 100 + 0 × 10 + 2 × 1

One can extend this idea to numbers upto lakh as seen in the following

table. (Let us call them placement boxes). Fill the entries in the blanks left.

Number TLakh Lakh TTh Th H T O Number Name Expansion

7,34,543 — 7 3 4 5 4 3 Seven lakh thirty

-----------------

four thousand five

hundred forty three

32,75,829 3 2 7 5 8 2 9

---------------------

3 × 10,00,000

+ 2 × 1,00,000

+ 7 × 10,000

+ 5 × 1000

+ 8 × 100

+ 2 × 10 + 9

Similarly, we may include numbers upto crore as shown below :

Number TCr Cr TLakh Lakh TTh Th H T O Number Name

2,57,34,543 — 2 5 7 3 4 5 4 3 ...................................

65,32,75,829 6 5 3 2 7 5 8 2 9 Sixty five crore thirty

two lakh seventy five

thousand eight hundred

twenty nine

You can make other formats of tables for writing the numbers in expanded form.

ATHEMATICS

Use of commas

You must have noticed that in writing large numbers in the

sections above, we have often used commas. Commas help us

in reading and writing large numbers. In our

Indian System

of Numeration

we use ones, tens, hundreds, thousands and

then lakhs and crores. Commas are used to mark thousands,

lakhs and crores. The first comma comes after hundreds place (three digits from the

right) and marks thousands. The second comma comes two digits later (five digits

from the right). It comes after ten thousands place and marks lakh. The third comma

comes after another

two

digits

(seven digits

from the right).

comes

after

ten

lakh

place

and marks crore.

For example, 5, 08, 01, 592

3, 32, 40, 781

7, 27, 05, 062

Try reading the numbers given above. Write five more numbers in this form and

read them.

International System of Numeration

In the International System of Numeration, as it is being used we have ones,

tens, hundreds, thousands and then millions. One million is a thousand

thousands. Commas are used to mark thousands and millions. It comes after

every three digits from the right. The first comma marks thousands and the

next comma marks millions. For example, the number 50,801,592 is read in

the International System as fifty million eight hundred one thousand five

hundred ninety two. In the Indian System, it is five crore eight lakh one thousand

five hundred ninety two.

How many lakhs make a million?

How many millions make a crore?

Take three large numbers. Express them in both Indian and International

Numeration systems.

Interesting fact :

To express numbers larger than a million, a billion is used in the

International System of Numeration: 1 billion = 1000 million.

While writing

number names,

we do not use

commas.

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How much was the increase in population

during 1991-2001? Try to find out.

Do you know what is India’s population

today? Try to find this too.

1. Read these numbers. Write them using placement boxes and then write their

expanded forms.

(i) 475320

(ii)

9847215

(iii)

97645310

(iv)

30458094

(a) Which is the smallest number?

(b) Which is the greatest number?

2. Read these numbers.

(i) 527864

(ii)

95432

(iii)

18950049

(iv)

70002509

(a) Write these numbers using placement boxes and then using commas in Indian

as well as International System of Numeration..

(b) Arrange these in ascending and descending order.

3. Take three more groups of large numbers and do the exercise given above.

Do you know?

India’s population increased by

about

27 million during 1921-1931;

37 million during 1931-1941;

44 million during 1941-1951;

78 million during 1951-1961

Can you help me write the numeral?

To write the numeral for a number you can follow the boxes again.

(a) Forty two lakh seventy thousand eight.

(b) Two crore ninety lakh fifty five thousand eight hundred.

1. You have the following digits 4, 5, 6, 0, 7 and 8. Using them, make five numbers

each with 6 digits.

(a) Put commas for easy reading.

(b) Arrange them in ascending and descending order.

2. Take the digits 4, 5, 6, 7, 8 and 9. Make any three numbers each with 8 digits.

Put commas for easy reading.

3. From the digits 3, 0 and 4, make five numbers each with 6 digits. Use commas.

ATHEMATICS

1. How many

centimetres make a

kilometre?

2. Name five large cities

in India. Find their

population. Also, find

the distance in

kilometres between

each pair of these cities.

1.3 Large Numbers in Practice

In earlier classes, we have learnt that we use centimetre (cm) as a unit of length.

For measuring the length of a pencil, the width of a book or

notebooks etc., we use centimetres. Our ruler has marks on each centimetre.

For measuring the thickness of a pencil, however, we find centimetre too big.

We use millimetre (mm) to show the thickness of a pencil.

(a) 10 millimetres = 1 centimetre

To measure the length of the classroom or

the school building, we shall find

centimetre too small. We use metre for the

purpose.

(b) 1 metre = 100 centimetres

= 1000 millimetres

Even metre is too small, when we have to

state distances between cities, say, Delhi

and Mumbai, or Chennai and Kolkata. For

this we need kilometres (km).

EXERCISE 1.1

1. Fill in the blanks:

(a) 1 lakh = _______ ten thousand.

(b) 1 million = _______ hundred thousand.

(d) 1 crore = _______ million.

(e) 1 million = _______ lakh.

2. Place commas correctly and write the numerals:

(a) Seventy three lakh seventy five thousand three hundred seven.

(b) Nine crore five lakh forty one.

(d) Fifty eight million four hundred twenty three thousand two hundred two.

(e) Twenty three lakh thirty thousand ten.

3. Insert commas suitably and write the names according to Indian System of

Numeration :

(a) 87595762 (b) 8546283 (c) 99900046 (d) 98432701

4. Insert commas suitably and write the names according to International System

of Numeration :

(a) 78921092 (b) 7452283 (c) 99985102 (d) 48049831

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How many millimetres make 1 kilometre?

Since 1 m = 1000 mm

1 km = 1000 m = 1000 × 1000 mm = 10,00,000 mm

We go to the market to buy rice or wheat; we buy it in

kilograms (kg). But items like ginger or chillies which

we do not need in large quantities, we buy in grams (g).

We know 1 kilogram = 1000 grams.

Have you noticed the weight of the medicine tablets

given to the sick? It is very small. It is in milligrams

(mg).

1 gram = 1000 milligrams.

What is the capacity of a bucket for holding water? It

is usually 20 litres (

). Capacity is given in litres. But

sometimes we need a smaller unit, the millilitres.

A bottle of hair oil, a cleaning liquid or a soft drink

have labels which give the quantity of liquid inside in

millilitres (ml).

1 litre = 1000 millilitres.

Note that in all these units we have some words

common like kilo, milli and centi. You should remember

that among these

kilo

is the greatest and

milli

is the

smallest; kilo shows 1000 times greater, milli shows

1000 times smaller, i.e. 1 kilogram = 1000 grams,

1 gram = 1000 milligrams.

Similarly, centi shows 100 times smaller, i.e. 1 metre

100 centimetres.

1. A bus started its journey and reached different places with a speed of

60 km/hour. The journey is shown below.

(i) Find the total distance covered by the bus from A to D.

(ii) Find the total distance covered by the bus from D to G.

(iii) Find the total distance covered by the bus, if it starts

from A and returns back to A.

(iv) Can you find the difference of distances from C to D and D to E?

1. How many

milligrams

make one

kilogram?

2. A box contains

2,00,000

medicine tablets

each weighing

20 mg. What is

the total weight

of all the

tablets in the

box in grams

and in

kilograms?

ATHEMATICS

The sales during the last year

Apples 2457 kg

Oranges 3004 kg

Combs 22760

Tooth brushes 25367

Pencils 38530

Note books 40002

Soap cakes 20005

(v) Find out the time taken by the bus to

reach

(a) A to B (b) C to D

Raman’s shop

Things Price

Apples

40 per kg

Oranges

30 per kg

Combs

3 for one

Tooth brushes

10 for one

Pencils

1 for one

Note books

6 for one

Soap cakes

8 for one

(a) Can you find the total weight of apples and oranges Raman sold last year?

Weight of apples = __________ kg

Weight of oranges = _________ kg

Therefore, total weight = _____ kg + _____ kg = _____ kg

Answer – The total weight of oranges and apples = _________ kg.

(b) Can you find the total money Raman got by selling apples?

together?

(d) Make a table showing how much money Raman received from selling

each item. Arrange the entries of amount of money received in

descending order. Find the item which brought him the highest amount.

How much is this amount?

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We have done a lot of problems that have addition, subtraction, multiplication

and division. We will try solving some more here. Before starting, look at these

examples and follow the methods used.

Example 1 :

Population of Sundarnagar was 2,35,471 in the year 1991. In the

year 2001 it was found to be increased by 72,958. What was the population of

the city in 2001?

Solution :

Population of the city in 2001

= Population of the city in 1991 + Increase in population

= 2,35,471 + 72,958

Now, 235471

+ 72958

308429

Salma added them by writing 235471 as 200000 + 35000 + 471 and

72958 as 72000 + 958. She got the addition as 200000 + 107000 + 1429 = 308429.

Mary added it as 200000 + 35000 + 400 + 71 + 72000 + 900 + 58 = 308429

Answer : Population of the city in 2001 was 3,08,429.

All three methods are correct.

Example 2 :

In one state, the number of bicycles sold in the year 2002-2003

was 7,43,000. In the year 2003-2004, the number of bicycles sold was 8,00,100.

In which year were more bicycles sold? and how many more?

Solution :

Clearly, 8,00,100 is more than 7,43,000. So, in that state, more

bicycles were sold in the year 2003-2004 than in 2002-2003.

Now, 800100

– 743000

057100

Can you think of alternative ways of solving this problem?

Answer : 57,100 more bicycles were sold in the year 2003-2004.

Example 3 :

The town newspaper is published every day. One copy has

12 pages. Everyday 11,980 copies are printed. How many total pages are

printed everyday?

Check the answer by adding

743000

+ 57100

800100 (the answer is right)

ATHEMATICS

Solution :

Each copy has 12 pages. Hence, 11,980 copies will have

12 × 11,980 pages. What would this number be? More than 1,00,000 or lesser.

Try to estimate.

Now, 11980

× 12

23960

+ 119800

143760

Answer:Everyday 1,43,760 pages are printed.

Example 4 :

The number of sheets of paper available for making notebooks

is 75,000. Each sheet makes 8 pages of a notebook. Each notebook contains

200 pages. How many notebooks can be made from the paper available?

Solution :

Each sheet makes 8 pages.

Hence, 75,000 sheets make 8 × 75,000 pages,

Now, 75000

× 8

600000

Thus, 6,00,000 pages are available for making notebooks.

Now, 200 pages make 1 notebook.

Hence, 6,00,000 pages make 6,00,000 ÷ 200 notebooks.

3000

Now, 200 600000

– 600

0000 The answer is 3,000 notebooks.

EXERCISE 1.2

1. A book exhibition was held for four days in a school. The number of tickets sold

at the counter on the first, second, third and final day was respectively 1094,

1812, 2050 and 2751. Find the total number of tickets sold on all the four days.

2. Shekhar is a famous cricket player. He has so far scored 6980 runs in test matches.

He wishes to complete 10,000 runs. How many more runs does he need?

3. In an election, the successful candidate registered 5,77,500 votes and his nearest

rival secured 3,48,700 votes. By what margin did the successful candidate win

the election?

4. Kirti bookstore sold books worth Rs 2,85,891 in the first week of June and books

worth Rs 4,00,768 in the second week of the month. How much was the sale for

)

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the two weeks together? In which week was the sale greater and by how much?

5. Find the difference between the greatest and the least number that can be written using

the digits 6, 2, 7, 4, 3 each only once.

6. A machine, on an average, manufactures 2,825 screws a day. How many screws did

it produce in the month of January 2006?

7. A merchant had Rs 78,592 with her. She placed an order for purchasing 40 radio sets

at Rs 1200 each. How much money will remain with her after the purchase?

8. A student multiplied 7236 by 65 instead of multiplying by 56. By how much was his

answer greater than the correct answer? (Hint: Do you need to do both the

multiplications?)

9. To stitch a shirt, 2 m 15 cm cloth is needed. Out of 40 m cloth, how many shirts can be

stitched and how much cloth will remain?

(

Hint:

convert data in cm.)

10. Medicine is packed in boxes, each weighing 4 kg 500g. How many such boxes can be

loaded in a van which cannot carry beyond 800 kg?

11. The distance between the school and the house of a student’s house is 1 km 875 m.

Everyday she walks both ways. Find the total distance covered by her in six days.

12. A vessel has 4 litres and 500 ml of curd. In how many glasses, each of 25 ml

capacity, can it be filled?

1.3.1 Estimation

News

1. India drew with Pakistan in a hockey match watched by 51,000 spectators

in the stadium and 40 million television viewers world wide.

2. Approximately, 2000 people were killed and more than 50000 injured in a

cyclonic storm in coastal areas of India and Bangladesh.

3. Over 13 million passengers are carried over 63,000 kilometre route of

railway track every day.

Can we say that there were exactly as many people as the numbers quoted

in these news items? For example,

In (1), were there exactly 51,000 spectators in the stadium? or did exactly

40 million viewers watched the match on television?

Obviously, not. The word

approximately

itself

shows that the number of people were near about these

numbers. Clearly, 51,000 could be 50,800 or 51,300

but not 70,000. Similarly, 40 million implies much

more than 39 million but quite less than 41 million

but certainly not 50 million.

ATHEMATICS

The quantities given in the examples above are not exact counts, but are

estimates to give an idea of the quantity.

Discuss what each of these can suggest.

Where do we approximate?

Imagine a big celebration at your home. The first

thing you do is to find out roughly how many guests may visit you. Can you get an

idea of the exact number of visitors? It is practically impossible.

The finance minister of the country presents a budget annually. The minister provides

for certain amount under the head ‘Education’. Can the amount be absolutely accurate?

It can only be a reasonably good estimate of the expenditure the country needs for

education during the year.

Think about the situations where we need to have the exact numbers and compare

them with situations where you can do with only an approximately estimated number.

Give three examples of each of such situations.

1.3.2 Estimating to the nearest tens by rounding off

Look at the following :

(a) Find which flags are closer to 260.

(b) Find the flags which are closer to 270.

Locate the numbers 10,17 and 20 on your ruler. Is 17 nearer to 10 or 20? The

gap between 17 and 20 is smaller when compared to the gap between 17 and 10.

So, we round off 17 as 20, correct to the nearest tens.

Now consider 12, which also lies between 10 and 20. However, 12 is

closer to 10 than to 20. So, we round off 12 to 10, correct to the nearest tens.

How would you

round off 76 to the nearest tens? Is it not 80?

We see that the numbers 1,2,3 and 4 are nearer to 0 than to 10. So, we

round off 1, 2, 3 and 4 as 0. Number 6, 7, 8, 9 are nearer to 10, so, we round

them off as 10. Number 5 is equidistant from both 0 and 10; it is a common

practice to round it off as 10.

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1.3.3 Estimating to the nearest hundreds by rounding off

Is 410 nearer to 400 or to 500?

410 is closer to 400, so it is rounded off to 400, correct to the nearest

hundred.

889 lies between 800 and 900.

It is nearer to 900, so it is rounded off as 900 correct to nearest hundred.

Numbers 1 to 49 are closer to 0 than to 100, and so are rounded off to 0.

Numbers 51 to 99 are closer to 100 than to 0, and so are rounded off to 100.

Number 50 is equidistant from 0 and 100 both. It is a common practice to round it off

as 100.

Check if the following rounding off is correct or not :

841

800; 9537

9500; 49730

49700;

2546

2500; 286

200; 5750

5800;

168

200; 149

100; 9870

9800.

Correct those which are wrong.

1.3.4 Estimating to the nearest thousands by rounding off

We know that numbers 1 to 499 are nearer to 0 than to 1000, so these numbers are

rounded off as 0.

The numbers 501 to 999 are nearer to 1000 than 0 so they are rounded off as

1000.

Number 500 is also rounded off as 1000.

Check if the following rounding off is correct or not :

2573

3000; 53552

53000;

6404

6000; 65437

65000;

7805

7000; 3499

4000.

Correct those which are wrong.

Round these numbers to the nearest tens.

28 32 52 41 39 48

64 59 99 215 1453 2936

ATHEMATICS

Round off the given numbers to the nearest tens, hundreds and thousands.

Given Number Approximate to

Nearest Rounded Form

75847 Tens ________________

75847

Hundreds

________________

75847

Thousands

________________

75847 Ten

thousands

________________

1.3.5 Estimating outcomes of number situations

How do we add numbers? We add numbers by following the algorithm (i.e. the

given method) systematically. We write the numbers taking care that the digits in the

same place (ones, tens, hundreds etc.) are in the same column. For example,

3946 + 6579 + 2050 is written as —

Th H T O

3946

6579

+ 2050

We add the column of ones and if necessary carry forward the appropriate

number to the tens place as would be in this case. We then add the tens

column and this goes on. Complete the rest of the sum yourself. This

procedure takes time.

There are many situations where we need to find answers more quickly.

For example, when you go to a fair or the market, you find a variety of attractive

things which you want to buy. You need to quickly decide what you can buy.

So, you need to estimate the amount you need. It is the sum of the prices of

things you want to buy.

A trader is to receive money from two sources. The money he is to receive

is Rs 13,569 from one source and Rs 26,785 from another. He has to pay

Rs 37,000 to someone else by the evening. He rounds off the numbers to their

nearest thousands and quickly works out the rough answer. He is happy that

he has enough money.

Do you think he would have enough money? Can you tell without doing

the exact addition/subtraction?

Sheila and Mohan have to plan their monthly expenditure. They know

their monthly expenses on transport, on school requirements, on groceries,

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on milk, and on clothes and also on other regular

expenses. This month they have to go for visiting

and buying gifts. They estimate the amount they

would spend on all this and then add to see, if what

they have, would be enough.

Would they round off to thousands as the

trader did?

Think and discuss five more situations where we have to estimate sums or

remainders.

Did we use rounding off to the same place in all these?

There are no rigid rules when you want to estimate the outcomes of numbers.

The procedure depends on the degree of accuracy required and how quickly

the estimate is needed. The most important thing is, how sensible the guessed

answer would be.

1.3.6 To estimate sum or difference

As we have seen above we can round off a number to any place. The trader

rounded off the amounts to the nearest thousands and was satisfied that he had

enough. So, when you estimate any sum or difference, you should have an idea

of why you need to round off and therefore the place to which you would round

off. Look at the following examples.

Example 5 :

Estimate: 5,290 + 17,986.

Solution :

You find 17,986 > 5,290.

Round off to thousands.

17,986 is rounds off to

18,000

+5,290 is rounds off to + 5,000

Estimated sum =

23,000

Does the method work? You may attempt to find the actual answer and

verify if the estimate is reasonable.

Example 6 :

Estimate: 5,673 – 436.

Solution :

To begin with we round off to thousands. (Why?)

5,673 rounds off to

6,000

– 436 rounds off to – 0

Estimated difference =

6,000

This is not a reasonable estimate. Why is this not reasonable?

ATHEMATICS

To get a closer estimate, let us try rounding each number to hundreds.

5,673 rounds off to

5,700

– 436 rounds off to – 400

Estimated difference =

5,300

This is a better and more meaningful estimate.

1.3.7 To estimate products

How do we estimate a product?

What is the estimate for 19 × 78?

It is obvious that the product is less than 2000. Why?

If we approximate 19 to the nearest tens, we get 20 and then approximate 78

to nearest tens, we get 80 and 20 × 80 = 1600

Look at 63 × 182

If we approximate both to the nearest hundreds we

get 100 × 200 = 20,000. This is much larger than the

actual product. So, what do we do? To get a more

reasonable estimate, we try rounding off 63 to the

nearest 10, i.e. 60, and also 182 to the nearest ten, i.e.

180. We get 60 × 180 or 10,800. This is a good

estimate, but is not quick enough.

If we now try approximating 63 to 60 and 182 to

the nearest hundred, i.e. 200, we get 60 × 200, and this

number 12,000 is a quick as well as good estimate of

the product.

The general rule that we can make is, therefore,

Round off each factor to its

greatest place, then multiply the rounded off factors

. Thus, in the above

example, we rounded off 63 to tens and 182 to hundreds.

Now, estimate 81 × 479 using this rule :

479 is rounded off to 500 (rounding off to hundreds),

and 81 is rounded off to 80 (rounding off to tens).

The estimated product = 500 × 80 = 40,000

An important use of estimates for you will be to check your answers.

Suppose, you have done the multiplication 37 × 1889, but

are not sure about your answer. A quick and reasonable estimate

of the product will be 40 × 2000 i.e. 80,000. If your answer

is close to 80,000, it is probably right. On the other hand, if

it is close to 8000 or 8,00,000, something is surely wrong in

your multiplication.

Same general rule may be followed by addition and

subtraction of two or more numbers.

Estimate the

following products :

(a) 87 × 313

(b) 9 × 795

(d) 958 × 387

Make five more

such problems and

solve them.

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EXERCISE 1.3

1. Estimate each of the following using general rule:

(a) 730 + 998 (b) 796 – 314 (c) 12,904 +2,888 (d) 28,292 – 21,496

Make ten more such examples of addition, subtraction and estimation of their outcome.

2. Give a rough estimate (by rounding off to nearest hundreds) and also a closer estimate

(by rounding off to nearest tens) :

(a) 439 + 334 + 4,317 (b) 1,08,734 – 47,599 (c) 8325 – 491

(d) 4,89,348 – 48,365

Make four more such examples.

3. Estimate the following products using general rule:

(a) 578 × 161 (b) 5281 × 3491 (c) 1291 × 592 (d) 9250 × 29

Make four more such examples.

1.4 Using Brackets

Suman bought 6 notebooks from the market and the cost was Rs 10 per notebook.

Her sister Sama also bought 7 notebooks of the same type. Find the total money

they paid.

Seema calculated the Meera calculated the

amount like this amount like this

6 × 10 + 7 × 10 6 + 7 =13

= 60 + 70

= 130 and 13 × 10 = 130

Ans. Rs 130 Ans. Rs 130

1. Write the expressions for each of the following using brackets.

(a) Four multiplied by the sum of nine and two.

(b) Divide the difference of eighteen and six by four.

2. Write three different situations for (5 + 8) × 6.

(One such situation is : Sohani and Reeta work for 6 days; Sohani

works 5 hours a day and Reeta 8 hours a day. How many hours do

both of them work in a week?)

3. Write

five sit

uations for the following where brackets would be

necessary.

(a) 7(8 – 3) (b) (7 + 2) (10 – 3)

ATHEMATICS

You can see that Seema’s and Meera’s ways to get the answer are a bit different.

But both give the correct result. Why?

Seema says, what Meera has done is 7 + 6 × 10.

Appu points out that 7 + 6 × 10 = 7 + 60 = 67. Thus, this is not what Meera

had done. All the three students are confused.

To avoid confusion in such cases we may use brackets. We can pack the numbers

6 and 7 together using a bracket, indicating that the pack is to be treated as a single

number. Thus, the answer is found by (6 + 7) × 10 = 13 × 10.

This is what Meera did. She first added 6 and 7 and then multiplied the sum

by 10.

This clearly tells us :

First, turn everything inside the brackets ( ) into a

single number and then do the operation outside which in this case is to

multiply by 10.

1.4.1 Expanding brackets

Now, observe how use of brackets allows us to follow our procedure

systematically. Do you think that it will be easy to keep a track of what steps we

have to follow without using brackets?

(i) 7 × 109 = 7 × (100 + 9) = 7 × 100 + 7 × 9 = 700 + 63 = 763

(ii)

102 × 103 = (100 + 2) × (100 + 3) = (100 + 2) × 100 + (100 + 2) × 3

= 100 × 100 + 2 × 100 + 100 × 3 + 2 × 3

= 10,000 + 200 + 300 + 6 = 10,000 + 500 + 6

= 10,506

(iii) 17 × 109 = (10 + 7) × 109 = 10 × 109 + 7 × 109

= 10 × (100 + 9) + 7 × (100 + 9)

= 10 × 100 + 10 × 9 + 7 × 100 + 7 × 9

= 1000 + 90 + 700 + 63 = 1,790 + 63

= 1,853

1.5 Roman Numerals

We have been using the Hindu-Arabic numeral system so far. This

is not the only system available. One of the early systems of

writing numerals is the system of Roman numerals. This system

is still used in many places.

For example, we can see the use of Roman numerals in clocks; it

is also used for classes in the school time table etc.

Find three other examples, where Roman numerals are used.

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The Roman numerals :

I, II, III, IV, V, VI, VII, VIII, IX, X

denote 1,2,3,4,5,6,7,8,9 and 10 respectively. This is followed by XI for 11, XII

for 12,... till XX for 20. Some more Roman numerals are :

IVXL C D M

1 5 10 50 100 500 1000

The rules for the system are :

(a) If a symbol is repeated, its value is added as many times as it occurs:

i.e. II is equal 2, XX is 20 and XXX is 30.

(b) A symbol is not repeated more than three times. But the symbols V, L and D

are never repeated.

value, its value gets added to the value of greater symbol.

VI = 5 + 1 = 6, XII = 10 + 2 = 12

and LXV = 50 + 10 + 5 = 65

(d) If a symbol of smaller value is written to the left of a symbol of greater

value, its value is subtracted from the value of the greater symbol.

IV = 5 – 1 = 4, IX = 10 – 1 = 9

XL = 50 – 10 = 40, XC = 100 – 10 = 90

(e) The symbols V, L and D are never written to the left of a symbol of greater

value, i.e. V, L and D are never subtracted.

The symbol I can be subtracted from V and X only.

The symbol X can be subtracted from L, M and C only.

Following these rules we get,

1 = I 10 = X 100 = C

2= II 20= XX

3 = III 3 0 = XXX

4= IV 40= XL

5= V 50= L

6= VI 60= LX

7 = VII 7 0 = LXX

8 = VIII 8 0 =

LXXX

9= IX 90= XC

(a) Write in Roman numerals the missing numbers in the table.

(b) XXXX, VX, IC, XVV are not written. Can you tell why?

Write in

Roman

numerals.

1. 73

2. 92

ATHEMATICS

Example 7 :

Write in Roman Numerals (a) 69 (b) 98.

Solution :

(a) 69 = 60 + 9 (b) 98 = 90 + 8

= (50 + 10) + 9 = (100 – 10) + 8

= LX + IX = XC + VIII

= LX IX = XCVIII

What have we discussed?

1. Given two numbers, one with more digits is the greater number. If the number of

digits in two given numbers is the same, that number is larger, which has a greater

leftmost digit. If this digit also happens to be the same, we look at the next digit and

so on.

2. In forming numbers from given digits, we should be careful to see if the conditions

under which the numbers are to be formed are satisfied. Thus, to form the greatest

four digit number from 7, 8, 3, 5 without repeating a single digit, we need to use all

four digits, the greatest number can have only 8 as the leftmost digit.

3. The smallest four digit number is 1000 (one thousand). It follows the largest three

digit number 999. Similarly, the smallest five digit number is 10,000. It is ten thousand

and follows the largest four digit number 9999.

Further, the smallest six digit number is 100,000. It is one lakh and follows the largest

five digit number 99,999. This carries on for higher digit numbers in a similar manner.

4. Use of commas helps in reading and writing large numbers. In the Indian system of

numeration we have commas after 3 digits starting from the right and thereafter every

2 digits. The commas after 3, 5 and 7 digits separate thousand, lakh and crore

respectively. In the International system of numeration commas are placed after every

3 digits starting from the right. The commas after 3 and 6 digits separate thousand

and million respectively.

5. Large numbers are needed in many places in daily life. For example, for giving number

of students in a school, number of people in a village or town, money paid or received

in large transactions (paying and selling), in measuring large distances say betwen

various cities in a country or in the world and so on.

6. Remember kilo shows 1000 times larger, Centi shows 100 times smaller and milli

shows 1000 times smaller, thus, 1 kilometre = 1000 metres, 1 metre = 100 centimetres

or 1000 millimetres etc.

7. There are a number of situations in which we do not need the exact quantity but need

only a reasonable guess or an estimate. For example, while stating how many spectators

watched a particular international hockey match, we state the approximate number,

say 51,000, we do not need to state the exact number.

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8. Estimation involves approximating a quantity to an accuracy required. Thus, 4117

may be approximated to 4100 or to 4000, i.e. to the nearest hundred or to the

nearest thousand depending on our need.

9. In number of situations, we have to estimate the outcome of number operations. This

is done by rounding off the numbers involved and getting a quick, rough answer.

10. Estimating the outcome of number operations is useful in checking answers.

11. Use of brackets allows us to avoid confusion in the problems where we need to

carry out more than one number operation.

12. We use the Hindu-Arabic system of numerals. Another system of writing numerals is

the Roman system.

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27

Hari's Random Thoughts by Hariharan Ramamurthy

Sunday, January 29, 2017

Mathematics basics from NCERT

Mathematics

1.1 introduction

Knowing our Numbers

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