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.
(c) 25286, 25245, 25270, 25210.
(d) 6895, 23787, 24569,
24659.
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9
867
1
027
4
4
9
9
1
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
(c) Digit 9 is always at Greatest
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?
(c) Arrange these numbers in ascending and
descending orders.
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.
(c) Seven crore sixty
thousand fifty five.
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.
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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.
(c) 1 crore
= _______ ten lakh.
(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.
(c) Seven crore fifty two lakh twenty one
thousand three hundred two.
(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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(c) 1 kilometre = 1000
metres
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?
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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
(c) E to G
(d) Total journey
2.
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?
(c) Can you find the total money Raman got by
selling apples and oranges
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)
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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.
M
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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
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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?
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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
(c) 898 × 785
(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.
(c) Forty five divided by three times the sum of
three and two.
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)
M
ATHEMATICS
24
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.
(c) If a symbol of
smaller value is written to the right of a symbol of greater
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
M
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26
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.
K
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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.
8
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
(c) 152700375 (d)
40350894
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
10
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).
It
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?
(c) Arrange these numbers in ascending and
descending orders.
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.
(c) Seven crore sixty
thousand fifty five.
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
12
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.
(c) 1 crore
= _______ ten lakh.
(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.
(c) Seven crore fifty two lakh twenty one
thousand three hundred two.
(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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(c) 1 kilometre = 1000
metres
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
14
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
(c) E to G
(d) Total journey
2.
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?
(c) Can you find the total money Raman got by
selling apples and oranges
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
16
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
18
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
20
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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21
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
22
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
(c) 898 × 785
(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.
(c) Forty five divided by three times the sum of
three and two.
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
24
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.
(c) If a symbol of
smaller value is written to the right of a symbol of greater
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
26
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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