integrated syllabus (free sample)iit foundation & olympiad explorer - mathematics class - vi...
TRANSCRIPT
CLASS - VI
IIT F
oundatio
n &
Olym
pia
d E
xplo
rer - M
ath
em
atic
s Cla
ss - VI
FOUNDATION OLYMPIAD&
IntegratedSyllabus
UNIQUE ATTRACTIONS●
● Cross word Puzzles
● Graded Exercise
Basic Practice■
Further Practice■
Brain Works■
● Multiple Answer Questions
● Paragraph Questions
Rs. 85Detailed solutionsfor all problems
of IIT Foundation &Olympiad Explorer
are available in this book
CLASS - X
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� Simple, clear and systematic presentation
� Concept maps provided for every chapter
� Set of objective and subjective questions at the
end of each chapter
� Previous contest questions at the end of each
chapter
� Designed to fulfill the preparation needs for
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and all competitive exams
` 250
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© Brain Mapping Academy
ALL RIGHTS RESERVEDNo part of this book may be reproduced, stored in a retrieval system or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise without the prior written permission of the publisher.
Publication Team
Authors: M. Gurunadham, Y.S. Srinivasu
Design & Typing: P. Sesha Chakravarthy
ISBN: 978-81-907285-1-5
Disclaimer
Every care has been taken by the compilers andpublishers to give correct, complete and updated information. In case there is any omission, printing mistake or anyother error which might have crept in inadvertently,neither the compiler / publisher nor any of thedistributors take any legal responsibility.
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Preface
Speed and accuracy play an important role in climbing the competitive ladder. Students
have to integrate the habit of being able to calculate and function quickly as well as efficiently
in order to excel in the learning culture. They need to think on their feet, understand basic
requirements, identify appropriate information sources and use that to their best advantage.
The preparation required for the tough competitive examinations is fundamentally different
from that of qualifying ones like the board examinations. A student can emerge successful in
a qualifying examination by merely scoring the minimum percentage of marks, whereas in a
competitive examination, he has to score high and perform better than the others taking the
examination.
This book provides all types of questions that a student would be required to tackle at the
foundation level. The questions in the exercises are sequenced as Basic Practice, Further Practice,
Brainworks, Multiple Answer Questions and Paragraph Questions. Simple questions involving
a direct application of the concepts are given in Basic Practice. More challenging questions
on direct application are given in Further Practice. Questions involving higher order thinking
or an open-ended approach to problems are given in Brainworks. These questions encourage
students to think analytically, to be creative and to come up with solutions of their own.
Constant practice and familiarity with these questions will not only make him/her
conceptually sound, but will also give the student the confidence to face any entrance
examination with ease.
Valuable suggestions as well as criticism from the teacher and student community are most
welcome and will be incorporated in the ensuing edition.
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1. Sets .................................................... 1
2. Natural and Whole Numbers ............ 21
3. Integers ............................................. 44
4. Factors and Multiples ........................ 66This page is intentionally left blank. 90
5. Fractions............................................. 91
6. Decimals ............................................ 113
7. Squares and square roots ................ 129This page is intentionally left blank. 149
8. Ratio,Proportion & Unitary method .. 150
9. Percentages ...................................... 172
10. Algebra .............................................. 191This page is intentionally left blank. 209
11. Lines and Angles ............................... 210
12. Triangles & Polygons ......................... 247
13. Circles................................................. 281This page is intentionally left blank. 293
14. Length & Mass ................................... 294
15. Area & Perimeter .............................. 318www.bmata
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© Brain Mapping Academy6. Decimals 113
1
Chapter
6DecimalsDecimals
Chapter
SYNOPSIS
DECIMAL AND FRACTIONS
Representing fractions of 1
10 and
1
100 as decimals and vice versa
• Decimals are fractions whose denominator is a multiple of 10, that is 10, 100, 1,000, ....and so on.
• In the figure below, the shaded areas represent 8 of 10 parts, that is 8
10 parts.
110
110
110
110
110
110
110
110
110
110
0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1 0.1
1 of 10 parts = 1
10 = 0.1
∴ 8
10 = 0.1 × 8 = 0.8
Hence, decimals and fractions are interchangeable.
Representing fractions with denominators 10, 100 and 1,000 as decimals
• Any fractions with denominators 10, 100 and 1,000 can be expressed in decimals.
81
100 = 0.81
273
1,000 = 2.073
• How to read and write decimals to thousandths?
410 = 0.4
3100 = 0.03
1,9871,000 = 1.987
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Changing fractions to decimals and vice versa
1. To change a fraction to a decimal:
Divide the numerator by its denominator.
2. To change a decimal to a fraction:
(a) Count the number of digits in the decimal.
(b) Then, convert into an equivalent fraction with a denominator that is a multiple of10.
(c) Simplify the answer to the lowest terms whenever possible.
PLACE VALUES AND VALUES OF EACH DIGIT IN DECIMALS
Stating place values and values of each digit in decimals
• Each digit in a decimal has a specific place value which determines the value of the digit.
• For the number 37.156, the table below shows the place value of each digit and the valueof each digit.
Place value Decimal Value of the digit
Tens (10) 3 30
Units (1) 7 7
Decimal point • •
Tenths 1
10 1 0.1
Hundredths 1
100
5 0.05
Thousands 1
1,000
6 0.006
• Each digit has only one place value and that place determines the value of the digit. Forexample, the place value of the digit 5 is hundredths. Therefore, the value of the digit is 0.05.
37.156 = 37 + 0.1 + 0.05 + 0.006
= 37 + 1 5 6
10 100 1,000+ +
Comparing the values of two decimals and arranging decimals in order
• Decimals can be arranged in ascending or descending order.
• To compare two decimals:
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SOLVED EXAMPLES
Example 1:
Express the following in decimal form.
(a) 7
10 (b) 34
100 (c) 565
1,000
Solution:
(a)7
10 = 0.7 (b)34
100 = 0.34 (c) 565
1,000 = 5.065
Example 2:
Change the following fractions into decimals.
(a) 18 (b)
158
Solution:
(a) 18 = 1 ÷ 8 (b)
158 = 15 ÷ 8
1.0008
0.125
8
20
16
40
40
15.0 0 08
1.8 7 5
8
7 06 4
6 0
5 6
4 0
4 0
∴ 18 = 0.125 ∴
158 = 1.875
Example 3:
Convert the following decimals into fractions.
(a) 0.02 (b) 0.075 (c) 1.228
Solution:
(a) 0.02 = 2
100 (b) 0.075 = 75
1,000
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(c) 1.228 = 1 + 0.228
= 1 + 228
1,000
= 1 + 57250
= 157250
Example 4:
Which of the decimals is greater, 485.760 or 485.670?
Solution:
Arranged the decimals according to their place values:
Comparetheir values in order from left to right485.760
In 485.760,thedigit in the tenths place is7,485.670
while in 485.670,thedigit in the tenth place is6.
←
Since, 0.7 > 0.6, therefore 485.760 is greater.
Example 5:
The figure below is a number line.
3.25 3.75 4 p
Find the value of p.
Solution:
First, determine the value of each portion.
4 � 3.75 = 0.25
∴ p = 4 + 2(0.25)
= 4 + 0.5
p = 4.5
Example 6:
Round off 87.4592 to
(a) the nearest whole number,
(b) 1 decimal place,
(c) 2 decimal places,
(d) 3 decimal places.
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Solution:
(a) 87.4592 = 87 ( )to nearest whole number
Digit 4 < 5. Therefore, keep digit 7 and omit all digits after 7.Place to
round off.
(b) 87.4592 = 87.5 (1 d.p.)
Digit 5 = 5. Therefore, add 1
to 4 and omit all digits after 4.Place to
round off.
(c) 87.4592 = 87.46 (2 d.p.)
Digit 9 > 5. Therefore, add 1
to 5 and omit all digits after 5.Place to
round off.
(d) 87.4592 = 87.459 (3 d.p.)
Digit 2 < 5. Therefore, keep
digit 9 and omit all digits after 9.Place to
round off.
Example 7:
Round off 13.539 to the nearest tenth.
Solution:
1 3 . 5 3 9 = 13.5
Digit 3 < 5. Keep digit 5
and omit all digits after 5.Place to
round off.
Example 8:
Solve 3.5 + 7.029 + 18.953.
Solution:
3 5 0 0
7 0 2 9
1 8 9 5 3
.
.
.
2 9 4 8 2.
1
1
Insert two zeros.
Line up digits of the decimal numbers
according to their place values
Add from right to left.
Align the decimal points in a straight line.
+
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3.5 + 7.029 + 18.953 = 29.482
3.5 47.029 7 to the nearest whole number
18.953 19
= = =
The sum of these 3 decimal numbers is about 30.
∴ The solution of 29.482 is reasonable.
Example 9:
In a grocery store, Bharati bought a tin of biscuits for Rs. 15.95, a bag of rice for Rs. 22.29and a box of sweets for Rs. 1.65. How much did Bharati pay altogether?
Solution:
Total amount paid
= Rs.15.95 + Rs. 22.29 + Rs. 1.65
1 1
1 5 . 9 5
2 2 . 2 9
+ 1 . 6 5
3 9 . 8 9
∴ Total amount paid = Rs. 39.89
Example 10:
Solve 49.81 � 10.19 � 3.54.
Solution:
3 9 . 6 2
3 . 5 4
3 6 . 0 8
5 12
Align the decimal points.
4 9 . 8 1
1 0 . 1 9
7 11
49.81 � 10.19 � 3.54 = 36.08
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Example 13:
Calculate the following.
(a) 0.054 × 10
(b) 1.796 × 100
(c) 48.38 × 1,000
Solution:
(a)
0.054 × 10 = 0.54
Move the decimal
point 1 place to the right.
(1 zero)
(b)
1.796 × 100 = 179.6
Move the decimal
point 2 places to the right.
(2 zeros)
(c)
48.38 × 1 000 = 48 380
Move the decimal
point 3 places to the right.
Add 1 zero to fill up the
empty space.
(3 zeros)
Example 14:
Find the product of the following.
(a) 9 × 0.1
(b) 7.5 × 0.01
(c) 89.4 × 0.001
Solution:
(a) 9 × 0 . 1 = 0.9
Move the decimal
point 1 place to the left.
(1 d.p.)
(b) 7.5 × 0.01 = 0.075
Move the decimal
point 2 places to the left.
(2 d.p.)
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(c) 89.4 × 0.001 = 0.0894
Move the decimal
point 3 places to the left.
(3 d.p.)
Example 15:
A table weighs 5.67 kg. What is the total mass of 6 identical tables?Solution:
Total mass = 5.67 kg × 6 5.67 = 34.02 kg × 6
34.02
Example 16:
Evaluate 75.38 ÷ 5.
Solution:
7 5 . 3 8 05
1 5 . 0 7 6
5
2 52 5
3 8
3 5
3 0
3 0
Align the decimal points.
Zero is added to
complete the division.
∴ 75.38 ÷ 5 = 15.076
Example 17:
Divide the following.
(a) 79.88 ÷ 100 (b) 22.128 ÷ 0.001
Solution:
(a) 79.88 100 = 0.7988÷
Move the decimal point
2 places to the left.
(2 zeros)
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CO
NC
EPT
MA
PD
ecim
al
nu
mb
ers
Th
e n
um
ber
s w
hic
h c
on
tain
deci
mal
poi
nt
are
call
ed d
eci
mal n
um
bers
. It h
as
two p
art
s,
On
e t
o t
he
left
(w
hole
nu
mber
or
inte
gra
l part
)
the o
ther
deci
mal p
art
De
cim
al
fra
cti
on
A fra
ctio
n w
hose
den
om
inato
r is
10, 100,
1000 ..
... i
s ca
lled a
dec
imal f
ract
ion
Co
mp
aris
on
of D
ecim
al n
um
bers:
(i)
Com
pare
th
e
inte
gra
l part
s (i
.e
wh
ole
nu
mber
part
s).
Th
e
decim
al
nu
mb
er
ha
vin
g
gre
ate
r in
tegra
l part
is
gre
ate
r
deci
mal n
um
ber.
(ii)
If
th
e
inte
gra
l part
s are
equ
al,
then
com
pare
th
e d
eci
mal
part
s.
For
it,
con
sid
er
the t
en
ths
dig
its.
Th
e
decim
al
nu
mb
er
ha
vin
g
gre
ate
r te
nth
s d
igit
is
gre
ate
r.
If
the t
en
ths
dig
its
are
als
o e
qu
al,
then
com
pa
re
the
dig
its
at
hu
ndre
dth
s pla
ce.
T
he deci
mal
nu
mber
havin
g g
reate
r h
un
dre
dth
dig
it is
gre
ate
r an
d s
o o
n.
Eg. 5
6.3
49, 5
6.3
71
Here
in
tegra
l p
art
s are
equ
al
(56)
als
o t
en
ths
dig
its
are
equ
al
(3)
on
com
pari
ng
hu
ndre
dth
dig
its
we
kn
ow
7 >
4 \
56.3
71 >
56.3
49
Ch
an
ge
de
cim
al
fra
cti
on
s i
nto
de
cim
al
nu
mb
ers
In t
he n
um
era
tor,
sta
rtin
g fro
m r
igh
t, m
ark
the d
eci
mal poin
t aft
er
as
man
y d
igit
s as
the
nu
mber
of ze
roes in
th
e d
en
om
inato
r
Eg. =
0.0
77
100
Ch
an
ge
fra
cti
on
s i
nto
de
cim
al
nu
mb
ers
Fir
st
chan
ge
the
giv
en
fr
act
ion
to
an
equ
ivale
nt
deci
mal
fract
ion
an
d
then
mark
th
e d
eci
mal p
oin
t as
above.
Eg. .
. 2
2
Ch
an
ge
de
cim
al
nu
mb
er i
nto
fra
cti
on
s
Rem
ove t
he d
eci
mal poin
t an
d w
rite
1 in
the
den
om
inato
r.
A
lso
wri
te
the
nu
mber
of
zero
s aft
er
this
, equ
al
to t
he
nu
mber
of dig
its
in th
e d
eci
mal part
.
Eg. 0
.35=
=
7 50
14
0.1
4100
==
7 20
35
100
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BASIC PRACTICE
1. Write each of the following fractions as a decimals.
(i) 7
100 (ii) 23
1000 (iii) 34
2. Write each of the following decimals as a fraction.
(i) 0.6 (ii) 0.13 (iii) 0.425
3. State the decimal represented by the shaded parts in the following figures.
(i) (ii)
4. Change the following fractions to decimals.
(i) 15 (ii)
82100 (iii)
21
5
(iv) 1
94
(v) 1
148
5. Change the following decimals to fractions.
(i) 1.5 (ii) 1.23 (iii) 0.625
(iv) 1.42 (v) 9.45
6. Complete the given number line
8.16 8.18 8.19 8.22
7. Round off each of the following decimals to the number of decimal places given in brackets.
(i) 2.1425 (3 d.p.) (ii) 0.01721 (2 d.p.) (iii) 52.167 (1 d.p.)
(iv) 1.0478 (2 d.p.)
8. Arrange the following decimals in ascending order.
(i) 0.42, 0.5, 0.39, 0.22 (ii) 0.042, 0.9, 0.03, 0.0099 (iii) 5.34, 5.43, 7.02, 5.099
9. Arrange the following decimals in descending order.
(i) 3.12, 3.21, 3.1, 3.09 (ii) 0.42, 1.01, 0.92, 0.63 (iii) 0.99, 10.1, 3.92, 0.097
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FURTHER PRACTICE
1. Express 18
100000 as a decimal.
(A) 0.0000018 (B) 0.00018 (C) 0.018 (D) 0.18
2. Express 0.07 as a fraction.
(A) 7
10 (B) 170 (C)
7100 (D)
1700
3. 0.0999 as a fraction is
(A) 99910 (B)
999100 (C)
9991,000 (D)
99910,000
4. The place value of the digit 4 in 0.01541 is
(A) tenths (B) hundredths
(C) thousandths (D) ten thousandths
5.0
P Q R S
0.1
On the above number line, which of the letters represents 0.075?
(A) P (B) Q (C) R (D) S
6.0.8 0.9y
On the above number line, y represents
(A) 0.83 (B) 0.85 (C) 0.86 (D) 0.88
7. Which of the following is the smallest decimal?
(A) 0.018 (B) 0.07 (C) 0.074 (D) 0.0054
10. Round off each of the following decimals correct to the number of decimal places given inthe brackets.
(i) 0.4192 (2 d.p.) (ii) 3.18 (1 d.p.) (iii) 9.186 (2 d.p.)
(iv) 0.9192 (3 d.p.) (v) 6.1995 (3 d.p.) (vi) 0.1495 (3 d.p.)
(vii) 14.178 (2 d.p.) (viii) 4.096 (1 d.p.) (ix) 15.972 (1 d.p.)
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Further practice
1. 2. 3. 4. 5. 6. 7. 8. 9. 10.
11. 12.
B C D D D C D C B B
C D
ANSWERS
Basic practice
1) (i) 0.07 (ii) 0.023 (iii) 0.75 2) (i) 35 (ii)
13100 (iii)
1740
3) (i) 1.7 7 7
1 110 10
← + = (ii) 2.3 3 3
2 210 10
← + =
4) (i) 0.2 (ii) 0.82 (iii) 1.4 (iv) 9.25 (v) 14.125
5) (i) 5
110 (ii)
231
100 (iii) 625
1000 (iv) 42
1100 (v)
459
100
6)8.16 8.18 8.19 8.228.17 8.20 8.21
+0.01 +0.01 +0.01 +0.01 +0.01 +0.01
7) (i) 2.143 (ii) 0.02 (iii) 52.2 (iv) 1.05
8) (i) 0.22, 0.39, 0.42, 0.5 (ii) 0.0099, 0.03, 0.042, 0.9
(iii) 5.099, 5.34, 5.43. 7.02
9) (i) 3.21, 3.12, 3.1, 3.09 (ii) 1.01, 0.92, 0.63, 0.42 (iii) 10.1, 3.92, 0.99, 0.097
10) (i) 0.42 (ii) 3.2 (iii) 9.19 (iv) 0.919 (v) 6.200
(vi) 0.150 (vii) 14.18 (viii) 4.1 (ix) 16.0
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CLASS - VI
IIT F
oundatio
n &
Olym
pia
d E
xplo
rer - M
ath
em
atic
s Cla
ss - VI
FOUNDATION OLYMPIAD&
IntegratedSyllabus
UNIQUE ATTRACTIONS●
● Cross word Puzzles
● Graded Exercise
Basic Practice■
Further Practice■
Brain Works■
● Multiple Answer Questions
● Paragraph Questions
Detailed solutionsfor all problems
of IIT Foundation &Olympiad Explorer
are available in this book
CLASS - X
www.bmatalent.com
� Simple, clear and systematic presentation
� Concept maps provided for every chapter
� Set of objective and subjective questions at the
end of each chapter
� Previous contest questions at the end of each
chapter
� Designed to fulfill the preparation needs for
international/national talent exams, olympiads
and all competitive exams
YOUR
COACH
India’s FIRST scientifically designed portalfor Olympiad preparation• Olympiad & Talent Exams preparation packages
Analysis Reports Previous question papers• •Free Demo Packages Free Android Mobile App• •
Get 15% discount on all packages by using the discount coupon code: KR157N
A unique opportunity to take about 50 tests per subject.
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om
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