integrated syllabus (free sample)iit foundation & olympiad explorer - mathematics class - vi...

CLASS - VI

IIT F

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ath

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ss - VI

FOUNDATION OLYMPIAD&

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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

international/national talent exams, olympiads

and all competitive exams

` 250

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COACH

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CLASS - VI

FOUNDATION & OLYMPIAD

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Published by:

Brain Mapping Academy#16–11–16/1/B, First Floor,Farhat Hospital Road,Saleem Nagar, Malakpet,Hyderabad–500 036Andhra Pradesh, India.✆ 040–65165169, 66135169E–mail: [email protected]: www.bmatalent.com

© 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.

Publisher

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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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IIT Foundation Explorer Class - VI

© 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

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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

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l part

s (i

.e

wh

ole

nu

mber

part

s).

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e

decim

al

nu

mb

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ha

vin

g

gre

ate

r in

tegra

l part

is

gre

ate

r

deci

mal n

um

ber.

(ii)

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th

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l part

s are

equ

al,

then

com

pare

th

e d

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mal

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s.

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it,

con

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er

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ths

dig

its.

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e

decim

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s d

igit

is

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r.

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the t

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dig

its

are

als

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qu

al,

then

com

pa

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dig

its

at

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s pla

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he deci

mal

nu

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havin

g g

reate

r h

un

dre

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dig

it is

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r an

d s

o o

n.

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6.3

49, 5

6.3

71

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in

tegra

l p

art

s are

equ

al

(56)

als

o t

en

ths

dig

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(3)

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we

kn

ow

7 >

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56.3

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49

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of ze

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0.0

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100

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above.

Eg. .

. 2

2

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fra

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Rem

ove t

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rite

1 in

the

den

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lso

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the

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this

, equ

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to t

he

nu

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of dig

its

in th

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eci

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.

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.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.

www.bmata

lent.c

om

(Free Sam

ple)

integrated syllabus (free sample)iit foundation & olympiad explorer - mathematics class - vi...

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