Simplification and Roots

7/21/2019 Simplification and Roots

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PREREQUISITES:

(I) SIMPLIFICATION

(i)'BODMAS' Rule:

This rule depicts the correct sequence in which the operations are to be executed, so as tofind out the value of given expression.

Here B - Bracket,O - of,D - Division,M - Multiplication,

A - Addition andS - Subtraction

Thus, in simplifying an expression, first of all the brackets must be removed, strictly inthe order (), {} and ||.

After removing the brackets, we must use the following operations strictly in the order:

(i) of (ii) Division (iii) Multiplication (iv) Addition (v) Subtraction.

(ii)Modulus of a Real Number:

Modulus of a real numbera is defined as

|a| =a, ifa > 0

-a, ifa < 0

Thus, |5| = 5 and |-5| = -(-5) = 5.

(iii)Virnaculum (or Bar):

When an expression contains Virnaculum, before applying the 'BODMAS' rule, wesimplify the expression under the Virnaculum.

(II)ROOTS

Roots" (or "radicals") are the "opposite" operation of applying exponents; you can "undo"

a power with a radical, and a radical can "undo" a power. For instance, if you square 2,

you get 4, and if you "take the square root of 4", you get 2; if you square 3, you get 9, and

if you "take the square root of 9", you get 3: Copyright Elizabeth Stapel 1999-2011

All Rights Reserved


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The " " symbol is called the "radical"symbol. (Technically, just the "check mark" part

of the symbol is the radical; the line across the top is called the "vinculum".) The

expression " " is read as "root nine", "radical nine", or "the square root of nine".

You can raise numbers to powers other than just 2; you can cube things, raise them to the

fourth power, raise them to the 100th power, and so forth. In the same way, you can take

the cube root of a number, the fourth root, the 100th root, and so forth. To indicate some

root other than a square root, you use the same radical symbol, but you insert a number

into the radical, tucking it into the "check mark" part. For instance:

The "3" in the above is the "index" of the radical; the "64" is "the argument of the

radical", also called "the radicand". Since most radicals you see are square roots, the

index is not included on square roots. While " " would be technically correct, I've

never seen it used.

a square (second) root is written as

a cube (third) root is written as

a fourth root is written as

a fifth root is written as:

You can take any counting number, square it, and end up with a nice neat number. Butthe process doesn't always work going backwards. For instance, consider , the square

root of three. There is no nice neat number that squares to 3, so cannot be simplified as

a nice whole number. You can deal with in either of two ways: If you are doing a word

problem and are trying to find, say, the rate of speed, then you would grab your calculator

and find the decimal approximation of :

Then you'd round the above value to an appropriate number of decimal places and use a

real-world unit or label, like "1.7 ft/sec". On the other hand, you may be solving a plain

old math exercise, something with no "practical" application. Then they would almostcertainly want the "exact" value, so you'd give your answer as being simply " ".

Simplifying Square-Root Terms

To simplify a square root, you "take out" anything that is a "perfect square"; that is, you

take out front anything that has two copies of the same factor:

(III)


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Note that the value of the simplified radical ispositive. While either of +2 and2 might

have been squared to get 4, "the square root of four" isdefinedto beonly the positive

option, +2. When you solve the equationx2 = 4, you are trying to findall possible values

that might have been squared to get 4. But when you are just simplifying the expression

, the ONLY answer is "2"; this positive result is called the "principal" root. (Other

roots, such as2, can be defined using graduate-school topics like "complex analysis"

and "branch functions", but you won't need that for years, if ever.)

Sometimes the argument of a radical is not a perfect square, but it may "contain" a square

amongst its factors. To simplify, you need to factor the argument and "take out" anything

that is a square; you find anything you've got a pair of inside the radical, and you move it

out front. To do this, you use the fact that you can switch between the multiplication ofroots and the root of a multiplication. In other words, radicals can be manipulated

similarly to powers:


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SOLVED PROBLEMS ON SIMPLIFICATION:

1. A man has Rs. 480 in the denominations of one-rupee notes, five-rupee notes and ten-rupeenotes. The number of notes of each denomination is equal. What is the total number of notesthat he has ?A.45 B.60C.75 D.90

Answer & Explanation

Answer: Option D

Explanation:

Let number of notes of each denomination bex.

Thenx + 5x + 10x = 480

16x = 480

x = 30.

Hence, total number of notes = 3x = 90.

2.There are two examinations rooms A and B. If 10 students are sent from A to B, then thenumber of students in each room is the same. If 20 candidates are sent from B to A, then thenumber of students in A is double the number of students in B. The number of students inroom A is:A.20 B.80C.100 D.200

Answer & ExplanationAnswer: Option C

Explanation:

Let the number of students in rooms A and B be x and y respectively.Then, x - 10 = y + 10 x - y = 20 .... (i)

and x + 20 = 2(y - 20) x - 2y = -60 .... (ii)Solving (i) and (ii) we get: x = 100 , y = 80.

The required answer A = 100.


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3. The price of 10 chairs is equal to that of 4 tables. The price of 15 chairs and 2 tables togetheris Rs. 4000. The total price of 12 chairs and 3 tables is:

A.Rs. 3500 B.Rs. 3750C.Rs. 3840 D.Rs. 3900


Answer: Option D

Explanation:

Let the cost of a chair and that of a table be Rs.x and Rs.y respectively.

Then, 10x = 4y or y =5x.2

15x + 2y = 4000

15x + 2 x5x = 4000

2

20x = 4000

x = 200.

So,y =5x 200 = 500.

2

Hence, the cost of 12 chairs and 3 tables = 12x + 3y

= Rs. (2400 + 1500)

= Rs. 3900.

4.If a - b = 3 and a2 + b2 = 29, find the value of ab.A.10 B.12C.15 D.18

Answer & ExplanationAnswer: Option AExplanation:2ab = (a2 + b2) - (a - b)2

= 29 - 9 = 20ab = 10.


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5. The price of 2 sarees and 4 shirts is Rs. 1600. With the same money one can buy 1 saree and 6shirts. If one wants to buy 12 shirts, how much shall he have to pay ?

A.Rs. 1200 B.Rs. 2400C.Rs. 4800 D.Cannot be determinedE.None of these


Answer: Option B

Explanation:

Let the price of a saree and a shirt be Rs.x and Rs.y respectively.

Then, 2x + 4y = 1600 .... (i)

andx + 6y = 1600 .... (ii)

Divide equation (i) by 2, we get the below equation.

=> x + 2y = 800. --- (iii)

Now subtract (iii) from (ii)

x + 6y = 1600 (-)x + 2y = 800----------------

4y = 800----------------

Therefore, y = 200.

Now apply value of y in (iii)

=> x + 2 x 200 = 800

=> x + 400 = 800

Therefore x = 400

Solving (i) and (ii) we getx = 400,y = 200.

Cost of 12 shirts = Rs. (12 x 200) = Rs. 2400.

6.A sum of Rs. 1360 has been divided among A, B and C such that A gets of what B gets and


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B gets of what C gets. B's share is:

A.Rs. 120 B.Rs. 160C.Rs. 240 D.Rs. 300


Answer: Option C

Explanation:

Let C's share = Rs.x

Then, B's share = Rs.x, A's share = Rs. 2xx = Rs.x4 3 4 6

x+x

+x = 13606 417x

= 136012

x =1360 x 12

= Rs. 96017

Hence, B's share = Rs.960

= Rs. 240.4

7.One-third of Rahul's savings in National Savings Certificate is equal to one-half of his

savings in Public Provident Fund. If he has Rs. 1,50,000 as total savings, how much has hesaved in Public Provident Fund ?A.Rs. 30,000 B.Rs. 50,000C.Rs. 60,000 D.Rs. 90,000

Answer & ExplanationAnswer: Option CExplanation:Let savings in N.S.C and P.P.F. be Rs. x and Rs. (150000 - x) respectively. Then,1x =

1(150000 -x)

3 2

x+x

= 750003 2

5x= 75000

6

x =75000 x 6

= 900005

Savings in Public Provident Fund = Rs. (150000 - 90000) = Rs. 600008.A fires 5 shots to B's 3 but A kills only once in 3 shots while B kills once in 2 shots. When B


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has missed 27 times, A has killed:A.30 birds B.60 birds

C.72 birds D.90 birds

Answer & ExplanationAnswer: Option AExplanation:Let the total number of shots be x. Then,

Shots fired by A =5x

8

Shots fired by B =3x8

Killing shots by A =1

of5x=

5x

3 8 24

Shots missed by B =1of

3x=

3x

2 8 16

3x= 27 orx =

27 x 16= 144.

16 3

Birds killed by A =5x

=5

x 144 = 30.24 24

9. Eight people are planning to share equally the cost of a rental car. If one person withdrawsfrom the arrangement and the others share equally the entire cost of the car, then the share ofeach of the remaining persons increased by:

A.17

B.18

C.19

D.78


Answer: Option A

Explanation:

Original share of 1 person =18

New share of 1 person =17

Increase = 1-1 =1


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7 8 56

Required fraction =(1/56)

=1

x8

=1

(1/8) 56 1 7

10. To fill a tank, 25 buckets of water is required. How many buckets of water will be required tofill the same tank if the capacity of the bucket is reduced to two-fifth of its present ?A.10 B.35C.62.5 D.Cannot be determinedE.None of these


Answer: Option C

Explanation:

Let the capacity of 1 bucket =x.

Then, the capacity of tank = 25x.

New capacity of bucket =2x5

Required number of buckets =25x

(2x/5)=

25xx5

2x

=1252

= 62.5

11.In a regular week, there are 5 working days and for each day, the working hours are 8. Aman gets Rs. 2.40 per hour for regular work and Rs. 3.20 per hours for overtime. If he earns

Rs. 432 in 4 weeks, then how many hours does he work for ?A.160 B.175C.180 D.195

Answer & ExplanationAnswer: Option BExplanation:Suppose the man works overtime for x hours.Now, working hours in 4 weeks = (5 x 8 x 4) = 160.

160 x 2.40 + x x 3.20 = 4323.20x = 432 - 384 = 48


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x = 15.Hence, total hours of work = (160 + 15) = 175.

12.Free notebooks were distributed equally among children of a class. The number ofnotebooks each child got was one-eighth of the number of children. Had the number ofchildren been half, each child would have got 16 notebooks. Total how many notebooks weredistributed ?A.256 B.432C.512 D.640E.None of these


Explanation:Let total number of children be x.

Then,x x1

=x

x 16 x = 64.8 2

Number of notebooks =1x

2 =1x 64 x 64 = 512.

8 8

13.A man has some hens and cows. If the number of heads be 48 and the number of feetequals 140, then the number of hens will be:A.22 B.23

C.24 D.26

Answer & ExplanationAnswer: Option DExplanation:Let the number of hens be x and the number of cows be y.Then, x + y = 48 .... (i)and 2x + 4y = 140 x + 2y = 70 .... (ii)

Solving (i) and (ii) we get: x = 26, y = 22.The required answer = 26.

14.(469 + 174) - (469 - 174) = ?(469 x 174)A.2 B.4C.295 D.643

Answer & ExplanationAnswer: Option BExplanation:

Given exp. =(a +b) - (a -b)ab


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=4abab

= 4 (where a = 469, b = 174.)

15.David gets on the elevator at the 11th floor of a building and rides up at the rate of 57floors per minute. At the same time, Albert gets on an elevator at the 51st floor of the samebuilding and rides down at the rate of 63 floors per minute. If they continue travelling atthese rates, then at which floor will their paths cross ?A.19 B.28C.30 D.37


Explanation:Suppose their paths cross after x minutes.Then, 11 + 57x = 51 - 63x 120x = 40

x =13

Number of floors covered by David in (1/3) min. =1x 57 = 19.

3

So, their paths cross at (11 +19) i.e., 30th floor.

SOLVED PROBLEMS ON ROOTS:

1.Simplify

There are various ways I can approach this simplification. One would be by factoring and then

taking two different square roots:


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The square root of 144 is 12.

You probably already knew that 122 = 144, so obviously the square root of 144 must be 12. But

my steps above show how you can switch back and forth between the different formats

(multiplication inside one radical, versus multiplication of two radicals) to help in the

simplification process.

2.Simplify

Neither of 24 and 6 is a square, but what happens if I multiply them inside one radical?

3.Simplify

This answer is pronounced as "five, root three". It is proper form to put the radical at the end of

the expression. Not only is " " non-standard, it is very hard to read, especially when hand-

written. And write neatly, because " " is not the same as " ".

You don't have to factor the radicand all the way down to prime numbers when simplifying. As

soon as you see a pair of factors or a perfect square, you've gone far enough.

4.Simplify

Since 72 factors as 236, and since 36 is a perfect square, then:

Since there had been only one copy of the factor 2 in the factorization 266, that left-over 2

couldn't come out of the radical and had to be left behind.

5.Simplify

Variables in a radical's argument are simplified in the same way: whatever you've got a pair of

can be taken "out front".

6.Simplify


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7.Simplify

The 12 is the product of 3 and 4, so I have a pair of 2's but a 3 left over. Also, I have two pairs of

a's; three pairs ofb's, with oneb left over; and one pair ofc's, with onec left over. So the root

simplifies as:

You are used to putting the numbers first in an algebraic expression, followed by any variables.

But for radical expressions, any variables outside the radical should go in front of the radical, as

shown above.

8.Simplify

Writing out the complete factorization would be a bore, so I'll just use what I know about

powers. The 20 factors as 45, with the 4 being a perfect square. Ther18 has nine pairs ofr's; the

s is unpaired; and thet21 has ten pairs oft's, with onetleft over. Then:

Technical point: Your textbook may tell you to "assume all variables are positive" when you

simplify. Why? The square root of the square of anegative number is not the original number.

For instance, you could start with2, square to get +4, and then take the square root (which is

definedto be thepositive root) to get +2. You plugged in a negative and ended up with a

positive. Sound familiar? It should: it's how the absolute value works: |2| = +2. Taking the

square root of the square is in fact the technical definition of the absolute value. But this

technicality can cause difficulties if you're working with values of unknown sign; that is, withvariables. The |2| is +2, but what is the sign on |x |? You can't know, because you don't know

the sign ofx itselfunless they specify that you should "assume all variables are positive", or

at least non-negative (which means "positive or zero").

Multiplying Square Roots

The first thing you'll learn to do with square roots is "simplify" terms that add or multiply roots.


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Simplifying multiplied radicals is pretty simple. We use the fact that the product of two radicals

is the same as the radical of the product, and vice versa.

9.Write as the product of two radicals:

Copyright Elizabeth Stapel 1999-2011 All Rights Reserved

Okay, so that manipulation wasn't very useful. But working in the other direction can be helpful:

10.Simplify by writing with no more than one radical:



The process works the same way when variables are included:


Just as with "regular" numbers, square roots can be added together. But you might not be able to

simplify the addition all the way down to one number. Just as "you can't add apples and

oranges", so also you cannot combine "unlike" radicals. To add radical terms together, they have

to have the same radical part.

14.Simplify:


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Since the radical is the same in each term (namely, the square root of three), I can combine the

terms. I have two copies of the radical, added to another three copies. This gives me five copies:

That middle step, with the parentheses, shows the reasoning that justifies the final answer. You

probably won't ever need to "show" this step, but it's what should be going through your mind.

15.Simplify:

The radical part is the same in each term, so I can do this addition. To help me keep track that the

first term means "one copy of the square root of three", I'll insert the "understood" "1":

Don't assume that expressions with unlike radicals cannot be simplified. It is possible that,after

simplifying the radicals, the expression can indeed be simplified.

EXERCISEPROBLEMS WITH SOLOUTION:

. The sum of the smallest six-digit number and the greatest five-digit number is:

a. 199999 b. 201110 c. 211110 d. 1099999

Answer with Explanation:

The smallest six digit number = 1,00,000

The greatest five digit number = 99,999


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Therefore the sum of the smallest = 1,00,000 + 99,999

six digit number and the greatestfive digit number = 1,99,999

2. Which of the following has most number of divisors?

a. 99 b. 101 c. 176 d. 182


Divisiors of 99 = 1, 3, 9, 11, 33, 99

Divisors of 101 = 1,101

Divisors of 176 = 1, 2, 4, 8, 11, 22, 44, 88, 176

Divisors of 182 = 1, 2, 7, 13, 14, 26, 91, 182

Therefore 176 has most number of divisors.

3. Which of the following has fractions in ascending order?

a.9

8,

11

9,

9

7,

5

3,

3

1b.

9

8,

9

7,

11

9,

3

2,

5

3c.

5

3,

3

2,

9

7,

11

9,

9

8d.

3

2,

5

3,

9

7,

11

9,

9

8


LCM of (1,3,7,9,8) = 1512 and

LCM of (2,3,7,8,9) = 3024

Therefore For (a) we have 4536

1512, 2515

1512, Not in ascending order

For (b) we have5040

3024,4536

3024, Not in ascending order

For (c) we have3402

3024,3696

3024,3888

3024,4536

3024,5040

3024

The fractions are in ascending order.


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For (d), we have3402

3024,3696

3024,3888

3024,5040

3024,4536

3024

Not in ascending order.

Therefore the answer is (c)

4. If one-third of one-fourth of a number is 15, then three-tenth of that number is:

a. 35 b. 36 c. 45 d. 54


Given one third of one fourth of a number (x) is 15.

(3

1(4

1xx)) = 15

x = 15 x 12

x = 180

Three benth ofx =10

3xx

=10

3x 180 = 54

5. (8 88) 8888088 =?

a. 808008 b. 808080 c. 808088 d. 8008008


88

8x 8888088 =

11

8888088

= 808008

Note :


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808008 x 11 = 8888088

6. The value of 2251541082510 is:

a. 4 b. 5 c. 8 d. 10


2251541082510

= 151541082510 Therefore 225 = 15

= 1691082510 Therefore 169 =13

= 1212510 Therefore 121 = 11

= 112510 Therefore 36 = 6

= 610

= 16

=4

7. 9548 + 7314 = 8362 + ?

a. 8230 b. 8410 c. 8500 d. 8600


9548 + 7314 = 9548

7314

___________

16862


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(-) 8362

___________8500

____________

Ans = 8500

(i.e) 9548 + 7314 = 8362 + 8500

8. The H.C.F of 24 33 55 24, 23 32 52 7 and 24 34 5 72 11 is

a. 22

32

5 b. 22 32 5 7 11 c. 24 34 54 d. 24 3455777


Among the three factors

HCF of 24 x 24, 23, 24 is = 23

HCF of 33, 32, 34 is = 32

HCF of 55, 52, 5 = 5

7 & 11 are not in three terms

So we leave the values 7, 72, 11

HCF of three terms = 23 x 32 x 5

9. What is the difference between the biggest and the smallest fraction among6

5

5

4,

4

3,

3

2and ?

a.6

1b.

12

1c.

20

1d.

30

1



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The biggest among3

2,4

3,5

4,6

5is =

6

5

The Smallest among3

2,

4

3,

5

4,

6

5is =

3

2

The difference between the biggest and the smallest fraction =6

5-

3

2

=18

1215

=18

3

=6

1

10. Find the number which when multiplied by 15 increased by 196.

a. 14 b. 20 c. 26 d. 28


15 x 14 = 210

Which is increased by 196 21014

196 since we multiply it by 15

So the number is 14

11. ?51466123

861844

a. 1 b. 2 c. 6.65 d. 7.75


51466123

86)184(4

xx

x

=

)5146()6123(

86724

xx


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=730738

86724

=8

62

= 7.75

12. In the following sum, ? stands for which digit?

? + 1? + 2? + ?3 + ?1 = 21?

a. 4 b. 6 c. 8 d. 9


8+81+28+83+81 = 218

Answer is 8

13. Which of the following is a pair of co-primes?

a. (16, 62) b. (18, 25) c. (21, 35) d. (23, 92)

Two numbers are said to be co-prime if their H.C.F is 1.


H.C.F of 16, 62 is 2

H.C.F of 18, 25 is 1

H.C.F. of 21, 35 is 7

Co-prime = (18, 25).

14. 34.95 + 240.016 + 23.98 = ?

a. 298.0946 b. 298.111 c. 298.946 d. 299.09

Ordinary Addition


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34.95

240.016

23.98 (+)

_________________

298.946

_________________

15. If the sum of one-half and one-fifth of a number exceeds one-third of that

number by3

17 , the number is:

a. 15 b. 18 c. 20 d. 30


Let the unknown number bex

2

1x +

5

1x(7

3

1) =

3

1x

2

x+

5

x-

3

22=

3

x

2

x+

5

x-

3

x=

3

22

30

10615 xxx =

3

22

30

11x=

3

22


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9

1

81

25 =

81

925

=81

16= 4/9

18. The value of 112 54 is:

a. 6700 b. 70000 c. 76500 d. 77200


54 = 5 x 5 x 5 x 5

= 25 x 25

= 625

112 x 54 = 112 x 625

_______________

560

224

672

________________

70000

_________________

The Answer is 70000

19. The L.C.M of 23 32 5 11, 24 34 52 7 and 25 33 53 72 11 is :


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5625.1 = 1.25

31. If -1 < x < 2 and 1 < x < 3, then least possible value of (2y3x) is:

a. 0 b. -3 c. -4 d. -5


-1 x 2 1 y 3

-3 3x 6 2 2y 6

3 -3x -6

i.e. -6 -3x 3

-4 2y - 3x 9

The least possible value of

(2y3x) is -4.

32. The G.C.D. of 1.08, 0.36 and 0.9 is

a. 0.03 b. 0.9 c. 0.18 d. 0.108


0.2 1.08, 0.36, 0.9 (or) 2 108, 36, 90

3 54, 18, 45

0.9 5.4, 1.8, 4.5 3 18, 6, 15

6, 2, 5 6, 2, 5

G.C.D = 0.2 x 0.9 = 0.18 2 x 3 x 3 =100

18= 0.18


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Basic Formula:

(x + y)2

= (x-y)2

+ 4xy


Givenx +y = 25 & x-y = 13

(x + y)2 = (x-y)2 + 4xy

252 = 132 + 4xy

4xy = 252 - 132

=625-169

4xy = 456

xy =4

456

xy = 114

40. On simplification, 3034(100220.04) is equal to:

a. 2543 b. 2984 c. 2993 d. 3029


3034(1002 -20.04)

= 3034 -04.20

1002

= 30341002 x 100

20.04 x 100

= 30341002 x 100

2004


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Since we have the sum of multiples of 33 are 528, we will combine these multiples as to get their

sum as 528.

we get (33, 495) (66, 462) (99, 429) (132, 396) (165, 363) (198, 330)

(231,297)

But we have a condition that HCF of (x, y) = 33 Eliminating the ones which does not satisfies

the condition we get (33, 495) (99, 429) (165, 363) (231, 297)

The number of pairs of numbers satisfying the above condition is 4 .

44. 1.0?

009.

a. .0009 b. 009 c. 9 d. 9


Letx

009.= 0.1

x =1.0

009.

x =1.0

009.x

10

10

x =1

09.0

x = 0.09

45. Two numbers differ by 5. If their product is 336, then the sum of the two numbers is:

a. 21 b. 28 c. 37 d. 51

Basic Formula:

(x+y)2 = (x-y)2 + 4xy


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Givenx-y = 5 andxy =336

We know that

(x+y)2 = (x-y)2 + 4xy

= 52 +4 (336)

= 25+1344

(x+y)2

= 1369

x+y = 37

46. 3[1.6{3.2(3.2 + 2.25 X)}] = .65. The value of X is:

a. 0.3 b. 0.7 c. 3 d. 7


Given 3-[1.6-{3.2-(3.2+2.25 x)}] = 0.65

-[1.6-{3.2-(3.2+2.25x)}] = 0.65-3

-1.6+{3.2-(3.2+2.25x)} = 0.65-3

3.2- (3.2+2.25x) = - 0.75

- 3.2-2.25x = -0.75-3.2

-2.25x = -0.75-3/2 + 3/2

-2.25x = - 0.75

x

25.2= -0.75

x =75.0

25.2


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x = 3

47. If 02.0441 x , then the value of x is:

a. 0.1764 b. 1.764 c. 1.64 d. 2.64


x 441 = 0.02

441

x= 0.02

Squaring on both sides

441

x= (0.02)2

441

x= 0.004

x = 0.004 x 441

x = 0.1764

48. In dividing a number by 585, a student employees the method of short division. He divided

the number successively by 5, 9 and 13 (factors of 585) and got the remainders 4, 8 and 12. If he

had divided the number by 585, the remainder would have been:

a. 24 b. 144 c. 292 d. 584

Basic Formula:

x

x+(x-1) =x where (x-1) is remainder here.


Given a unknown numberx when divided by 5,9,13. We get remainders 4, 8 &12 .


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(i.e.)5

x+4 =x x = 5 & 4 =x-1 (remainder)

9

x+8 =x x = 9 & 8 =x-1 (remainder)

13

x+12 =x x =13 & 12 =x-1 (remainder)

IIIly

585

x+(585-1) =x (i.e.)

585

x+(x-1) =x x-1 = 584 (remainder)

The remainder would have been 584.

49. The least number which when divided by 5, 6, 7 and 8 leaves a remainder 3, but when

divided by 9 leaves no remainder, is:

a. 1677 b. 1683 c. 2523 d. 3363


336 28 24 21 187

5 1683 6 1683 7 1683 8 1683 9 1683

15 12 14 16 9

18 48 28 8 78

15 48 28 8 72


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33 3 3 3 63

30 63

3 0

Ans: L.C.M of 5,6,7,8 is 840

Let the required number be 840k+3,

Which is divisible by 9.

Least value of k for which (90k+4)

is divisible by 9 is k=2

840 x 2 +3 = 1680 +3

= 1683

50. The value of15.009.05.05.0

027.0125.0

is:

a. 0.08 b. 0.2 c. 0.8 d. 1

Basic Formula:

22

33

baba

ba

=

22

22 ))((

baba

bababa

= a+b


15.009.05.05.0

027.0125.0

x=

22

33

)3.0(3.05.0)5.0(

)3.0()5.0(

x

=22

22

)3.0(3.05.0)5.0(

])3.0(3.05.0)5.0)[(3.05.0(

x

x


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= 0.5 + 0.3

= 0.8

51.Simplify:

To simplify a radical addition, I must first see if I can simplify each radical term. In this

particular case, the square roots simplify "completely" (that is, down to whole numbers):

52.Simplify:

I have three copies of the radical, plus another two copies, giving meWait a minute! I cansimplify those radicals right down to whole numbers:

Don't worry if you don't see a simplification right away. If I hadn't noticed until the end that the

radical simplified, my steps would have been different, but my final answer would have been the

same:

53.Simplify:

I can only combine the "like" radicals, so I'll end up with two terms in my answer:

There is not, to my knowledge, any preferred ordering of terms in this sort of expression, so the

expression should also be an acceptable answer.

54.Simplify: Copyright Elizabeth Stapel 1999-2011 All Rights Reserved

I can simplify the radical in the first term, and this will create "like" terms:

55.Simplify:


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63.Simplify:

64.Simplify:

This looks very similar to the previous exercise, but this is the "wrong" answer. Why? Because

the denominator contains a radical. The denominator must contain no radicals, or else it's

"wrong". (Why "wrong" in quotes? Because this issue may matter to your instructor right now,

but it probably won't later on. It's like when you were in elementary school and improper

fractions were "wrong" and you had to convert everything to mixed numbers instead. But now

that you're in algebra, improper fractions are fine, even preferred. Once you get to calculus or

beyond, they won't be so uptight about where the radicals are.)

To get the "right" answer, I must "rationalize" the denominator. That is, I must find some way to

convert the fraction into a form where the denominator has only "rational" (fractional or whole

number) values. But what can I do with that radical-three? I can't take the 3 out, because I don't

have a pair of threes.

Thinking back to those elementary-school fractions, you couldn't add them unless they had the

same denominators. To create these "common" denominators, you would multiply, top and

bottom, by whatever the denominator needed. Anything divided by itself is just 1, and

multiplying by 1 doesn't change the value of whatever you're multiplying by the 1. But

multiplying that "whatever" by a strategic form of 1 could make the necessary computations

possible, such as:

We can use the same technique to rationalize radical denominators.

I could take a 3 out of the denominator if I had two factors of 3 inside the radical. I can create

this pair of 3's by multiplying by another copy of root-three. If I multiply top and bottom by root-


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This multiplication made the radical terms cancel out, which is exactly what I want. This "same

numbers but the opposite sign in the middle" thing is the "conjugate" of the original expression.

By using the conjugate, I can do the necessary rationalization.

Do not try to reach inside the numerator and rip out the 6 for "cancellation". The only thing that

factors out of the numerator is a 3, but that won't cancel with the 2 in the denominator. Nothing

cancels!

67.Simplify:

I'll multiply by the conjugate in order to "simplify" this expression. The denominator's

multiplication results in a whole number (okay, a negative, but the point is that there aren't any

radicals):

The numerator's multiplication looks like this:


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Then the simplified (rationalized) form is:

It can be helpful to do the multiplications separately, as shown above. Don't try to do too much at

once, and make sure to check for any simplifications when you're done with the rationalization

Operations with cube roots, fourth roots, and other higher-index roots work similarly to squareroots.

Simplifying Higher-Index Terms

68.Simplify

Just as I can pull from a square (or second) root anything that I have two copies of, so also I can

pull from a fourth root anything I've got four of:

If you have a cube root, you can take out any factor that occurs in threes; in a fourth root, take

out any factor that occurs in fours; in a fifth root, take out any factor that occurs in fives; etc.

69.Simplify the cube root:

70.Simplify the cube root:

71.Simplify: Copyright Elizabeth Stapel 1999-2011 All Rights Reserved

72.Simplify:


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QUESTION BANK PROBLEMS:

I. Among the two numbers, the bigger number is greater than the smaller number by 6

II. 40% of the smaller number is equal to 30% of the bigger number

III. The ratio between half of the bigger number and one-third of the smaller number is 2 : 1

a. I and II only b. II and III only c. All I, II and III d. None of these

2. The taxi charges in a city comprise of a fixed charge, together with the charge of the distance

covered. For a journey of 16 km, the charges paid are Rs.156 and for a journey of 24 km, the

charges paid are Rs.204. What will a person have to pay for traveling a distance of 30 km?

a. Rs.236 b. Rs.240 c. Rs.248 d. Rs.252

3. ?125

1243

a.521 b.

531 c.

541 d.

522

4. A number when divided by 114 leaves the remainder 21. If the same number is divided by 19,

then the remainder will be:

a. 1 b. 2 c. 7 d. 21

5. The least multiple of 7, which leaves a remainder of 4, when divided by 6, 9, 15 and 18 is:

a. 74 b. 94 c. 184 d. 364

6. The value of09.69.3.2

027.3.22

3

is:

a. 0 b. 1.6 c. 2 d. 3.4


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7. What is the two-digit number?

I. Sum of the digits is 7.

II. Difference between the number and the number obtained by interchanging the digits is 9.

III. Digit in the tens place is bigger than the digit in the units place by 1.

a. I & II only b. II & III only c. All I, II & III d. None of these

8. In and examination, a student scores 4 marks for every correct answer and loses 1 mark for

every wrong answer. If he attempts in all 60 questions and secures 130 marks, the number of

questions he attempts correctly, is:

a. 35 b. 38 c. 40 d. 42

9. ?503212235

63

a. 3 b. 23 c. 6 d. None of these

10. On dividing a number by 999, the quotient is 366 and the remainder is 103. The number is:

a. 364724 b. 365387 c. 365737 d. 366757

11. The smallest number which when diminished by 7, is divisible by 12, 16, 18, 21 and 28 is:

a. 1008 b. 1015 c. 1022 d. 1032

12. )5.25.25.375.75.7( is equal to:

a. 30 b. 60 c. 80 d. 100

13. 54 is to be divided into two parts such that the sum of 10 times the first and 22 times the

second is 780. The bigger part is:

a. 24 b. 34 c. 30 d. 32


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a. 5 b. 6 c. 15 d. 30

22. If x and y are positive integers such that (3x+7y) is a multiple of 11, then which of the

following will also be divisible by 11?

a. 4x + 6y b. x + y + 4 c. 9x + 4y d. 4x9y

23. Which of the following fraction is the largest?

a.8

7b.

16

13c.

40

31d.

80

63

24. The value of (4.7 13.26 + 4.7 9.43 + 4.7 7731) is:

a. 0.47 b. 47 c. 470 d. 4700

25. The denominator of a fraction is 3 more than the numerator. If the numerator as well as the

denominator is increased by 4, the fraction becomes5

4. What was the original fraction?

a. 11

8

b. 8

5

c. 13

10

d. 10

7

26. A sum of Rs.1360 has been divided among A B and C such that A gets3

2of what B gets and

B gets4

1of what C gets. Bs share is:

a. Rs.120 b. Rs.160 c. Rs.240 d. Rs.300

27. if 5 = 2.236, then the value of 1255

10

2

5 is equal to :

a. 5.59 b. 7.826 c. 98.994 d. 10.062

28. The sum of three consecutive odd numbers is always divisible by:

I. 2 II. 3 III. 5 IV. 6

a. Only I b. Only II c. III. Only I & II d. Only II and IV


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29. A rectangular courtyard 3.78 metres long and 5.25 metres wide is to be paved exactly with

square tiles, all of the same size. What is the largest size of the tile which could be used for the

purpose?

a. 14 cms b. 21 cms c. 42 cms d. None of these

30. ?7.05.140073.0

92.20203.0

a. 0.8 b. 1.45 c. 2.40 d. 3.25

31. The different between a tow-digit number and the number obtained by interchanging the

digits is 36. What is the difference between the sum and the difference of the digits of the

number if the ratio between the digits of the number is 1 : 2 ?

a. 4 b. 8 c. 16 d. 18

32. A person travels 3.5 km form place A to place B. Out of this distance, he travels3

21 km on

bicycle,

6

11 km on scooter and the rest on foot. What portion of the whole distance does he

cover on foot?

a.19

3b.

11

4c.

21

4d.

6

5

33. The value of1.01

01.01

is close to:

a. 0.6 b. 1.1 c. 1.6 d. 1.7

34. The digits indicated * and $ in 3422213*$ so that this number is divisible by 99, are

respectively:

a. 1, 9 b. 3, 7 c. 4, 6 d. 5, 5

35. The greatest possible length which can be used to measure exactly the lengths 7 m, 3 m 85

cm, 12 m 95 cm is :


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a. 15 cm b. 25 cm c. 35 cm d. 42 cm

36. If 1.5 x = 0.04y, then the value of

xy

xyis:

a.77

730b.

77

73c.

77

3.7d. None of these

37. The difference between a two-digit number and the number obtained by interchanging the

two digits is 63. Which is the smaller of the two numbers?

a. 29 b. 70 c. 92 d. None of these

38. A pineapple costs Rs.7 each. A watermelon costs Rs.5 each. X spends Rs.38 on these fruits.

The number f pineapples and purchased is:

a. 2 b. 3 c. 4 d. 5

39. Which one of the following numbers has rational square root?

a. 0.4 b. 0.09 c. 0.9 d. 0.025

Ans is 0.09

40. Which of the following numbers is divisible by 3, 7, 9 and 11?

a. 639 b. 2079 c. 3791 d. 37911

41. About the number of pairs which have 16 as their H.C.F. and 136 as their L.C.M., we can

definitely say that:

a. No such pair exists b. Only one such pair exists

c. Only two such pairs exist d. Many such pairs exist

42. ?963.8654.9

3.8964.965

63.8954.96

63.8954.96

a. 10-2 b. 10-1 c. 10 d. None of these


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Simplification and Roots