Reasoning and Quantitative aptitude Simplification & Roots

PREREQUISITES: 

(I) SIMPLIFICATION 

(i)'BODMAS' Rule: 

This rule depicts the correct sequence in which the operations are to be executed, so as to 

find out the value of given expression. 

Here B - Bracket, 

O - of, 

D - Division, 

M - Multiplication, 

A - Addition and 

S - Subtraction 

Thus, in simplifying an expression, first of all the brackets must be removed, strictly in 

the 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 number a is defined as 

|a| = 

a, if a > 0 

-a, if a < 0 

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

(iii)Virnaculum (or Bar): 

When an expression contains Virnaculum, before applying the 'BODMAS' rule, we 

simplify 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 

Page 1

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 " 

never seen it used. 

a square (second) root is written as 

a cube (third) root is written as 

a fourth root is written as 

" would be technically correct, I've 

a fifth root is written as: 

You can take any counting number, square it, and end up with a nice neat number. But 

the process doesn't always work going backwards. For instance, consider 

root of three. There is no nice neat number that squares to 3, so 

, the square 

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 almost 

certainly want the "exact" value, so you'd give your answer as being simply " ". 

 

(III) 

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: 

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Note that the value of the simplified radical is positive. While either of +2 and –2 might 

have been squared to get 4, "the square root of four" is defined to be only the positive 

option, +2. When you solve the equation x 2 = 4, you are trying to find all 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 as –2, 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 of 

roots and the root of a multiplication. In other words, radicals can be manipulated 

similarly to powers: 

Page 3

SOLVED PROBLEMS ON SIMPLIFICATION: 

1. A man has Rs. 480 in the denominations of one-rupee notes, five-rupee notes and ten-rupee 

notes. The number of notes of each denomination is equal. What is the total number of notes 

that he has ? 

A.45 B.60 

C.75 D.90 

Answer & Explanation 

Answer: Option D 

Explanation: 

Let number of notes of each denomination be x. 

Then x + 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 the 

number of students in each room is the same. If 20 candidates are sent from B to A, then the 

number of students in A is double the number of students in B. The number of students in 

room A is: 

A.20 B.80 

C.100 D.200 


Answer: 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 together 

is Rs. 4000. The total price of 12 chairs and 3 tables is: 

A.Rs. 3500 B.Rs. 3750 

C.Rs. 3840 D.Rs. 3900 



Explanation: 

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

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

2 

15x + 2y = 4000 

15x + 2 x 5 x = 4000 

2 

20x = 4000 

x = 200. 

So, y = 5 x 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.12 

C.15 D.18 


Answer: Option A 

Explanation: 

2ab = (a2 + b2) - (a - b)2 

= 29 - 9 = 20 

ab = 10. 

Page 5

5. The price of 2 sarees and 4 shirts is Rs. 1600. With the same money one can buy 1 saree and 6 

shirts. If one wants to buy 12 shirts, how much shall he have to pay ? 

A.Rs. 1200 B.Rs. 2400 

C.Rs. 4800 

D.Cannot be determined 

E.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) 

and x + 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 get x = 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 

Page 6

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

A.Rs. 120 B.Rs. 160 

C.Rs. 240 D.Rs. 300 



Explanation: 

Let C's share = Rs. x 

Then, B's share = Rs. x 2 x x 

, A's share = Rs. x = Rs. 

4 3 4 6 

x x + + x = 1360 

6 4 

17x = 1360 

12 

x = 1360 x 12 = Rs. 960 

17 

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 he 

saved in Public Provident Fund ? 

A.Rs. 30,000 B.Rs. 50,000 

C.Rs. 60,000 D.Rs. 90,000 



Explanation: 

Let savings in N.S.C and P.P.F. be Rs. x and Rs. (150000 - x) respectively. Then, 

1 x = 

1 (150000 - x) 

3 2 

x + 

x = 75000 

3 2 

5x = 75000 

6 

x = 75000 x 6 = 90000 

5 

Savings in Public Provident Fund = Rs. (150000 - 90000) = Rs. 60000 

8.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 

Page 7

has missed 27 times, A has killed: 

A.30 birds B.60 birds 

C.72 birds D.90 birds 



Explanation: 

Let the total number of shots be x. Then, 

Shots fired by A = 5 x 

8 

Shots fired by B = 3 x 

8 

Killing shots by A = 1 of 5 x= 5 x 

3 8 24 

Shots missed by B = 1 of 3 x= 3 x 

2 8 16 

3x = 27 or x = 

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 withdraws 

from the arrangement and the others share equally the entire cost of the car, then the share of 

each of the remaining persons increased by: 

A. 1 7 

C. 1 9 

B. 1 8 

D. 7 8 



Explanation: 

Original share of 1 person = 1 8 

New share of 1 person = 1 7 

Increase = 1-1 =1 

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7 8 56 

Required fraction = (1/56) = 1 x 8 = 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 to 

fill the same tank if the capacity of the bucket is reduced to two-fifth of its present ? 

A.10 B.35 

C.62.5 

D.Cannot be determined 




Explanation: 

Let the capacity of 1 bucket = x. 

Then, the capacity of tank = 25x. 

New capacity of bucket = 2 x 

5 

Required number of buckets = 25x 

(2x/5) 

= 25x 

x 5 2x 

= 125 

2 

= 62.5 

11.In a regular week, there are 5 working days and for each day, the working hours are 8. A 

man 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.175 

C.180 D.195 



Explanation: 

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 = 432 

3.20x = 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 of 

notebooks each child got was one-eighth of the number of children. Had the number of 

children been half, each child would have got 16 notebooks. Total how many notebooks were 

distributed ? 

A.256 B.432 

C.512 D.640 




Explanation: 

Let total number of children be x. 

Then, x x 1 x = x x 16 x = 64. 

8 2 

Number of notebooks = 1 x 2 = 1 x 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 feet 

equals 140, then the number of hens will be: 

A.22 B.23 

C.24 D.26 



Explanation: 

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) 2 - (469 - 174) 2 = ? 

(469 x 174) 

A.2 B.4 

C.295 D.643 



Explanation: 

Given exp. = (a + b)2 - (a - b) 2 

ab 

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= 4ab 

ab 

= 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 57 

floors per minute. At the same time, Albert gets on an elevator at the 51st floor of the same 

building and rides down at the rate of 63 floors per minute. If they continue travelling at 

these rates, then at which floor will their paths cross ? 

A.19 B.28 

C.30 D.37 



Explanation: 

Suppose their paths cross after x minutes. 

Then, 11 + 57x = 51 - 63x 120x = 40 

x = 1 3 

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

Page 11

The square root of 144 is 12. 

You probably already knew that 12 2 = 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 handwritten. 

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 2×36, and since 36 is a perfect square, then: 

Since there had been only one copy of the factor 2 in the factorization 2×6×6, 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 

Page 12

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 of b's, with one b left over; and one pair of c's, with one c 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 4×5, with the 4 being a perfect square. The r 18 has nine pairs of r's; the 

s is unpaired; and the t 21 has ten pairs of t's, with one t left 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 a negative number is not the original number. 

For instance, you could start with –2, square to get +4, and then take the square root (which is 

defined to be the positive 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, with 

variables. The |–2| is +2, but what is the sign on | x |? You can't know, because you don't know 

the sign of x itself — unless 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. 

Page 13

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: 

Page 14

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 

Page 15

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

six digit number and the greatest 

five 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. 

1 

, 

3 

3 7 9 8 

, , , 

5 9 11 9 

b. 

3 

, 

5 

2 

, 

3 

9 

11 

7 

, , 

9 

8 

9 

c. 

8 

, 

9 

9 

11 

7 2 

, , , 

9 3 

3 

5 

d. 

8 

, 

9 

9 

11 

7 3 

, , , 

9 5 

2 

3 


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

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

1512 1512 

Therefore For (a) we have , , Not in ascending order 

4536 2515 

3024 3024 

For (b) we have , , Not in ascending order 

5040 4536 

3024 3024 3024 3024 3024 

For (c) we have , , , , 

3402 3696 3888 4536 5040 

The fractions are in ascending order. 

Page 16

3024 3024 3024 3024 3024 

For (d), we have , , , , 3402 3696 3888 5040 4536 

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 

1 x x)) = 15 

x = 15 x 12 

x = 180 

Three benth of x = 10 

3 x x 

= 

10 

3 x 180 = 54 

5. (8 ÷ 88) × 8888088 =? 

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


8 8888088 

x 8888088 = 

88 

11 

= 808008 

Note : 

Page 17

808008 x 11 = 8888088 

6. The value of 10 25 108 154 225 

is: 

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


10 25 108 154 

225 

= 10 25 108 154 15 

Therefore 225 = 15 

= 10 25 108 169 

Therefore 169 =13 

= 10 25 121 


= 10 25 11 


= 10 6 

= 16 

=4 

7. 9548 + 7314 = 8362 + ? 

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


9548 + 7314 = 9548 

7314 

___________ 

16862 

Page 18

(-) 8362 

___________ 

8500 

____________ 

Ans = 8500 

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

8. The H.C.F of 2 4 × 3 3 × 5 5 × 2 4 , 2 3 × 3 2 × 5 2 × 7 and 2 4 × 3 4 × 5 × 7 2 × 11 is 

a. 2 2 ×3 2 × 5 b. 2 2 ×3 2 ×5 ×7 ×11 c. 2 4 ×3 4 ×5 4 d. 2 4 ×3 4 ×5 5 ×7×77 


Among the three factors 

HCF of 2 4 x 2 4 , 2 3 , 2 4 is = 2 3 

HCF of 3 3 , 3 2 , 3 4 is = 3 2 

HCF of 5 5 , 5 2 , 5 = 5 

7 & 11 are not in three terms 

So we leave the values 7, 7 2 , 11 

HCF of three terms = 2 3 x 3 2 x 5 

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

2 

, 

3 

3 4 5 

, and ? 

4 5 6 

a. 6 

1 

b. 12 

1 

1 

c. 20 

d. 30 

1 


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The biggest among 3 

2 , 4 

3 , 5 

4 , 6 

5 is = 6 

5 

The Smallest among 3 

2 , 4 

3 , 5 

4 , 6 

5 is = 3 

2 

5 2 

The difference between the biggest and the smallest fraction = - 

6 3 

= 

15 12 

18 

= 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 196 210 – 14 

196 since we multiply it by 15 

So the number is 14 

4 4 18 

6 8 

11. ? 

123 

6 146 

5 

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


4 (4x18) 

6 8 

123x6 

146x5 

= 

4 72 6 8 

(123x6) 

(146x5) 

Page 20

= 

4 72 6 8 

738 730 

= 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 

Page 21


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 by 3 

1 

7 , the number is: 

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


Let the unknown number be x 

1 1 1 1 x + x – (7 ) = x 

2 5 3 3 

x x 22 x 

+ - = 

2 5 3 3 

x x x 22 

+ - = 

2 5 3 3 

15x 

6x 

10x 

30 

= 

3 

22 

11x 22 = 

30 3 

Page 22

x 

10 

= 2 

x = 20 

16. 

1 1 is equal to: 

1 3 

2 1 

3 4 

a. 14 

7 


1 1 + 

1 3 

2 1 

3 4 

= 

1 1 + 

7 7 

3 4 

= 

7 

3 + 

7 

4 

12 

b. 49 

c. 

1 

1 d. None of these 

12 

= 7 

7 

= 1 

The Answer is None of these. 

25 1 

17. ? 

81 9 

a. 3 

2 

b. 9 

4 

16 

c. 81 

25 

d. 81 

Page 23


25 1 

25 9 

= 

81 9 

81 

= 

16 

81 

= 4/9 

18. The value of 112 × 5 4 is: 

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


5 4 = 5 x 5 x 5 x 5 

= 25 x 25 

= 625 

112 x 5 4 = 112 x 625 

_______________ 

560 

224 

672 

________________ 

70000 

_________________ 

The Answer is 70000 

19. The L.C.M of 2 3 × 3 2 × 5 × 11, 2 4 ×3 4 × 5 2 × 7 and 2 5 × 3 3 × 5 3 × 7 2 × 11 is : 

Page 24

a. 2 3 × 3 2 × 5 b. 2 5 ×3 4 × 5 3 

c. 2 3 ×3 2 × 5 × 7 × 11 d. 2 5 ×3 4 × 5 3 ×7 2 × 11 


Since the multipliers are all in multiplication we can take the L.C.M. of each number separately. 

LCM of 2 3 , 2 4 , 2 5 = 2 5 

LCM of 3 2 , 3 4 , 3 3 = 3 4 

LCM of 5, 5 2 , 5 3 = 5 3 

LCM of 7, 7 2 = 7 2 

LCM of 11, 11 = 11 

LCM of 2 3 x 3 2 x 5 x 11, 2 4 x 3 4 x 5 2 x 7, 

2 5 x 3 3 x 5 3 x 7 2 x 11 is 

= 2 5 x 3 4 x 5 3 x 7 2 x 11 

20. 12.1212 + 17.0005 – 9.1102 = ? 

a. 20.0015 b. 20.0105 c. 20.0115 d. 20.1015 


12.1212 + 

17.0005 

______________ 

29.1217 

9.1102 

______________ 

20.0115 

_____________ 

Page 25

21. The sum of a number and its reciprocal is one-eight of 34. What is the product of the number 

and its square root? 

a. 8 b. 27 c. 32 d. None of these 


Let the unknown number be x 

The sum of a number x & its reciprocal is 8 

1 (34) 

2 

1 1 - b b - 4ac 

Given x + = (34) x = 

x 8 2a 

x 2 1 17 = 

x 4 

a = 4, b= -17, c=4 

4x 2 – 17x + 4 = 0 x = 

17 289 64 

8 

Taking x = 4 = 

17 225 

8 

The product of x & it’s square root = x x x = 

17 15 

8 

= 4 x 4 = 

17 15 

, 

8 

17 15 

8 

= 4 x 2 = x = 4, 

1 

4 

= 8 

22. 8 

3 of 168 × 15 ÷ 5 + ? = 549 ÷ 9 + 235 

Page 26

a. 107 b. 174 c. 189 d. 296 


549 

R.H.S. = + 235 9 

= 61 + 235 

R.H.S. = 296 

3 15 

L.H.S. = ( x168) x + x Where x has to be found 

8 5 

L.H.S. = 189 = x 

L.H.S. = R.H.S. 189 + x = 296 

x = 296 – 189 

x = 107 

23. 0. 

00004761 equals: 

a. 0.00069 b. 0.0069 c. 0.0609 d. 0.069 


0.00004761x10 

0 .00004761 = 

8 

10 

8 

4761 

= 

8 

10 

( 4761 = 69 ) 

2 

69 

(10 ) 

= 

4 2 

Page 27

69 

= 

4 

10 

= 0.0069 

24. The average of five consecutive odd numbers is 61. What is the difference between of these 

highest and lowest numbers? 



Let the Consecutive odd numbers be 

x, x +2, x +4, x +6, x +8 

Given 

x x 2 x 4 x 6 x 8 

5 

= 61 

5x 

20 

= 61 

5 

x+4 =61 

x = 57 

The numbers are 57, 59, 61, 63, 65 

Difference between the highest and lowest number is 65 – 57 = 8. 

25. A positive integer, which when added to 1000, gives a sum which is greater than when it is 

multiplies by 1000. This positive integer is: 

a. 1 b. 3 c. 5 d. 7 


Let us take the positive integes as x. 

Page 28

Then x+1000 1000 x (Given) 

1000 x x+1000 

1000 x- x 1000 

(1000-1) x 1000 

999 x 1000 

x 

1000 

999 

x 1.00 

x = 1 

26. The H.C.F of 

9 12 

, 

10 25 

18 

, , and 

35 

21 

40 

is: 

a. 5 

3 

b. 

252 

5 

3 

c. 2800 

63 

d. 700 


The numerators are : 9, 12, 18, 21. 

3, 9, 12, 18, 21 

3, 4, 6, 7 Here common divisor = 3 

Page 29

The denominators are : 10, 25, 35, 40 

5 10, 25, 35, 40 

2, 5, 7, 8 

5 x 2 x 5 x 7 x 8 = 2800 

3 

H.C.F = 2800 

27. Which of the following is equal to 3.14 × 10 6 ? 



3.14 x 10 6 

= 3.14 x 10,00,000 

= 31,40,000 

28. The product of two natural numbers is 17. Then, the sum of the reciprocals of their squares 

is: 

1 

a. 189 

289 

b. 290 

290 

c. 289 

d. 289 


Let the take the natural numbers as 1 and 17. 

Given that, xy = 17 where x = 1 y = 17 

Page 30

1 1 + 

2 

17 

2 

1 

2 

17 1 

= 

2 

17 

289 1 

= 289 

290 

= 289 

29. If a * b = 

ab 

, fine the value of 3 * (3 * -1). 

a b 

a. -3 b. -1.5 c. -1 d. 2 / 3 

a*b= 

ab 

a b 

(given) 


Now 3 * (3 * -1) = ? 

Here a = 3 b=3 *-1 = 

3.( 1) 

3 ( 1) 

3 

= 2 

3* (3*-1) = 

3.( 3/ 

2) 

3 ( 3/ 

2) 

( by given) 

= 

9/ 2 

3/ 2 

= - 

3 

9 

= - 3 

30. 1 . 5625 = ? 

a. 1.05 b. 1.25 c. 1.45 d. 1.55 


Page 31

1 .5625 = 1.25 

31. If -1 < x < 2 and 1 < x < 3, then least possible value of (2y – 3x) 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 - 3 x 9 

The least possible value of 

(2y – 3x) 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 

18 

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

100 

Page 32

33. 40.83 × 1.02 × 1.2 = ? 

a. 41.64660 b. 49.97592 c. 42.479532 d. 58.7952 


4083 102 12 x x 

100 100 10 

4083 1224 

= x 100 1000 

4997592 

= 100000 

= 49.97592 

34. The sum of three numbers is 264. If the first number be twice the second and third number be 

one-third of the first, then the second number is: 

a. 48 b. 54 c. 84 d. 72 


x+y+z = 264 

2y+y+ 3 

1 (2y) = 264 

3y+ 

3 

2 y = 264 

11y 

3 

= 264 

11y =792 

y=72 

Page 33

35. A student was asked to solve the fraction 

was his answer wrong? 

7 

3 

1 

1 

of 

2 

2 1 

2 

3 

5 

3 

and his answer was 4 

1 . By how much 

1 

a. 1 b. 55 

1 

c. 220 

d. None of these 


7 /3 3/ 2(5/3) 

2 5/3 

= 

7 /3 5/ 2 

11/3 

= 

14 15 

6 

3 29 

x = 11 22 

The answer is 

4 

1 

29 1 116 22 

i.e. - = 22 4 88 

94 47 

= = 88 44 

47 

Ans: 

44 

None of these. 

36. Which number can replace both the question marks in the equation? 

1 

4 

2 

? 

? 

? 

32 



9/ 2 

x 

= 

32 

x 

X 2 = 2 

9 x 32 16 

X 2 = 144 

x = 12 

Page 34

37. For the integer n, if n 3 is odd, then which of the following statements are true? 

I. n is odd II. n 2 is odd II. n 2 is even 

a. I only b. II only c. I and II only d. I and III only382. Three 

numbers are in the ratio 1 : 2 : 3 and their H.C.F. is 12. The numbers are: 

a. 4, 8, 12 b. 5, 10, 15 c. 10, 20, 30 d. 12, 24, 36 

Basic Formula: 

Product of two odd number is odd 

Product of an old and an even number is even 

Product of two even numbers is even 


n 3 = n .n 2 

n 3 is odd = n .n 2 is odd 

= both n & n 2 are odd. 

38. How many digits will be there to the right of the decimal point in the product of 95.75 and 

0.2554? 

a. 5 b. 6 c. 7 d. 9 


95.75 x 0.2554 

= 24.45455 

Ans: 5 

39. The sum of two numbers is 25 and their difference is 13. Find their product. 

a. 104 b. 114 c. 315 d. 325 

Page 35


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


Given x +y = 25 & x-y = 13 

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

25 2 = 13 2 + 4xy 

4xy = 25 2 - 13 2 

4xy = 456 

456 

xy = 4 

=625-169 

xy = 114 

40. On simplification, 3034 – (1002÷20.04) is equal to: 

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


3034 – (1002 -20.04) 

= 3034 - 

1002 

20.04 

= 3034 – 1002 x 100 

20.04 x 100 

= 3034 – 1002 x 100 

2004 

Page 36

= 3034 - 

100 

2 

= 3034 – 50 

= 2984 

n 

41. If 3 729 , then the value of n is: 

a. 6 b. 8 c. 10 d. 12 


Given 

n 

3 = 729 

i.e) 3 n/2 = 729 

Taking log on both sides, 

log 3 n/2 = log 729 

n/2 log 3 = log 729 

n/2 = 

n/2 = 6 

n= 2 x 6 

n = 12 

log729 

= 

log3 

log3 

6 = 

log3 

6log3 

log3 

42. There are four prime numbers written in ascending order. The product of the first three is 385 

and that of the last three is 1001. The last number is: 

a. 11 b. 13 c. 17 d. 19 


Let a, b, c, d be the Four Prime numbers 

Page 37

Then abc = 385 1 & bcd = 1001 2 

Now gcd of abc & bcd i.e., (abc, bcd) 

= bc (a,d) 

= bc x 1 ( a & d are prime) 

Also (abc, bcd) = (gcd 385, 1001 ) = 77 

bc = 77 

From 2 bcd = 1001 

d = 

1001 1001 = = 13 

bc 77 

III ly 385 385 

From 1, a = = = 5 

bc 77 

Four Prime no’s are 5, 7, 11, 13 

The last no. is 13 

43. The sum of two numbers is 528 and their H.C.F. is 33. The number of pairs of numbers 

satisfying the above conditions is: 

a. 4 b. 6 c. 8 d. 12 


Given x + y = 528 

Greatest common divisor of x, y i.e HCF of (x, y) = 33 


Considering the multiples of 33 we have (33, 66, 99, 132, 165, 198, 231, 264, 297, 330, 363, 

396, 429, 462, 495) we can stop here because the next factor of 33 is 528. 

Page 38

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) = 33Eliminating the one’s 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 . 

.009 

44. 0. 1 

? 

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


Let 

x = 

x = 

.009 

= 0.1 

x 

.009 

0.1 

0. 09 

x = 

1 

.009 10 

x 

0.1 10 

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 


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

Page 39


Given x-y = 5 and xy =336 

We know that 

(x+y) 2 

= (x-y) 2 + 4xy 

= 5 2 +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+2.25÷ x)}] = 0.65-3 

-1.6+{3.2-(3.2+2.25÷ x)} = 0.65-3 

3.2- (3.2+2.25÷ x) = - 0.75 

- 3.2-2.25÷ x = -0.75-3.2 

-2.25÷ x = -0.75-3/2 + 3/2 

-2.25÷ x = - 0.75 

2.25 

x 

= -0.75 

2.25 

x = 

0.75 

Page 40

x = 3 

47. If x 441 0. 02 , then the value of x is: 

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


x ÷ 441 = 0.02 

x = 0.02 

441 

Squaring on both sides 

x = (0.02) 

2 

441 

x = 0.004 

441 

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 


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

x 


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

Page 41

(i.e.) 5 

x +4 = x 

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

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

9 

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

13 

III ly x x +(585-1) = x (i.e.) +(x-1) = x x-1 = 584 (remainder) 

585 

585 

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 

Page 42

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 of 

0.125 0.027 

0.5 

0.5 0.09 0.15 

is: 

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


3 3 

a b 

2 

2 

a ab b 

2 

( a b)( 

a ab b 

a ab b 

= 

2 

2 

2 

) 

= a+b 


0.125 0.027 

0.5x0.5 

0.09 0.15 

3 

3 

(0.5) (0.3) 

(0.5) 0.5x0.3 

(0.3) 

= 

2 

2 

2 

2 

(0.5 0.3)[(0.5) 0.5x0.3 

(0.3) ] 

(0.5) 0.5x0.3 

(0.3) 

= 

2 

2 

Page 43

= 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 me— Wait a minute! I can 

simplify 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: 


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

55.Simplify: 

Page 44

I can simplify most of the radicals, and this will allow for at least a little simplification: 

56.Simplify: 

These two terms have "unlike" radical parts, and I can't take anything out of either radical. Then 

I can't simplify the expression any further and my answer has to be: 

(expression is already fully simplified) 

57.Expand: 

To expand (that is, to multiply out and simplify) this expression, I first need to take the square 

root of two through the parentheses: 

As you can see, the simplification involved turning a product of radicals into one radical 

containing the value of the product (being 2×3 = 6). You should expect to need to manipulate 

radical products in both "directions". 



Page 45

It will probably be simpler to do this multiplication "vertically". 

Simplifying gives me: 

By doing the multiplication vertically, I could better keep track of my steps. You should use 

whatever multiplication method works best for you. 

60.Simplify 

I do the multiplication: 

 

61.Simplify: 

I do the multiplication: 

Page 46

Then I simplify: 

Note in the last example above how I ended up with all whole numbers. (Okay, technically 

they're integers, but the point is that the terms do not include any radicals.) I multiplied two 

radical "binomials" together and got an answer that contained no radicals. You may also have 

noticed that the two "binomials" were the same except for the sign in the middle: one had a 

"plus" and the other had a "minus". This pair of factors, with the second factor differing only in 

the one sign in the middle, is very important; in fact, this "same except for the sign in the 

middle" second factor has its own name: 

Given the radical expression , the "conjugate" is the expression . 

The conjugate (KAHN-juh-ghitt) has the same numbers but the opposite sign in the middle. So 

not only is the conjugate of , but is the conjugate of . 

When you multiply conjugates, y 

We will see shortly why this matters. To get to that point, let's take a look at fractions containing 

radicals in their denominators. 

. 

62.Simplify: 

I can simplify this by working inside, and then taking the square root: 

...or else by splitting the division into two radicals, simplifying, and cancelling: 

Page 47

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- 

Page 48

three, then I will have multiplied the fraction by a strategic form of 1. I won't have changed the 

value, but simplification will now be possible: 

This last form, "five, root-three, divided by three", is the "right" answer they're looking for. 

65.Simplify: 

Don't stop once you've rationalized the denominator. As the above demonstrates, you should 

always check to see if something remains to be simplified. 

66.Simplify: 

This expression is in the "wrong" form, due to the radical in the denominator. But if I try to 

multiply through by root-two, I won't get anything useful: 

Multiplying through by another copy of the whole denominator won't help, either: 

But look what happens when I multiply by the same numbers, but with the opposite sign in the 

middle: Copyright © Elizabeth Stapel 1999-2011 All Rights Reserved 

Page 49

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: 

Page 50

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 square 

roots. 

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: 


72.Simplify: 

Page 51

73.Simplify: 

Multiplying Higher-Index Roots 

74.Simplify: 

75.Simplify: 

Adding Higher-Index Roots 

76.Simplify: 

77.Simplify: 

Dividing Higher-Index Roots 

78.Simplify: 

79.Simplify: 

Page 52

I can't simplify this expression properly, because I can't simplify the radical in the denominator 

down to whole numbers: 

To rationalize a denominator containing a square root, I needed two copies of whatever factors 

were inside the radical. For a cube root, I'll need three copies. So that's what I'll multiply onto 

this fraction: 

80.Simplify: 

Since 72 = 8 × 9 = 2 × 2 × 2 × 3 ×3, I won't have enough of any of the denominator's factors to 

get rid of the radical. To simplify a fourth root, I would need four copies of each factor. For this 

denominator's radical, I'll need two more 3s and one more 2: 

A Special Case of Rationalizing 

If your class has covered the formulas for factoring the sums and differences of cubes, then you 

might encounter a special case of rationalizing denominators. The reasoning and methodology 

are similar to the "difference of squares" conjugate process for square roots. 

81.Simplify: 

I would like to get rid of the cube root, but multiplying by the conjugate won't help much: 

Page 53

But I can "create" a sum of cubes, just as using the conjugate allowed me to create a difference 

of squares earlier. Using the fact that a 3 + b 3 = (a + b)(a 2 – ab + b 2 ), and letting a = 1 and b equal 

the cube root of 4, I get: 

If I multiply, top and bottom, by the second factor in the sum-of-cubes formula, then the 

denominator will simplify with no radicals: 

Naturally, if the sign in the middle of the original denominator had been a "minus", I'd have 

applied the "difference of cubes" formula to do the rationalization. This sort of "rationalize the 

denominator" exercise almost never comes up. But if you see this in your homework, expect one 

of these on your next test. 

Radicals Expressed With Exponents 

Radicals can be expressed as fractional exponents. Whatever is the index of the radical becomes 

the denominator of the fractional power. For instance: 

The second root became a one-half power. A cube root would be a one-third power, a fourth root 

would be a one-fourth power, and so forth.This conversion process will matter a lot more once 

you get to calculus. For now, it allows you to simplify some expressions that you might 

otherwise not have been able to. Copyright © Elizabeth Stapel 1999-2011 All Rights Reserved 

82.Express 

as a single radical term. 

I will convert the radicals to exponential expressions, and then apply exponent rules to combine 

the factors: 

Page 54

83.Simplify: 

A Few Other Considerations 

Usually, we cannot have a negative inside a square root. (The exception is for "imaginary" 

numbers. If you haven't done the number "i" yet, then you haven't done imaginaries.) So, for 

instance, is not possible. Do not try to say something like " ", because it's not true: 

. You must have a positive inside the square root. This can be important for 

defining and graphing functions. 

84.Find the domain of the following: 

The fact that I have the expression x – 2 inside a square root requires that x – 2 be zero or greater, 

so I must have x – 2 > 0. Solving, I get: 

domain: x > 2 

On the other hand, you CAN have a negative inside a cube root (or any other odd root). For 

instance: 

...because (–2) 3 = –8. 

85.Find the domain of the following: 

For , there is NO RESTRICTION on the value of x, because x – 2 is welcome to be 

negative inside a cube root. Then the domain is: 

domain: all x 

Page 55

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 

12 

3. 3 4 ? 

125 

a. 

2 

1 b. 

5 

3 

1 c. 

5 

4 

1 d. 

5 

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: 

2 

2 

5 

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


3 

2.3 .027 

is: 

2 

2.3 .69 .09 

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

Page 56

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 ten’s place is bigger than the digit in the unit’s 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 

3 6 

9. ? 

5 3 2 12 32 50 

a. 3 b. 3 2 

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. ( 7.5 

7.5 37.5 2.5 

2.5) 

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 

Page 57

14. 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 

has missed 27 times, A has killed: 

a. 30 birds b. 60 birds c. 72 birds d. 90 birds 

15. The greatest four-digit perfect square number is: 

a. 9000 b. 9801 c. 9900 d. 9981 

16. What least number must be subtracted from 427398 so that the remaining number is divisible 

by 15? 

a. 3 b. 6 c. 11 d. 16 

17. The greatest number of hour digits which is divisible by 15, 25, 40 and 75 is: 

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

36.54 

2 

3.46 

? 

18. 40 

2 

a. 3.308 b. 4 c. 33.08 d. 330.8 

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

4 

denominator is increased by 4, the fraction becomes . What was the original fraction? 

5 

a. 11 

8 

b. 8 

5 

10 

c. 13 

d. 10 

7 

3 4 

20. A boy read th of a book on one day and th of the remainder on another day. If there 

8 5 

were 30 pages unread, how many pages did the book contain? 


21. The least number by which 1470 must be divided to get a number which is a perfect square, 

is: 

Page 58

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. 4x – 9y 

23. Which of the following fraction is the largest? 

a. 8 

7 

13 

b. 16 

31 

c. 40 

63 

d. 80 

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 

4 

denominator is increased by 4, the fraction becomes . What was the original fraction? 

5 

a. 11 

8 

b. 8 

5 

10 

c. 13 

d. 10 

7 

26. A sum of Rs.1360 has been divided among A B and C such that A gets 3 

2 of what B gets and 

B gets 4 

1 of what C gets. B’s share is: 

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

5 10 

27. if 5 = 2.236, then the value of 125 is equal to : 

2 5 

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 

Page 59

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 

0.0203 

2.92 

30. ? 

0.007314.5 

0.7 

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 

2 

32. A person travels 3.5 km form place A to place B. Out of this distance, he travels 1 km on 

3 

1 

bicycle, 1 km on scooter and the rest on foot. What portion of the whole distance does he 

6 

cover on foot? 

a. 19 

3 

b. 11 

4 

4 

c. 21 

d. 6 

5 


1 

1 

0.01 

0.1 

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 : 

Page 60

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

y x 

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

y x 

is: 

a. 

730 

77 

73 

b. 77 

c. 

7.3 

77 

d. 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? 


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 

96.54 89.63 965.4 896.3 

42. 

? 

96.54 89.63 9.654 8.963 

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

Page 61

43. The sum of the digits of a two-digit number is 15 and the difference between the digits is 3. 

What is the two-digit number? 

a. 69 b. 78 c. 96 d. 87 

44. Income of a company doubles after every one year. If the initial income was Rs.4 lakhs, 

what would be the income after 5 years? 

a. Rs.1.24 crores b. Rs.1.28 crores c. Rs.2.56 crores d. None of these 

45. If a = 0.1039, then the value of 4a 2 4a 

1 

3a 

is: 

a. 0.1039 b. 0.2078 c. 1.1039 d. 2.1039 

46. What least value must be assigned to * so that the number 63576*2 is divisible by 8? 

a. 1 b. 2 c. 3 d. 4 

47. The product of two numbers is 1320 and their H.C.F. is 6. The L.C.M. of the numbers is: 

a. 220 b. 1314 c. 1326 d. 7920 

48. If 213 × 16 = 3408, then the 1.6 × 21.3 is equal to: 

a. 0.3408 b. 3.408 c. 34.08 d. 340.8 

49. The sum of the squares of three consecutive natural numbers is 2030. What is the middle 

number? 

a. 25 b. 26 c. 27 d. 28 

50. How many times are the hands of a clock at right angel in a day? 

a. 22 b. 44 c. 24 d. 48 

Page 62

Reasoning and Quantitative aptitude Simplification & Roots

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