Mixed ratio and proportion

 

Ratio

Introduction:

Ratio is the relation which one quantity bears to another of the same kind. The ratio of two quantities a and b is the fraction a/b and we write it as a: b.

In the ratio a: b, we call a as the first term or antecedent and b, the second term or consequent.

 

Note: The multiplication or division of each term of a ratio by the same non- zero number does not affect the ratio.

 

Compound Ratio: – It is obtained by multiplying together the numerators for new numerator and denominators for new denominator.

 

 

Example 1. If the ratios are 4:3, 15:20, 2:6 and 3:5 find the compound ratio?

 

 

 

 

 

 

 

 

Example2. If we divide 4185 into two parts such that they are in ratio 7:2, then find the values of both the parts?

Sol 2. Let the actual variable be 7x and 2x.

So, the 1st part = 7 ×465=3255

The 2nd part = 2 ×465=930

 

 

Note:

The ratio of first , second and third quantities is given by

ac : bc : bd

 

If the ratio between first and second quantity is a:b and third and fourth is c:d .

Similarly, the ratio of first, second, third and fourth quantities is given by
ace : bce : bde : bdf
If the ratio between first and second quantity is a: b and third and fourth is c:d.

 

                                                 Proportion

 
Introduction:-
Four quantities are said to be proportional if the two ratios are equal i.e.  the A, B, C and D are proportion. It is denoted by “::” it is written as A : B : C : D where A and D are extremes and B and C are called means .
                             Product of the extreme = Product of the means

 

 

Direct proportion: – The two given quantities are so related that if one quantity increases (or decreases) then the other quantity also increases (or decreases).

Example 1. If 5 pens cost Rs 10 then 15 pen cost?

Sol 1. It is seen that if number of pens increases then cost also increases. So,

5 pens: 15 pens:: Rs 10 : required cost

 

 

Inverse proportion: – The two given quantities are so related that if one quantity increases (or decreases) then the other quantity also decreases (or increases).

Example 2.If 10 men can do a work in 20 days then in how many days 20 men can do that work?

Sol 2. Here if men increase then days should decrease, so this is a case of inverse proportion, so

10 men: 20 men :: required days : 20 days

 

 

Rule of three: It Is the method of finding 4th term of a proportion if all the other three are given, if ratio is a:b :: c:d then ,

 

 

 

                                             ALLIGATION

Introduction:-

The word allegation means linking. It is used to find:

  1. The proportion in which the ingredients of given price are mixed to produce a new mixture at a given price.
  2. The mean or average value of mixture when the price of the two or more ingredients and the proportion in which they are mixed are given.

Mathematical Formula:

 

For two ingredient:-

 

 

Example 1: If the rice at Rs 3.20 per kg and the rice at Rs 3.50 per kg be mixed then what should be their proportion so that the new mixture be worth Rs 3.35 per kg ?

Sol 1: CP of 1 kg of cheaper rice                          CP of 1 kg of dearer rice

Hence they must be mixed in equal proportion i.e. 1:1

 

 

Example 2: Find out the ratio of new mixture so that it will cost Rs 1.40 per kg from the given three kinds of rice costing Rs 1.20, Rs 1.45 and Rs 1.74?

 

Sol 2: 1st rice cost = 120, 2nd rice cost = 145 and 3rd rice cost = 174 paisa.

From the above rule: we have,

Therefore, three rice must be mixed in 39: 20: 20 ratios to have a new mixture of rice.

 

 

Questions

Level-I

 

..

1.   A  and B together have Rs. 1210. If  of A’s amount is equal to  of B’s amount, how much amount does B have?
A. Rs. 460
B. Rs. 484
C. Rs. 550
D. Rs. 664

 

2. Two numbers are respectively 20% and 50% more than a third number. The ratio of the two numbers is:
A. 2 : 5
B. 3 : 5
C. 4 : 5
D. 6 : 7

 

 

3. A sum of money is to be distributed among A, B, C, D in the proportion of 5 : 2 : 4 : 3. If C gets Rs. 1000 more than D, what is B’s share?
A. Rs. 500
B. Rs. 1500
C. Rs. 2000
D. None of these

 

 

 

 

4. Seats for Mathematics, Physics and Biology in a school are in the ratio 5 : 7 : 8. There is a proposal to increase these seats by 40%, 50% and 75% respectively. What will be the ratio of increased seats?
A. 2 : 3 : 4
B. 6 : 7 : 8
C. 6 : 8 : 9
D. None of these

 

 

5. In a mixture 60 litres, the ratio of milk and water 2 : 1. If this ratio is to be 1 : 2, then the quanity of water to be further added is:
A. 20 litres
B. 30 litres
C. 40 litres
D. 60 litres
 

6.

 

The ratio of the number of boys and girls in a college is 7 : 8. If the percentage increase in the number of boys and girls be 20% and 10% respectively, what will be the new ratio?

A. 8 : 9
B. 17 : 18
C. 21 : 22
D. Cannot be determined

 

7. Salaries of Ravi and Sumit are in the ratio 2 : 3. If the salary of each is increased by Rs. 4000, the new ratio becomes 40 : 57. What is Sumit’s salary?
A. Rs. 17,000
B. Rs. 20,000
C. Rs. 25,500
D. Rs. 38,000

 

8. If 0.75 : x :: 5 : 8, then x is equal to:
A. 1.12
B. 1.2
C. 1.25
D. 1.30

 

 

9. The sum of three numbers is 98. If the ratio of the first to second is 2 :3 and that of the second to the third is 5 : 8, then the second number is:
A. 20
B. 30
C. 48
D. 58

 

 

  10 .If Rs. 782 be divided into three parts, proportional to  :  : , then the first part is:
A. Rs. 182
B. Rs. 190
C. Rs. 196
D. Rs. 204

 

 

 

 

Answers

  1. Answer:Option B

 

Explanation:

4 A = 2 B
15 5

 

 A = 2 x 15 B
5 4

 

 A = 3 B
2

 

A = 3
B 2

A : B = 3 : 2.

 B’s share = Rs. 1210 x 2 = Rs. 484.
5

 

 

 

 

2 .Answer: Option C

 

Explanation:

Let the third number be x.

Then, first number = 120% of x = 120x = 6x
100 5

 

Second number = 150% of x = 150x = 3x
100 2

 

 Ratio of first two numbers = 6x : 3x = 12x : 15x = 4 : 5.

 

 

3 .Answer: Option C

Explanation:

Let the shares of A, B, C and D be Rs. 5x, Rs. 2x, Rs. 4x and Rs. 3x respectively.

Then, 4x – 3x = 1000

x = 1000.

B’s share = Rs. 2x = Rs. (2 x 1000) = Rs. 2000.

 

 

4 .Answer: Option A

 

Explanation:

 

Originally, let the number of seats for Mathematics, Physics and Biology be 5x, 7x and 8x respectively.

 

Number of increased seats are (140% of 5x), (150% of 7x) and (175% of 8x).

 

140 x 5x , 150 x 7x and 175 x 8x
100 100 100
                             

 

 7x, 21x and 14x.
2
     

 

 The required ratio = 7x : 21x : 14x
2

 

14x : 21x : 28x

 

2 : 3 : 4.

 

 

 

 

 

 

 

 

 

 

 

5 .Answer: Option D

 

Explanation:

Quantity of milk = 60 x 2 litres = 40 litres.
3

Quantity of water in it = (60- 40) litres = 20 litres.

New ratio = 1 : 2

Let quantity of water to be added further be x litres.

 

 

Then, milk : water = 40 .
20 + x

 

Now, 40 = 1
20 + x 2

 

20 + x = 80

 

x = 60.

Quantity of water to be added = 60 litres.

 

6 .Answer: Option C

 

Explanation:

 

Originally, let the number of boys and girls in the college be 7x and 8x respectively.

 

Their increased number is (120% of 7x) and (110% of 8x).

 

120 x 7x and 110 x 8x
100 100
                   

 

42x and 44x
5 5
       

 

The required ratio = 42x : 44x = 21 : 22

 

7 .Answer: Option D

 

Explanation:

Let the original salaries of Ravi and Sumit be Rs. 2x and Rs. 3x respectively.

 

Then, 2x + 4000 = 40
3x + 4000 57
       

57(2x + 4000) = 40(3x + 4000)

 

6x = 68,000

 

3x = 34,000

 

Sumit’s present salary = (3x + 4000) = Rs.(34000 + 4000) = Rs. 38,000.

 

 

 

8 .Answer: Option B

 

Explanation:

(x x 5) = (0.75 x 8)    x = 6 = 1.20
5
         

 

 

 

 

 

 

 

 

9 .Answer: Option B

 

Explanation:

Let the three parts be A, B, C. Then,

 

A : B = 2 : 3 and B : C = 5 : 8 = 5 x 3 : 8 x 3 = 3 : 24
5 5 5
                       

 

 A : B : C = 2 : 3 : 24 = 10 : 15 : 24
5
     

 

 B = 98 x 15 = 30.
49

 

 

 

10 .Answer: Option D

 

 

Explanation:

 

Given ratio =  :  :  = 6 : 8 : 9.

 

 1st part = Rs. 782 x 6 = Rs. 204

 

 

 

 

Level-II

11. The salaries A, B, C are in the ratio 2 : 3 : 5. If the increments of 15%, 10% and 20% are allowed respectively in their salaries, then what will be new ratio of their salaries?
A. 3 : 3 : 10
B. 10 : 11 : 20
C. 23 : 33 : 60
D. Cannot be determined

 

 

Answer: Option C

 

Explanation:

Let A = 2k, B = 3k and C = 5k.

A’s new salary = 115 of 2k = 115 x 2k = 23k
100 100 10

 

B’s new salary = 110 of 3k = 110 x 3k = 33k
100 100 10

 

C’s new salary = 120 of 5k = 120 x 5k = 6k
100 100

 

 New ratio 23k : 33k : 6k = 23 : 33 : 60
10 10

 

12. If 40% of a number is equal to two-third of another number, what is the ratio of first number to the second number?
 

A. 2 : 5
B. 3 : 7
C. 5 : 3
D. 7 : 3

Answer: Option C

 

Explanation:

Let 40% of A = 2 B
3

 

Then, 40A = 2B
100 3

 

2A = 2B
5 3

 

A = 2 x 5 = 5
B 3 2 3

A : B = 5 : 3.

 

13. The fourth proportional to 5, 8, 15 is:
A. 18
B. 24
C. 19
D. 20

 

 

Answer: Option B

 

Explanation:

Let the fourth proportional to 5, 8, 15 be x.

Then, 5 : 8 : 15 : x

5x = (8 x 15)

 

x = (8 x 15) = 24.
5

 

 

 

 

14.

 

 

 

Two number are in the ratio 3 : 5. If 9 is subtracted from each, the new numbers are in the ratio 12 : 23. The smaller number is:

A. 27
B. 33
C. 49
D. 55

Answer: Option B

 

Explanation:

Let the numbers be 3x and 5x.

Then, 3x – 9 = 12
5x – 9 23

23(3x – 9) = 12(5x – 9)

9x = 99

x = 11.

The smaller number = (3 x 11) = 33.

 

 

15.

 

 

In a bag, there are coins of 25 p, 10 p and 5 p in the ratio of 1 : 2 : 3. If there is Rs. 30 in all, how many 5 p coins are there?

A. 50
B. 100
C. 150
D. 200

Answer: Option C

 

Explanation:

Let the number of 25 p, 10 p and 5 p coins be x, 2x, 3x respectively.

Then, sum of their values = Rs. 25x + 10 x 2x + 5 x 3x = Rs. 60x
100 100 100 100

 

60x = 30     x = 30 x 100 = 50.
100 60

Hence, the number of 5 p coins = (3 x 50) = 150.

SIMPLIFICATION

Simplification

Simplification is one of the most important part of Quantitative Aptitude section of any competitive exam. Today I am sharing all the techniques to solve Simplification questions quickly.

Rules of Simplification

V  Vinculum

B  Remove Brackets – in the order ( ) , { }, [ ]

O  Of
D  Division

M  Multiplication

A  Addition

S  Subtraction

 

Classification

Types Description
Natural Numbers: all counting numbers ( 1,2,3,4,5….∞)
Whole Numbers: natural number + zero( 0,1,2,3,4,5…∞)
Integers: All whole numbers including Negative number + Positive number(∞……-4,-3,-2,-1,0,1,2,3,4,5….∞)
Even & Odd Numbers : All whole number divisible by 2 is Even (0,2,4,6,8,10,12…..∞) and which does not divide by 2 are Odd (1,3,5,7,9,11,13,15,17,19….∞)
Prime Numbers: It can be positive or negative except 1, if the number is not divisible by any number except the number itself.(2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,61….∞)
Composite Numbers: Natural numbers which are not prime
Co-Prime: Two natural number a and b are said to be co-prime if their HCF is 1.

 

Divisibility

Numbers IF A Number Examples
Divisible by 2 End with 0,2,4,6,8 are divisible by 2 254,326,3546,4718 all are divisible by 2
Divisible by 3 Sum of its digits  is divisible by 3 375,4251,78123 all are divisible by 3.  [549=5+4+9][5+4+9=18]18 is divisible by 3  hence 549 is divisible by 3.
Divisible by 4 Last two digit divisible by 4 5648 here last 2 digits are 48 which is divisible by 4 hence 5648 is also divisible by 4.
Divisible by 5 Ends with 0 or 5 225 or 330 here last digit digit is 0 or 5 that mean both the numbers are divisible by 5.
Divisible by 6 Divides by Both 2 & 3 4536 here last digit is 6 so it divisible by 2 & sum of its digit (like 4+5+3+6=18) is 18 which is divisible by 3.Hence 4536 is divisible by 6.
Divisible by 8 Last 3 digit divide by 8 746848 here last 3 digit 848 is divisible by 8 hence 746848 is also divisible by 8.
Divisible by 10 End with 0 220,450,1450,8450 all numbers has a last digit zero it means all are divisible by 10.
Divisible by 11 [Sum of its digit in
odd places-Sum of its digits
in even places]= 0 or multiple of 11
Consider the number 39798847

(Sum of its digits at odd places)-(Sum of its digits at even places)(7+8+9+9)-(4+8+7+3)

(23-12)

23-12=11, which is divisible by 11. So 39798847 is divisible by 11.

Division & Remainder Rules

Suppose we divide 45 by 6

 

hence ,represent it as:

dividend = ( divisorquotient ) + remainder

or

divisior= [(dividend)-(remainder] / quotient

could be write it as

x = kq + r where (x = dividend,k = divisor,q = quotient,r = remainder)

 

 

Rules

  1. 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.

  1. Virnaculum (or Bar):

When an expression contains Virnaculum, before applying the ‘BODMAS’ rule, we simplify the expression under the Virnaculum.

 

 

Example:

On dividing a certain number by 342, we get 47 as remainder. If the same number is divided by 18, what will be the remainder ?

Number = 342k + 47

( 18 ✘19k ) + ( 18 ✘2 ) + 11

18 ✘( 19k + 2 ) +11.

Remainder = 11

 

Sum Rules

(1+2+3+………+n) = 1/n(n+1)

(12+22+32+………+n2) = 1/n (n+1) (2n+1)

(13+23+33+………+n3) = 1/4 n2 (n+1)2

 

Questions:

Level-I:

 

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

 

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

 

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

 

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

 

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

 

 

6.

A sum of Rs. 1360 has been divided among A, B and C such that A gets  of what B gets and B gets  of what C gets. B’s share is:
A. Rs. 120
B. Rs. 160
C. Rs. 240
D. Rs. 300

 

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

 

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 has missed 27 times, A has killed:
A. 30 birds
B. 60 birds
C. 72 birds
D. 90 birds

 

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
B.
1
8
C.
1
9
D.
7
8

 

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
E. None of these

 

Level-II:

1. 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

 

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
E. None of these

 

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

 

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

 

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

 

  1. Find the value of 1/(3+1/(3+1/(3-1/3)))
A.) 3/10 B.) 10/3
C.) 27/89 D.) 89/27

  1. Find the value of
A.) 3½ 99; B.) 34/99
C.) 2.131313 D.) 3.141414

 

18.Find the value of

((0.1)3 + (0.6)3 + (0.7)3 − (0.3)(0.6)(0.7))/((0.1)2 + (0.6)2 + (0.7)2 − 0.006 − 0.42 − 0.07)

 

A.) 14/10 B.) 1.35
C.) 13/10 D.) 0

 

 

 

 

  1. Solve(0.76 × 0.76 × 0.76 − 0.008)/(0.76 × 0.76 + 0.76 × 0.2 + 0.04)
A.) 0.56 B.) 0.65
C.) 0.54 D.) 0.45
   
  1. Find the value of
A.) 1.5 B.) -1.5
C.) 1 D.) 0

 

Answers:

Level-I

 

Answer:1 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.

 

 

Answer:2 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.

 

 

Answer:3 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 = 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.

 

 

 

 

 

Answer:4 Option A

 

Explanation:

2ab = (a2 + b2) – (a – b)2

= 29 – 9 = 20

ab = 10.

 

 

 

Answer:5 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.

 

 

Answer:6 Option C

 

Explanation:

Let C’s share = Rs. x

Then, B’s share = Rs. x ,   A’s share = Rs. 2 x x = Rs. x
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.

 

 

Answer:7 Option C

 

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

 

 

Answer:8 Option A

 

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

 

 

 

 

 

 

 

Answer:9 Option A

 

Explanation:

Original share of 1 person = 1
8

 

New share of 1 person = 1
7

 

Increase = 1 1 = 1
7 8 56

 

 Required fraction = (1/56) = 1 x 8 = 1
(1/8) 56 1 7

 

 

Answer:10 Option C

 

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

 

Level-II:

Answer:11 Option B

 

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

x = 15.

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

 

Answer:12 Option C

 

Explanation:

Let total number of children be x.

Then, x x 1 x = x x 16     x = 64.
8 2

 

 Number of notebooks = 1 x2 = 1 x 64 x 64 = 512

 

Answer:13 Option D

 

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.

 

Answer:14 Option B

 

Explanation:

Given exp. = (a + b)2 – (a – b)2
ab

 

   = 4ab
ab

= 4 (where a = 469, b = 174.)

 

Answer:15 Option C

 

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.

 

Answer:16 Option ‘C’

Explanation:

1/[3 + (1/(3+1/(3 – 1/3)))]

=> 1/[3 + 1/(3 + 1/(8/3))]

=> 1/[3 + 1/(3 + 3/8)]

=> 1/[3 + 8/27]

=> 1/(89/27)

=> 27/89

 

Answer:17 Option ‘D’

Explanation:

6/9 + 7/9 + 9/9 + 69/99

2/3 + 7/9 + 1 + 69/99

(66 + 77 + 99 + 69)/99

311/99 => 3.141414

 

Answer:18 Option ‘A’

Explanation:

((0.1)3 + (0.6)3 + (0.7)3 − (0.3)(0.6)(0.7))/((0.1)2 + (0.6)2 + (0.7)2 − 0.006 − 0.42 − 0.07)

=> (0.1 + 0.6 + 0.7)3/(0.1 + 0.6 + 0.7)2

=> 0.1 + 0.6 + 0.7 => 1.4 = 14/10

 

Answer:19 Option ‘A’

 

Answer:20 Option ‘D’

11/30 − [1/6 + 1/5 + [7/12 − 7/12]]

11/30 − [1/6 + 1/5 + [0]]

11/30 − [(5 + 6)/30]

11/30 − 11/30 = 0.