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71. If the numerator of a fraction is increased by 2 and the denominator is increased by 3, the fraction becomes 79 and if both the numerator as well as he denominator are decreased by 1, the fraction becomes 45. What is the original fraction ?

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Solution:
Let the fraction be xy
Then,
⇔x+2y+3=79⇔9x−7y=3.....(i)And,⇔x−1y−1=45⇔5x−4y=1.....(ii)
Solving (i) and (ii), we get :
x = 5 and y = 6
Hence, the original fraction is 56
72. The difference of two numbers is 20% of the larger number. If the smaller number is 12, the larger one is :

Discuss
Solution:
Let the number be x
Then,
⇔x−12=20% of x⇔x−x5=12⇔4x5=12⇔x=12×54⇔x=15
73. Two numbers are such that the ratio between them is 4 : 7. If each is increased by 4, the ratio becomes 3 : 5. The lager number is ?

Discuss
Solution:
Let the numbers be 4x and 7x
Then,
⇔4x+47x+4=35⇔5(4x+4)=3(7x+4)⇔x=8
∴ Larger number :
= 7x
= 7 × 8
= 56
74. the difference between two numbers is 3 and the difference between their squares is 63. Which is the larger number ?

Discuss
Solution:
Let the number be x and y
Then,
x2−y2=63&x−y=3
On dividing, we get: x + y = 21
Solving x + y = 21 and x - y = 3,
We get: x = 12 and y = 9
∴ Larger number = 12
75. twenty times a positive integer is less than its square by 96. What is the integer ?

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Solution:
Let the integer be x
Them,
⇔x2−20x=96⇔x2−20x−96=0⇔(x+4)(x−24)=0⇔x=24
76. There are two numbers such that the sum of twice the first number and thrice the second number is 100 and the sum of thrice the first number and twice the second number is 120. Which is the larger number ?

Discuss
Solution:
Let the numbers be x and y
Then,
2x + 3y = 100 ..... (i)
And,
3x + 2y = 120 ..... (ii)
Adding (i) and (ii), we get :
5x + 5y = 220
or x + y = 44 ..... (iii)
Subtracting (i) from (ii): we get :
x - y = 20 ..... (iv)
Adding (iii) and (iv), we get :
2x = 64
or x = 32
Putting x = 32 in (iii), we get :
y = 12
Hence, larger number = 32
77. If one-seventh of a number exceeds its eleventh part by 100, then the number is :

Discuss
Solution:
Let the number be x
Then,
⇒17x−111x=100⇒4x77=100⇒x=77004⇒x=1925
78. The sum of three numbers is 264. If the first number be twice the second and third number be be one-third of the first, then the second number is :

Discuss
Solution:
Let the second number be x
Then, first number = 2x,
And third number = 2x3
∴2x+x+2x3=264⇔11x3=264⇔x=(264×311)⇔x=72
79. A, B, C, D and E are five consecutive odd numbers. The sum of A and C is 146. What is the value of E ?

Discuss
Solution:
Let A = x
B = x + 2
C = x + 4
D = x + 6 and
E = x + 8
Then,
⇒A+C=146⇒x+(x+4)=146⇒2x=142⇒x=71∴E=x+8=71+8=79
80. The difference between a two-digit number and the number obtained by interchanging the positions of its digits is 36. What is the difference between the two digits of the number ?

Discuss
Solution:
Let the ten's digit be x and unit's digit be y
Then,
⇔(10x+y)−(10y+x)=36⇔9(x−y)=36⇔x−y=4

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