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Direction: Study the following questions carefully and choose the right answer.
1
If a two-digit number is added to a number obtained by reversing the digits of the given number, then the sum is always divisible by which one of the following numbers?
» Explain it
D
Let a two-digit number be (10x + y) and reversing number be (10y + x) 
 
∴    Required sum = 10x + y + 10y + x = 11x + 11y = 11(x + y) 
 
Hence, it's divisible by 11.

Hence, option D is correct.
 
2
What is the sum of two numbers whose differences is 45 and the quotient of the greater number by the lesser number is 4?
» Explain it
D
Let the greater number be x and smaller number be y

∴   x – y = 45                       ..... (i)

and   x = 4y                           ..... (ii)

From eqs. (i) and (ii)

4y – y = 45

⇒  y =  45  = 15
3

On putting the value of y in Eq. (ii), we get

x = 4 × 15 = 60

Hence, required sum = x + y = 60 + 15 = 75.

Hence, option D is correct.

3
If one-third of a two-digit number exceeds its one-fourth by 8, then what is the sum of the digits of the number?
» Explain it
C
Let the number be y,

∴    y  =  y  + 8   ⇒   4y – 3y  = 8
3 4 12

⇒ y = 12 × 8 = 96      

∴   Sum of digits = 9 + 6 = 15.

Hence, option C is correct.

4
A person bought 5 tickets from a station P to a station Q and 10 tickets from the station P to a station R. He paid Rs. 350. If the sum of a ticket from P to Q and a ticket from P to R is Rs. 42, then what is the fare from P to Q?
» Explain it
B
Let the fare from station P to station Q is ₹ x and the fare from station P to station R is ₹ y. 
 
By given condition, x + y = 42                .....(i) 
 
and 5x + 10y = 350                                 .... (ii)
 
On solving Eqs. (i) and (ii), we get
 
x = 14 and y = 28
 
Hence, fare from station P to station Q is ₹ 14. 

Hence, option B is correct.
 
5
If 6x – 5y = 13, 7x + 2y = 23 then 11x + 18y = ?
» Explain it
B
6x – 5y = 13                    ...... (i)

7x + 2y = 23                   ...... (ii)

By equation (i) × 2 & (ii) × 5,

12x – 10y = 26                   ..... (iii)

35x + 10y = 115                 ...... (iv)

by solving these equation we get

x = 3,    y = 1

∴    11x + 18y  =  11 × 3 + 18 × 1  =  33 + 18 = 51.

Hence, option B is correct.