Directions: Study the following questions carefully and choose the right answer.
Important for :
1
The least number by which 1470 must be divided to get a number which is a perfect square is
» Explain it
D
1470 = 7 × 7 × 6 × 5 
 
To make it a perfect square, it must be divided by 5 × 6 = 30.

Hence, option D is correct.
 
2
Of three numbers, the sum of the first two is 45, the sum of the second and the third is 55, and the sum of the third and thrice the first is 90. Find the third number.
» Explain it
C
Let the number be x, y and z.
 
Then x + y = 45      ....(i)                y + z = 55     ....(ii)               z + 3x = 90    .... (iii) 
 
Now, from (i) 
 
y = 45 – x 
 
And from (ii) z = 55 – y 
 
∴     z = 55 – (45 – x) 
 
z = 10 + x
 
Putting the value of z in equn. (iii) , we have
 
3x + 10 + x = 90
 
∴      x = 20 
 
y = 45 – 20 = 25   and   z = 10 + 20 = 30.

Hence, option C is correct.
 
3
A girl finds the average of 6 positive integers, each integers containing two digits. By mistake she interchanges the digit of one number say xy for yx. Due to this the average becomes 1.5 less than the previous one. What was the difference between two digits x and y.
» Explain it
A
No. of observations = 6,  Increased average = 1.5,  Total increased = 6 × 1.5 = 9 
 
Now, let the unit digit of the number whose digits have been interchanged = x and ten's digit of the number = y
 
Original number = (10y + x),  Number after interchanging the digits = (10x + y) 
 
So, (10x + y) – (10y + x) = 9    OR    9x – 9y = 9
 
∴    Required difference = x – y = 1. 

Smart Trick Method:

For such questions you can refer to the table below
  Difference in no. after interchanging the digits     Difference in the digits of the original no.  
9 (9 × 1) 1
18 (9 × 2) 2
27 (9 × 3) 3
36 (9 × 4) 4
 

Hence, option A is correct.

4
The sum of the squares of three numbers is 532 and the ratio of the first and the second as also of the second and the third is 3 : 2. Then the first number is
» Explain it
C
Let the three numbers be x, y and z.

   x2 + y2 + z2 = 532

x  =  3 y  =  3
y 2 z 2

⇒  z =  2y  and y =  2x
3 3

⇒  z =  4 x  and  y =  2x
9 3

∴    x2 + y2 + z2 = 532

⇒ x2 4x2  +  16x2  = 532
9 81

⇒  x2 = 324  ⇒ x = 18,  y = 12,  z = 8

Hence, option C is correct.

5
If the product of two successive positive integers is 7482 then, which is the greater integer?
» Explain it
A
Method I: Let the two successive positive integers be x and x + 1

Then, x(x + 1) = 7482

or, x2 + x – 7482 = 0

or, x2 + 87x – 86x – 7482 = 0

or, (x + 87)(x – 86) = 0

or,   x = 86     (Ignore negative value)

∴  (x + 1) = 86 + 1 = 87

Hence, the greater positive integer is 87.

Method II:
 
Calculate square root of 7482 = 86.49
 
Therefore, the first no. will be 86 & the larger no. be (86 + 1) = 87 
 
PS: Apply the shortcut method to find square root of 7482.

Hence, option A is correct.