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Numbers

Aptitude

Numbers -> IMPORTANT FACTS AND FORMULAE

1. Natural Numbers :
Counting numbers 1, 2, 3, 4, 5, .. are called natural numbers.
II. Whole Numbers :
All counting numbers together with zero form the set of whole numbers. Thus,
I. 0 is the only whole number which is not a natural number.
II. Every natural number is a whole number.
III.Some Important Formulae :
I. ( 1 + 2 + 3 + .....+ n) = n (n + 1 ) / 2
II. (1 2 + 22 + 32 + ..... + n2) = n ( n + 1 ) (2n + 1) / 6
III. (1 3 + 23 + 33 + ..... + n3) = n2 (n + 1)2 / 4

Numbers -> SOLVED EXAMPLES

1. 217 × 217 + 183 × 183 = ?
  Sol. Let the speed of the motorboat in still water be x kmph. Then,
Speed downstream = (x + 2) kmph; Speed upstream = (x - 2) kmph.
∴ 6/x+2 + 6/x-2 = 33/60
⇔ 11x² - 240x - 44 = 0
⇔ 11x² - 242x + 2x - 44 = 0
⇔ (x - 22) (11x + 2) = 0 ⇔ x = 22.
Hence, speed of motorboat in strill water = 22 kmph.
2. A man can row 18 kmph in still water. It takes him trice as long to row up as to row down the river. Find the rate of stream.
  Sol. Let man’s rate upsream be x kmph. Then, his rate downstream = 3x kmph.
∴ Rate in still water = 1/2(3x + x) kmph = 2x kmph.
So, 2x = 18 or x = 9.
∴ Rate upstream = 9 km/hr, Rate downstream = 27 km/hr.
Hence, rate of stream = 1/2(27 - 9) km/hr = 9 km/hr.
3. A man can row upstream at 7 kmph and downstream at 10 kmph. Find man’s rate in still water and the rate of current.
  Sol. Rate in still water = 1/2 (10 + 7) km/hr = 8.5 km/hr,
Rate of current = 1/2 (10 - 7) km/hr = 1.5 km/hr.

Numbers -> Exercise

10. Which of the following numbers is exactly divisible by 99?
 
  • A. 114345
  • B. 135792
  • C. 913464
  • D. 3572404
Ans: A.
Sol.
The required number should be divisible by both 9 and 11.
Clearly, 114345 is divisible by both 9 and 11.
So, it is divisible by 99.
 
 
11. A number when divided by 899 gives a remainder 63. If the same number is divided by 29, the remainder will be
 
  • A. 3
  • B. 4
  • C. 5
  • D. 6
Ans: C.
Sol.
Number = (899 x Q) + 63
= (29 x 31 x Q) + (29 x 2) + 5
= 29 x (31Q + 2) + 5.
∴ Required remainder = 5.
 
 
12. The difference between the local value and face value of 7 in the numeral 657903 is
 
  • A. 0
  • B. 7896
  • C. 6993
  • D. 903
Ans: C.
Sol.
(Local Value) - (Face Value) = (7000 - 7) = 6993.