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## 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. 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. 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

1. 587×999 = ?

• A. 615173
• B. 587523
• C. 586413
• D. 614823
 Ans: C. Sol. 587×999 = 587 ×(1000 – 1) = (587 × 1000) – (587 × 1) = (587000 – 587) = 586413.

2. 461 + 462 + 463 + 464 is divisible by

• A. 5
• B. 10
• C. 11
• D. 15
 Ans: B. Sol. = 461 + 462 + 463 + 464 = 461 ( 1 + 4 + 42 + 43) = 461 × 85 = 460 × 340, which is clearly divisible by 10.

3. In a division sum, the divisor is 10 times the quotient and 5 times the remainder. If the remainder is 46, the dividend is

• A. 5336
• B. 4306
• C. 4236
• D. 4336
 Ans: A. Sol. Divisor = ( 5 × 46 ) = 230. Also, 10 × Q = 230 ⇒ Q = 23. And, R = 46.Dividend = (230 × 23 + 46 ) = 5336.