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## PROBLEMS ON NUMBERS -> DESCRIPTION

 Types of Numbers: Natural Numbers : Counting numbers 1,2,3,4,5,..... are called natural numbers. Whole Numbers : All counting numbers together with zero from 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. Even Numbers : A number divisible by 2 is called an even number. e.g. 2,4,6,7,10,etc. Odd Numbers : A number is not divisible by 2 is called an odd number. e.g. 1,3,5,6,7,9,11, etc.

## PROBLEMS ON NUMBERS -> SOLVED EXAMPLES

1. 50 is divided into tow parts such that the sum of their reciprocals is 1/12 Find the two parts. Sol. Let the two parts be x and (50 - x) Then, 1/x + 1/50-x = 1/12 ⇔ 50 - x + x/ x(50-x) = 1/12 ⇒ x² - 50x + 600 = 0 ⇒ (x - 30) (x - 20) = 0 ⇒ x = 30 or x = 20. So, the parts are 30 and 20. A number is as much greater than 36 as is less than 86. Find the number. Sol. Let the number be x. Then, x - 36 = 86 - x ⇔ 2x = 86 + 36 = 122 ⇔ x = 61. Hence, the required number is 61. Find a number such that when 15 is subtracted from 7 times the number, the result is 10 more than twice the number. Sol. Let the number be x. Then, 7x - 15 = 2x + 10 ⇔ 5x = 25 ⇔ x = 5. Hence, the required number is 5. The sum of two numbers is 184. If one-third of the one exceeds one-seventh of the other by 8, find the smaller number. Sol. Let the numbers be x and (184 - x). Then, x / 3 - (184-x)/7 = 8 ⇔ 7

## PROBLEMS ON NUMBERS -> Exercise

1. The sum of three consecutive multiples of 3 is 72. What is the larget number?

• A. 26
• B. 27
• C. 46
• D. 57
 Ans: B. Sol. Let the numbers be 3x, 3x + 3 and 3x + 6. Then, 3x + (3x + 3) + (3x + 6) = 72 ⇔ 9x = 63 ⇔ x = 7. ∴ Largest number = 3x +6 = 27.

2. The sum of three consecutive odd numbers is 20 more than the first of these numbers. What is the middle number?

• A. 3
• B. 6
• C. 9
• D. 12
 Ans: C. Sol. Let the numbers be x, x + 2 and x + 4. Then, x + (x + 2) + (x + 4) = x + 20 ⇔ 2x = 14 ⇔ x = 7. ∴ Middle number = x + 2 = 9.

3. The sum of four consecutive even integers is 1284. The greatest of them is:

• A. 324
• B. 342
• C. 364
• D. 382
 Ans: A. Sol. Let the four integers be x, x + 2, x + 4 and x + 6 Then, x + (x + 2) + (x + 4) + (x + 6) = 1284 ⇔ 4x = 1272 ⇔ x = 318 ∴ Greatest integer = x + 6 = 324.