Number Problems Based Question Fraction Problem

⇒ Natural Numbers :  The counting numbers 1, 2, 3, ……… are called natural numbers. Natural numbers are beginning with 1 and increasing one by one. The set of natural numbers is denoted by the  symbol N. In the set of natural numbers, 2 is called the successor of 1. 1 is called the predecessor of 2 and 3 is called successor of 2 and 2 is called predecessor of 3. Can you give the last natural number?  

                                    Thus N = {1, 2, 3, ……… } 

⇒ Whole Numbers :  If we  include the number 0 to the set of natural numbers, we get 0, 1, 2, 3, ……… . This is called the set of whole numbers and is denoted by the symbol W.  

                                    Thus W = {0, 1, 2, 3, ……… } 

⇔   Numbers ending with 0 are divisible by 10  
 
 For example, the numbers 10, 20, 50, 580, 2700  end with 0 are exactly divisible by 10.  

⇔   Numbers ending with 0 or 5 are divisible by 5    

For example, the numbers 10, 15, 35, 175, 825 (ending with 0 or 5) are divisible by 5.  

⇔   Numbers ending with 0‚ 2‚ 4‚ 6 or 8 are divisible by 2    

For example 20, 32, 64, 76, 28 are divisible by 2.

Discuss :  If a number is divisible by 2 and 5, then  it  will be divisible by 10.

⇔   If the number formed by the last two digits of a given number is divisible by 4‚ then the number will be divisible by 4.

Example :  Are the numbers 216, 1900, 1788 divisible by 4?

Solution :  Each of the numbers 216, 1900, 1788 are divisible by 4 because  The last two digists of 216 is 16 and 16 is divisible 4.

 ∴  216 is divisible by 4.  The last two digits of 1900 is 00 and 0 is divisible by 4.  

 ∴  1900 is divisible by 4.  The last two digits of 1788 is 88 and 88 is divisible by 4.

 ∴  1788 is divisible by 4.

Example  :  Is 1930 divisible by 4?

Solution :  The last two digists of 1930 is 30 and 30 is not divisible 4.  

∴  1930 is not divisible by 4.

⇔    If the number formed by the last three digits of a given number is divisible by 8‚ then the number will be divisible by 8.

Example  :  Is 78848 divisible by 8?

Solution :  The last three digits of 78848 is 848. Here 848 is divisible by 8.  

∴ 78848 is divisible by 8.

Example :  Is 23102 divisible by 8?

Solution :  The last three digits of 23102 is 102. Here 102 is not divisible by 8.

 ∴ 23102 is not divisible by 8. If the sum of the digits of a number is divisible by 3, the number is divisible by 3.

Example  :  Is 6234 divisible by 3? 

Solution :  The sum of the digits = 6 + 2 + 3 + 4 = 15 and 15 is divisible by 3.

 ∴ 6234 is divisible by 3. If the sum of the digits of a number is divisible by 9, the number is divisible by 9.

Example  :  Is 7623 divisible by 9?

Solution :  The sum of the digits 7 + 6 + 2 + 3 = 18 and 18  is divisible by 9.  

∴ 7623 is divisible by 9. Discuss :  If a number is divisible by 3 and also by 2, will it be divisible by 6?

If the difference of the sums of the digits in alternate places is divisible by 11, the number is divisible by 11. 

Objective Question On Number Problems

1. Which one of the following is not a prime number?

1. A. 31     
2. B. 61      
3. C. 71      
4. D. 91        Correct Answer

2.  

    

   A number is as much greater than 36 as is less than 86. Find the number.

1. A. 61       Correct Answer
2. B. 67      
3. C. 64      
4. D. 69      

3.  

    

   Find a number such that when 15 is subtracted from 7 times the number, the Result is 10 more than twice the number.

1. A. 7     
2. B. 11      
3. C. 5        Correct Answer
4. D. 6      

4.  

    

   The sum of a rational number and its reciprocal is 13/6. Find the number.

1. A. 1/3 or 2/3     
2. B. 3/2 or 1/3      
3. C. 2/3 or 3/2        Correct Answer
4. D. 1/2 or 1/3      

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.

1. A. 76     
2. B. 77      
3. C. 83      
4. D. 72        Correct Answer

6.  

    

   The difference of two numbers is 11 and one-fifth of their sum is 9. Find the numbers.

1. A. 30 and 12     
2. B. 12 and 17      
3. C. 28 and 17        Correct Answer
4. D. 32 and 11      

7.  

    

   The sum of two numbers is 16 and the sum of their squares is 113. Find the numbers.

1. A. 7 and 8       Correct Answer
2. B. 8 and 9      
3. C. 11 and 7      
4. D. 11 and 17      

8.  

    

   The average  of  four consecutive even numbers is 27. Find the largest of these numbers.

1. A. 32     
2. B. 36      
3. C. 38      
4. D. 30        Correct Answer

9.  

    

   The sum of the squares of three consecutive odd numbers is 2531.Find the numbers.

1. A. 7, 29 and 31     
2. B. 27, 29 and 31        Correct Answer
3. C. 17, 29 and 31      
4. D. 37, 29 and 31      

10.  

     

    If two numbers, 4 times the smaller one is less then 3 times the 1arger one by 5. If the sum of  the numbers is larger than 6 times their difference by 6, find the two numbers.

1. A. 59 and 41     
2. B. 54 and 41      
3. C. 44 and 448      
4. D. 59 and 43        Correct Answer

11.  

     

    The ratio between a two-digit number and the sum of the digits of that number is 4 : 1.If the digit in the unit's place is 3 more than the digit in the ten’s place, what is the number?

1. A. 36       Correct Answer
2. B. 31      
3. C. 32      
4. D. 38      
5. E. 42      

12.  

     

    If there are 6 terms in a series, then find the sum of geometric series 2, 6, 18, 54, ----

1. A. 758     
2. B. 728        Correct Answer
3. C. 754      
4. D. 738      

13.  

     

    Find the number of terms in geometric progression 3, 6, 12, 24, ---- , 384.

1. A. 10     
2. B. 11      
3. C. 9      
4. D. 8        Correct Answer

14.  

     

    If 6 + 12 + 18 + 24 + --- = 1800, then find the number of terms in the series.

1. A. 21     
2. B. 22      
3. C. 23      
4. D. 24        Correct Answer

15.  

     

    1 + 2 + 3 + ---- 50 = ?

1. A. 1275       Correct Answer
2. B. 1350      
3. C. 1575      
4. D. 1455      

16.  

     

    Find which of the following number is divisible by 11?

1. A. 246542     
2. B. 415624        Correct Answer
3. C. 146532      
4. D. 426513      

17.  

     

    Which of the following number is divisible by 24 ?

1. A. 35718     
2. B. 63810      
3. C. 537804      
4. D. 3125736        Correct Answer

18.  

     

    (?) + 3699 + 1985 - 2047 = 31111

1. A. 34748     
2. B. 27474        Correct Answer
3. C. 30154      
4. D. 27574      

19.  

     

    The difference between the local value and the face value of 7 in the numeral 32675149 is

1. A. 75142     
2. B. 64851      
3. C. 5149      
4. D. 69993        Correct Answer

20.  

     

    On dividing a number by 56, we get 29 as remainder. On dividing the same number by 8, what will be the remainder ?

1. A. 4     
2. B. 5        Correct Answer
3. C. 6      
4. D. 7      

21.  

     

    107 x 107 + 93 x 93 = ?

1. A. 19578     
2. B. 19418      
3. C. 20098        Correct Answer
4. D. 21908      
22. Direction To Solve:
    Direction: What will come in place of the question mark (?) in the following number series?

23.  

     

    17, 98, 26, 89, 35, ?

1. A. 80       Correct Answer
2. B. 75      
3. C. 67      
4. D. 53      
5. E. None of these      

24.  

     

    2, 17, 89, 359, 1079, ?

1. A. 2137     
2. B. 2121      
3. C. 2377      
4. D. 2159        Correct Answer
5. E. None of these      

25.  

     

    3, 5, 15, 45, 113, ?

1. A. 250     
2. B. 243        Correct Answer
3. C. 247      
4. D. 261      
5. E. None of these      

26.  

     

    4.5, 5.5, 12, 49, ?

1. A. 393       Correct Answer
2. B. 396      
3. C. 354      
4. D. 367      
5. E. None of these      

27.  

     

    3240, 540, 108, 27, ?, 4.5

1. A. 7     
2. B. 10      
3. C. 9        Correct Answer
4. D. 8      
5. E. None of these      

28.  

     

    The difference between a number and its three - fifth is 50. What is the number?

1. A. 75     
2. B. 100      
3. C. 125        Correct Answer
4. D. None of these      

29.  

     

    If a number is decreased by 4 and divided by 6, the result is 8. What would be the result if 2 is subtracted from the number and then it is divided by 5?

1. A. 9(2/3)     
2. B. 10        Correct Answer
3. C. 10(1/5)      
4. D. 11(1/5)      

30.  

     

    A number is doubled and 9 is added. If the resultant is trebled, it becomes 75. What is that number?

1. A. 3.5     
2. B. 6      
3. C. 8        Correct Answer
4. D. None of these      

31.  

     

    Three-fourth of a number is 60 more than its one-third. The number is

1. A. 84     
2. B. 108      
3. C. 144        Correct Answer
4. D. None of these      

32.  

     

    When 24 is subtracted from a number, it reduces to its four-seventh. What is the sum of the digits of that number?

1. A. 1     
2. B. 9      
3. C. 11        Correct Answer
4. D. Data Inadequate      
5. E. None of these      

33.  

     

    Find the number which when multiplied by 15 is increased by 196.

1. A. 14       Correct Answer
2. B. 20      
3. C. 26      
4. D. 28      

34.  

     

    If a number, when divided by 4, is reduced by 21, the number is

1. A. 18     
2. B. 20      
3. C. 28        Correct Answer
4. D. 38      

35.  

     

    A number whose fifth part increased by 4 is equal to its fourth part diminished by 10, is :

1. A. 240     
2. B. 260      
3. C. 270      
4. D. 280        Correct Answer

36.  

     

    The difference of two numbers is 20% of the larger number. If the smaller amount is 12, the larger one is :

1. A. 16     
2. B. 18      
3. C. 20      
4. D. 15        Correct Answer

37.  

     

    If one-seventh of a number exceeds its eleventh part by 100, then the number is 

1. A. 770     
2. B. 1100      
3. C. 1825      
4. D. 1925        Correct Answer

38.  

     

    If the sum of one-half and one-fifth of a number exceeds one-third of that number by 7(1/3), the number is

1. A. 15     
2. B. 18      
3. C. 20        Correct Answer
4. D. 30      

39.  

     

    If doubling a number and adding 20 to the reult gives the same answer as multiplying the number by 8 and taking away 4 from the product, the number is

1. A. 2     
2. B. 3      
3. C. 4        Correct Answer
4. D. 6      

40.  

     

    If 50 is subtracted from two-third of a number, the result is equal to sum of 40 and one-fourth of that number. What is the number?

1. A. 174     
2. B. 216        Correct Answer
3. C. 246      
4. D. 336      

41.  

     

    if the sum of a number and its square is 182. What is the number?

1. A. 26     
2. B. 28      
3. C. 91      
4. D. 15      
5. E. None of these        Correct Answer

42.  

     

    Twenty times a positive integer is less than its square by 96. What is the integer?

1. A. 24     
2. B. 30        Correct Answer
3. C. Cannot be Determined      
4. D. None of these      

43.  

     

    Thrice the square of a natural number decreased by 4 times the number is equal to 50 more than the number. The number is:

1. A. 5     
2. B. 60        Correct Answer
3. C. 100      
4. D. 41      

44.  

     

    The sum of a number and its reciprocal  is one-eighth of 34. What is the product of the number and its square root ?

1. A. 8       Correct Answer
2. B. 27      
3. C. 31      
4. D. None of These      

45.  

     

    Two-third of a positive number and 25 / 216 of its reciprocal are equal. The number is

1. A. 5 / 12       Correct Answer
2. B. 12 / 5      
3. C. 25 / 144      
4. D. 144 / 25      

46.  

     

    Find a positive number which when increased by 17 is equal to 60 times the reciprocal of the number:

1. A. 3       Correct Answer
2. B. 10      
3. C. 17      
4. D. 20      

47.  

     

    A positive number when decreased by 4 is equal to 21 times the reciprocal of the number. The number is

1. A. 3     
2. B. 5      
3. C. 7        Correct Answer
4. D. 9      

48.  

     

     The sum of a positive number and its reciprocal is thrice the difference of the number and its reciprocal. The number is

1. A. Sqrt (2)       Correct Answer
2. B. 1/ Sqrt (2)      
3. C. Sqrt (3)      
4. D. 1/Sqrt (3)      

49.  

     

    The product of two natural numbers is 17. Then the sum of the reciprocals of their square is

1. A. 1/289     
2. B. 289/290      
3. C. 290/289        Correct Answer
4. D. 289      

50.  

     

    Three numbers are in the ratio 4 : 5 : 6 and their average is 25. The largest number is

1. A. 30       Correct Answer
2. B. 32      
3. C. 36      
4. D. 42      

51.  

     

    Two numbers are such that the square of 1 is 224 less than 8 times the square of the other. If the number be in the ratio of 3:4, the numbers are

1. A. 6,8       Correct Answer
2. B. 9,12      
3. C. 12,16      
4. D. None of these      
5. 
   

Notes:

(1). Numbers ending with 0 are divisible by 10.  

(2). Numbers ending with 0 or 5 are divisible by 5.  

(3). Numbers ending with 0, 2, 4, 6 or 8 are divisible by 2.  

(4). If the number formed by the last two digits of a given number is divisible by 4, the given number will be divisible by 4.  

(5). If the number formed by the last three digits of a given number is divisible by 8, the given number will be divisible by 8.  

(6). If the sum of the digits of a number is divisible by 3, the number is divisible by 3.  

(7). If the sum of the digits of a number is divisible by 9, the number is divisible by 9.  

(8). If the difference of sums of the digits in alternate places is divisible by 11, the number is divisible by 11.

THINGS TO REMEMBER  

•  A number, which divides a given number leaving no remainder is called a divisor of the given number.  
•  Some numbers have only two divisors, that is 1 and the number itself.  
•  The divisors of a number other than ‘1’ and the number itself are called the factors of that number.  
•  The whole numbers which are multiples of 2 are called even numbers.  
•  The whole numbers which are not multiples of 2 are called odd numbers.  
•  The natural numbers which have only two divisors are called prime numbers.  
•  The natural numbers which have more than two divisors are called composite numbers.  
•  Composite number can be written as a product of prime numbers. 
•  The product of a number, when repeated more than once can be expressed in a simple way. It is known as exponential form.  
•  To denote power of a number, the index is written on the top, to the right of the base.  
•  When a number is multiplied by the same number, the result is known as square of that number.  (12) When a number is multiplied repeatedly thrice by itself, the result is the cube of that number. 

Problems on Calendar Based Question and shortcuts

What is Odd Day in Calendar Problem:

In a given days, the no. of days extra than weeks are called odd day.

For Example : How Many Odd Days in 9 day's?

Solutions: In 9 Days it contain 7 days ( 1 week ) + 2 days

: :  Here extra 2 Day's is treated as Odd Day's

Leap Year: [ calendar word problems ]

(i). Year which is divisible by 4 is called a leap year, if it is not a century.

(ii). Every 4th century ( 4*100, 4*100*2, 4*100*3 ) is a leap year and no other century is a leap year.

Note: A leap year Contain 366 days  [ Calendar Problem Questions ].

Examples:

This 1948, 2008, 1680 etc. is a leap year.

This 2000, 1600, 1200, 800, 400 etc. is a leap year.

This 2005, 2007, 2009, 1800, 2100 is  not a leap year.

Ordinary Year:

The year which is not a leap year is called an ordinary years. 

Note: An ordinary year Contain 365 days.  [ Word Problems on Calendar ]

Counting of Odd Days in calendar word problems:

1 year [ ordinary year ] = 365 days = (52 weeks + 1 day.)

1 ordinary year Contain 1 odd day.

1 year [ leap year ] = 366 days = (52 weeks + 2 days)

 1 leap year contain 2 odd days.

In 100 years = 76 ordinary years + 24 leap years

  = (76 x 1 + 24 x 2) odd days = 124 odd days.

  = (17 weeks + days)  5 odd days.

No. of odd days in 100 years = 5.

No. of odd days in 200 years = (5 x 2 = 10 days) [ 1 week + 3 days]  ⇒ 3 odd days.

No. of odd days in 300 years = (5 x 3 = 15 Days) [ 2 week's + 1 days]  ⇒ 1 odd day.

No. of odd days in 400 years = (5 x 4 + 1 = 21 Days) [ 3 Week's ]  ⇒  0 odd day.

Similarly, 800 years, 1200 years, 1600 years, 2000 years etc. contain 0 odd days.

Day of the Week Related to Odd Days in calendar math

No. of days:	0	1	2	3	4	5	6
Day:	Sunday	Monday	Tuesday	Wednesday	Thursday	Friday	Saturday

---

Objective Questions on Problems on Calendar

1. 
   

   It was Sunday on Jan 1, 2006. What was the day of the week Jan 1, 2010?

   or

   If it was sunday on jan 1st 2006. What was the day of the week Jan 1, 2010?

1. A. Sunday     
2. B. Saturday      
3. C. Friday        Correct Answer
4. D. Wednesday      

2.  

   
   

   What was the day of the week on 28th May, 2006?

1. A. Thursday     
2. B. Friday      
3. C. Saturday      
4. D. Sunday        Correct Answer

3.  

   
   

   What was the day of the week on 17th June, 1998?

1. A. Monday     
2. B. Tuesday      
3. C. Wednesday        Correct Answer
4. D. Thursday      

4.  

   
   

   What will be the day of the week 15th August, 2010?

1. A. Sunday       Correct Answer
2. B. Monday      
3. C. Tuesday      
4. D. Friday      

5.  

   
   

   Today is Monday. After 61 days, it will be:

1. A. Wednesday     
2. B. Saturday        Correct Answer
3. C. Tuesday      
4. D. Thursday      

6.  

   
   

   26th January, 1996 was a Friday. What day of the week lies on 26th January, 1997?

1. A. Saturday     
2. B. Sunday        Correct Answer
3. E. Thursday      

7.  

   
   

   How many days are there in x weeks x days?

1. A. 7x²     
2. B. 8x        Correct Answer
3. C. 14x      
4. D. 7      

8.  

   
   

   The calendar for the year 2001 is same for which of the following year?

1. A. 2005     
2. B. 2007        Correct Answer
3. C. 2011      
4. D. 2006      

9.  

   
   

   The last day of a century cannot be

1. A. Monday     
2. B. Wednesday      
3. C. Tuesday        Correct Answer
4. D. Friday      

10.  

    
    

    What day of the week was 31st July, 1993?

1. A. Monday     
2. B. Sunday      
3. C. Saturday        Correct Answer
4. D. Tuesday      

11.  

    
    

    January 1, 2008 is Tuesday. What day of the week lies on Jan 1, 2009?

1. A. Monday     
2. B. Wednesday      
3. C. Thursday        Correct Answer
4. D. Sunday      

12.  

    
    

    Which of the following year is not a leap year?

1. A. 1960     
2. B. 2080      
3. C. 2024      
4. D. 2100        Correct Answer

13.  

    
    

    The calendar for the year 2007 will be the same for the year:

1. A. 2014     
2. B. 2016      
3. C. 2017      
4. D. 2018        Correct Answer

14.  

    
    

    Which of the following is not a leap year?

1. A. 700       Correct Answer
2. B. 800      
3. C. 1200      
4. D. 2000      

15.  

    
    

    On 8th Dec, 2007 Saturday falls. What day of the week was it on 8th Dec, 2006?

1. A. Sunday     
2. B. Sunday      
3. C. Tuesday      
4. D. Friday        Correct Answer