Problems On Number

Problems On Number
Example 1.
                A number is as much less than 54 is more than 24.Find the number?
Explanation :-Let the number be x.
                        Given 54-x = x-24
                                        2x = 54 + 24= 78
                                         x  = 78/2= 39
Example 2.

                 If the digits of a two-digit number are interchanged, the number so formed is 18

                 more than the original number. The sum of digits is 8. What is the thrice value of

                 original number?

Explanation :-(1) Let the unit digit of number is y and tens place digit of number is x

                          Then original number is 10x + y.

                          Interchanging digit number is 10y + x.

                           10y + x = 10x + y +18.

                            -9x + 9y = 18

                            -x + y = 2     ............(1)

                             x + y = 8      ...........(2)

                             2y = 10

                               y = 5

                               x = 3

                     Hence the original number is 10x + y = 10✖ 3 + 5 = 35.
                   (2) Let the unit digit of number is x
                         then tens digit f number is 8-x
                        Then original number is 10(8-x) + x = 80-9x.
                         Interchanging digit number is 10x + (8-x) = 8 + 9x
                         8 + 9x = 80 - 9x + 18
                         18x = 90
                             x = 5
                      Hence the original number is 80 - 9x = 80 -9✖ 5 = 35
Example 3.

                 Sum of two numbers is 24 and their product is 140, then find the numbers? 

Explanation :- Let one number be x then another number be 24-x.
                              x✖(24-x) = 140
                              24x-x²= 140                               

                            x²-24x+140=0
                             x²-14x-10x+140=0
                             x(x-14)-10(x- 14)=0
                             (x-14)(x-10) =
                             x=14,     x = 10
Example 4.

                The  ten’s place digit of a two digit number is twice  the unit’s place digit.
                If the sum of this number and the number formed by interchanging  the digits is 66. 
                Find the number?

Explanation :- Let unit’s place digit of number be x 
                        Ten’s place digit = 2x .
                        Then original number is 10(2x) + x = 21x.
                         Interchanging digit number is 10x + 2x = 12x
                        21x + 12x = 66
                        33x = 66
                           x = 2
                     Hence original number is = 21x= 42
Example 5.

                The sum of the digits of a two digit number is 10. 
                The tens place digit is 4 times the unit place digit. Find the number?

Explanation :- Let unit’s place digit of number be x 
                          and tens place digit of number is 10-x

                          Then original number is 10(10-x)x + x = 100-9x.

                          10-x = 4x
                           5x = 10
                             x = 2
                         Then original number is 10(10-x)x + x = 100-9x
                                                                                        = 100 -18
                                                                                        = 82
Example 6.

                The sum of the digits of a two digit number is 9.
                 If the digits are reversed, the new number is 9 less than 3 times
                 the original number. Find the original number.

Explanation :- Let unit’s place digit of number be x 
                          and tens place digit of number is 9-x

                          Then original number is 10(9-x)x + x = 90-9x.
                           Interchanging digit number is 10x +( 9-x )= 9x + 9
                           9x+9 = 3(90-9x) - 9
                             36x = 252
                                x = 7
                        Hence original number is = 90-9x = 90-63= 27.
Example 7.             

              A two-digit number is such that the product of the digit 8. 

              When 18 is added to the number the digit are reversed. What is the number?

Explanation :- Let unit’s place digit of number be x 

                          then  ten’s place digit of number be 8/x
                          Then original number is 10✖8/x + x = 80/x+x.
                           Interchanging digit number is 10x +8/x
                           80/x + x +18 = 10x +8/x
                           80 +x²+18x = 10x²+8
                           9x²-18x-72 = 0
                            x²-2x-8 = 0
                             x²-4x+2x-8 = 0
                             x(x-4)+2(x-4)=0
                             x=4,x=-2
                        Hence original number is = 80/4 +4 .
Example 8.

                 The digits of a two-digit number differ by 5. what is the difference between

                  the number and the number formed by reversing its digits?

                

Explanation :- Let unit’s place digit of number be x 
                          and tens place digit of number is 5+x

                          Then original number is 10(5+x)x + x = 50+11x.
                          Interchanging digit number is 10x +5+x=5+11x
                          (50+11x)-(5+11x)=45
                          Hence the number and interchanging number differ by 45.

Chain Rule

 Chain Rule :-

1. Direct Proportion

                               Two quantities are said to be directly proportional,

                               if on the increase or decrease of one, the other increases or decreases

                               the same extent.

Examples 

              (a) Cost of the goods is directly proportional to the number of goods.

                   (More goods, More cost)

              (b) Amount of work done is directly proportional to the number of persons

                   who did the work.     (More persons, More Work)

2. Indirect Proportion (inverse proportion)

                          Two quantities are said to be indirectly proportional

                          (inversely proportional) if on the increase of one, the other

                           decreases to the same extent and vice-versa.

Examples

             (a) Number of days needed to complete a work is indirectly proportional

                (inversely proportional) with the number of persons who does the work

                (More Persons, Less Days needed)

             (b) The time taken to travel a distance is indirectly proportional

                (inversely proportional) with the speed in which one is travelling 

                (More Speed, Less Time)

Note :-In solving questions by chain rule, we compare every item with the term to be find out.

Example 1.If  the price of 12 toys is Rs. 270, what will be the price of 10 toys? 

Explanation:
                  More Toys,More price            (Direct Proportion )

                 12 : 10 ::270 : x
                 12✖x = 270✖10

                  x =(270✖10)/12 =225.

Example 2.If 36 men can do a piece of work in 25 days, then in how many days will 15 men do it? 

Explanation:
                  More men,Less days            (Indirect Proportion )

                 15 : 36 ::25 : x
                 15✖x = 36✖25

                  x =(36✖25)/15 =60
Example 3.39 persons can repair a road in 12 days, working 5 hours a day.
                  In how many days will 30 persons, working 6 hours a day, complete the work?

Explanation:

           Let the required number of days be x.

            Less Persons, More days                          (Indirect Proportion )

             More working hour per day, Less days   (Indirect Proportion )

Persons	30	:	39		:: 12 : x
Working hours/day	6	:	5

              30 ✕ 6 ✕ x = 39 ✕ 5 ✕ 12

x =	(39 ✕ 5 ✕ 12)
	(30 ✕6)

                          x = 13.

Example 4.If 30 men can build a wall 56 m long in 4 days, what length of a similar
                  wall can be built by 25 men in 3 days?

Explanation:

                 
      Let the required length be x meter.
      More Persons, More length built                          (Direct Proportion )
      Less days, Less length built                                  (Direct Proportion )
     

Persons	30	:	25		:: 56 : x
days	4	:	3

              30 ✕ 4 ✕ x = 25 ✕ 3 ✕ 56

x =	(25 ✕ 3 ✕ 56)
	(30 ✕4)

                          x = 35 meters.

Example 5.A soft drink machine fills 980 bottles of 250 g in 7 hours.
                   How many bottles of 300 g can be filled in 6 hours?

Explanation:

                 Let the required number of bottles be x.

                 More hours, More fill bottles            (Direct Proportion )

                 More g in Bottles , Less fill bottles   (Indirect Proportion )

                

Hours	7	:	6		:: 980 : x
g	300	:	250

          300 ✕ 7 ✕ x = 250✕ 6 ✕ 980

x =	(250 ✕ 6✕ 980)
	(300 ✕ 7)

          x = 700.

Example 6. If 6 persons can do a piece of work in 10 days working 7 hours a day.
                   How many days will it take to complete a piece of work twice 14 persons
                   working together for 8 hours a day

Explanation:
                     Let the required number of days be x.
                    More work, More days                      (Direct Proportion )
                    More boys, Less days                        (Indirect Proportion )
                     More hours per ay, Less days            (Indirect Proportion )

           

1✕14✕8✕x = 2✕6✕7✕10

                                                       

x =	(2 ✕ 6✕ 7✕10)
	(1 ✕14✕8)

                                                           x = 15/2 days

Problems on Ages

Problems on Ages :-
Example 1. :- The ratio of present ages of Ram and his father is 2:9 . The father's age
                    at time of Ram's birthday was 28 years.the present ages of the father and Ram.
Explanation :- Let the present ages of Ram and his father is 2x and 9x
                        Ram age = father age -28
                            2x=   9x-28
                            7x = 28
                              x = 4
                        Hence  present ages of Ram and his father is 8 and 36.
     
Example 2. :-The age of the father 3 years ago was 7 times the age of his son. 

                       At present the father’s age is five times that of his son. What are the

                       present ages of the father and son?

Explanation :- Let the present age of son and father are x  and 5x . 

                       Three years ago  age of son and father are x-3  and 5x-3 .                      

                        7(x – 3) = 5x – 3

                         7x – 21 = 5x – 3

                         2x = 18

                       Hence, son’s age = 9 yrs.  Fathers’ age = 45 years.                        

Example 3. :-The age of father is thrice that of his son. After 12 years,

                       the age of the father will be twice that of his son. What is the

                        present age of the son and father?

Explanation :- Let the present age of son and father are x  and 3x . 

                       After 12 years age of son and father are x+12  and 3x+12.                      

                        3x+12 = 2(x+12)

                         x = 12

                       Hence  present ages of son and his father is 12 and 36.

Example 4. :-The ratio between Ram and Shyam ages are 5:6 respectively.

                       After 10 years the ratio between age of Ram and  Shyam’s age will be

                       7:8. Then what is Shyam’s present age?

Explanation :- Let the present ages of Ram and Shyam are 5x and 6x

                        After 10 years age of Ram and Shyam are 5x + 10 and 6x + 10

                         (5x+10)/(6x + 10) = 7/8

                          40x + 80 = 42x + 70

                           2x = 10

                             x = 5 

                         Hence  present ages of Ram and Shyam are 25 and 30.

Example 5. :-The product of the ages of Ram an Shyam is 240.If the twice the age of Shyam is 4                                year more to Ram age.Then what is Ram’s present age?

Explanation :- Let the present ages of Ram is x 

                        Then Shyam age 240/x

                        2*240/x = x + 4

                         x²+4x = 480

                         x²+4x -480 = 0

                         x²+24x - 20x -480= 0

                         x(x+24) - 20(x+24) = 0

                         x=-24, x=20

                         Hence  present ages of Ram and Shyam are 20 and 12.

Example 6. :-The sum of ages of 5 children born at the intervals of 3 years each is 50 years. 

                       What is the age of the youngest child?

Explanation :- Let the  ages of children are x,x+3,x+6,x+9,x+12.

                       x+(x+3)+(x+6)+(x+9)+(x+12) = 50

                       5x + 30 = 50

                       5x = 20

                       x = 4

                       Hence age of the youngest child = 4

Example 7. :-A father said to his son, "I was as old as you are at the present 

                      at the time of your birth". If the father's age is 38 years now, the son's age

                      five years back was?

Explanation :-Let the son's present age be x years. 

                      Then x years before father age  (38 - x)

                        (38-x)   = x

                           2x = 38.

                             x = 19.

                       Son's age 5 years back (19 - 5) = 14 years.

Example 8. :-Six years ago, the ratio of the ages of Kunal and Sagar was 6 : 5.

                       Four years hence, the ratio of their ages will be 11 : 10. 

                      What is Sagar's age at present?

Explanation :-Let the ages of Kunal and Sagar  6 years ago be 6x and 5x .

                       Present ages of Kunal and Sagar  are 6x +6 and 5x+6 years

                       4 years hence ages of Kunal and Sagar will be 6x +6+4 and 5x+6 + 4.

Then,	(6x + 6) + 4	=	11
	(5x + 6) + 4		10

                       10(6x + 10) = 11(5x + 10)

                         5x = 10

                         x = 2.

                       Hence Sagar's present age = (5x + 6) = 16 years.