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.