Question

four digit number has first digit is half of third digit, total of second and third digit is eight, forth digit is multiply of first and third digit and total of all four digit is twelve

Answer 1

Given :

For a four digit number

First digit is half of third digit

The sum of second digit and third digit = 8

Fourth digit = first digit × third digit

The sum of total four digits = 11

To Find :

The four digit number

Solution :

Let The First digit = a

The second digit = b

The third digit = c

The fourth digit = d

According to question

Statement I

First digit is half of third digit

i.e     a = $\dfrac{c}{2}$                      .........1

Statement II

The sum of second digit and third digit = 8

i.e   b + c = 8                 ...........2

Statement III

Fourth digit = first digit × third digit

i.e   d  =  a  ×  c            ..........3

Statement IV

The sum of total four digits = 12

i.e  a + b + c + d = 11             ...........4

Now,

From eq 2 and eq 4

  a + 8 + d = 11

Or,  a + d = 11 - 8

Or, a + d = 3

∵  from eq 3 ,

  a + a c = 3

And from eq 1

   c = 2 a

So,  a + a × 2 a = 3

Or,  2 a² + a - 3 = 0

Or,   2 a² - 2 a + 3 a - 3 = 0

Or,  2 a ( a - 1 ) + 3 ( a - 1 ) = 0

So, ( a - 1 ) ( 2 a + 3 ) = 0

∴     a = 1

So, c = 2 a = 2 × 1 = 2

i.e  c = 2

∵  a + d = 3

So,    d = 3 - a

           = 3 - 1

           = 2

i.e d = 2

∵  b + c = 8

So,  b = 8 - c

          = 8 - 2

           = 6

i.e  b = 6

So, number = 1000 × a + 100 × b + 10 × c + 1 × d

                   = 1000 × 1 + 100 × 6 + 10 × 2 + 1 × 2

                   = 1000 + 600 + 20 + 2

                   = 1622

Hence, The four digit number is  1622  . Answer