## Number Question

Consider a real number x such that x = ab.cd, where a, b, c and d are the digits of the number . Also, a and d is not equal to 0.

[x] = The greatest integer less than or equal to x and {x} = x – [x]

## Solution

Consider a real number x such that x = ab.cd, where a, b, c and d are the digits of the number . Also, a and d is not equal to 0.

[x] = The greatest integer less than or equal to x and {x} = x – [x]

if [10{p}] = 8 and {10{p}}=9

Also, [p] is 5 less than the sum of the digits of the quantity 100{p}. Find p

a. 21.89

b. 89.12

c. 12.89

d. None of the above

We can always work with options.

Suppose option (a) is the correct answer.

x = 21.89

[x] = [21.89] = 21

{x} = x - [x] = 21.89 - 21 = 0.89

Now, note that:

[10{p}] = [10(0.89)] = [8.9] = 8

{10{p}} = {10(0.89)} = {8.9} = 8.9 - [8.9] = 8.9 - 8 = 0.9

which does not satisfy the given condition of the problem.

Hence option (a) is not the correct answer.

Next suppose option (b) is the correct answer.

x = 89.12

[x] = [89.12] = 89

{x} = x - [x] = 89.12 - 89 = 0.12

Now, note that:

[10{p}] = [10(0.12)] = [1.2] = 1

which does not satisfy the given condition of the problem.

Hence option (b) is not the correct answer.

Now suppose option (a) is the correct answer.

x = 12.89

[x] = [12.89] = 12

{x} = x - [x] = 12.89 - 12 = 0.89

Now, note that:

[10{p}] = [10(0.89)] = [8.9] = 8

{10{p}} = {10(0.89)} = {8.9} = 8.9 - [8.9] = 8.9 - 8 = 0.9

which does not satisfy the given condition of the problem.

Hence option (c) is also not the correct answer.

Therefore, option (d) is correct.

Ravi Raja