IBM Model Problems On Decimal Fraction

 Below are three problems which are based on simplification of decimal fractions.

 Question 1

 The expression {(1.21 x 1.1 + 3 x 1.21 x 0.2 + 3 x 0.2 x X + 0.008) / 1.3 } will be a perfect square for X equal to ___

 a)0.22
 b)1.69
 c)1.21
 d)1.1

 Answer : a)0.22

 Solution:

 Given expression can be rewritten as (1.1 x 1.1 x 1.1 + 3 x 1.1 x 1.1 x 0.2 + 3 x 0.2 x X + 0.2 x 0.2 x 0.2)/1.3

 The key to solving these types of problems is to try to match a part/entire expression to any familiar formula we already know.

 Coming to our problem, the numerator is closely resembling the form a3 + 3(a2)b + 3a(b2) + b3 which is the expansion of (a+b)3 .

 Hope you remember the familiar formula (a+b)3 = a3 + 3(a2)b + 3a(b2) + b3. (Keep this formula and expansion in mind right through solving the problem. This will help a lot in taking correct decisions while simplifying.)

 For such problems, it is wise to rewrite the denominator to contain the most commonly used terms in the numerator. In numerator, we predominantly find two terms, 1.1 and 0.2.

 Therefore, let us rewrite the denominator 1.3 as (1.1 + 0.2)

 Then the expression becomes (1.13 + 3 x (1.12) x 0.2 + 3 x 0.2 x X + 0.23) / (1.1+0.2).

 For the expression to be a perfect square then numerator must be a perfect cube. i.e if the numerator is (1.1+0.2)3, then when divided by (1.1+0.2), the expression will become (1.1+0.2)2 which is a perfect square.

 Therefore, finding the value for X, for which numerator is (1.1 + 0.2)3 is our goal.

 Let us equate numerator to (1.1 + 0.2)3

 (1.13 + 3 x(1.12) x 0.2 + 3 x 0.2 x X + 0.23) = (1.1 + 0.2)3 ...(1)

 (1.1 + 0.2)3 can be expanded as (1.1 x 1.1 x 1.1 + 3 x 1.1 x 1.1 x 0.2 + 3 x 0.1 x 0.2 x 0.2 + 0.2 x 0.2 x 0.2)

 Therefore, equation 1, becomes

 (1.13 + 3 x(1.12) x 0.2 + 3 x 0.2 x X + 0.23) = (1.1 x 1.1 x 1.1 + 3 x 1.1 x 1.1 x 0.2 + 3 x 1.1 x 0.2 x 0.2 + 0.2 x 0.2 x 0.2)

 1st,2nd and 4th terms are equal on Left and Right hand sides. Hence they will cancel out.

 Therefore, we have, 3 x 0.2 x X = 3 x 0.1 x 0.2 x 0.2

 Or X = 3 x 1.1 x 0.2 x 0.2 / 3 x 0.2

 = 1.1 x 0.2 = 0.22

 Thus, the given expression will be a perfect square if X=0.22.

 Question 2

 Evaluate [(1.386 x 0.643 + (2.921-1.535)x 0.357)/(0.6 x 0.015 + 0.6 x 0.985)]x(3 x 2.7 + 3 x 0.3)

 a)20.00
 b)18.19
 c)20.79
 d)19.19

 Answer : c)20.79

 Solution:

 The given expression can be simplified as follows;

 [(1.386 x 0.643 + (1.386) x 0.357)/(0.6 x 0.015 + 0.6 x 0.985)]x(3 x 2.7 + 3 x 0.3)

 = {[1.386 x (0.643 + 0.357)]/[0.6(0.015 + 0.985)]} x [3(2.7 + 0.3)]

 = {[1.386 x 1.000]/[0.6 x 1.000]} x [3x3]

 = [1.386 / 0.6] x [3 x 3] = 1.386 x 3/0.2 = 6.93 x 3 = 20.79

 Question 3

 The value of [(88.53 - 66.98)/(88.53 + 66.98)] / [(885.3 - 669.8)/(8.853 + 6.698)] is:

 a)1/10
 b)1/100
 c)100
 d)./1

 Answer : b)1/100

 Solution:

 The given expression can be simplified as follows;
  (88.53-66.98)   (8.853+6.698)
  ------------- x -------------
  (88.53+66.98)  (885.3-669.8)

  (88.53-66.98)   (8.853+6.698)
  ------------- x ------------- (We have just rearranged the order of denominators)
 (885.3-669.8)   (88.53+66.98)

(8.853+6.698) can be rewritten as (88.53-66.98) x 10

(885.3-669.8) can be rewritten as (88.53-66.98)/10


Therefore, above expression becomes,

    (88.53-66.98)   (88.53+66.98)/10
    ------------- x -------------
    (88.53-69.98)10  88.53+66.98)

    Cancelling out the terms same in numerator and denominators, the above expression becomes 1/10 x 1/10 = 1/100

    Hence 1/100 is the required answer.