Problem Solving

(0.99999999/1.0001) - (0.99999991/1.0003) =

a) 10^-8
b) 3(10^-8)
c) 3(10^-4)
d) 2(10^-4)
e) 10^-4

Many thanks,

Vitto

0.99999999/1.0001 - .99999991/1.0003 =

10^-8

3(10^-8)

3(10^-4)

2(10^-4)

10^-4

Approach 1:
In each answer choice, exactly ONE DIGIT is a positive integer.
Try to determine the value of this digit.

Let x = .99999999/1.0001
Then:
1.0001x = .99999999
10001x = 9999.9999.
The digits in red imply that the rightmost digit of x must be 9, since 1*9 = 9.

Let y = .99999991/1.0003
Then:
1.0003y = .99999991
10003y = 9999.9991.
The digits in red imply that the rightmost digit of y must be 7, since 3*7 = 21.

Thus:
0.99999999/1.0001 - .99999991/1.0003
= x-y
= (value with a rightmost digit of 9) - (value with a rightmost digit of 7)
= value with a rightmost digit of 2.
The correct answer choice must include a digit of 2.

The correct answer is D.

Approach 2:
(x+y)(x-y) = x² - y².
In the identity above, x+y and x-y are called CONJUGATES.

It is possible to rephrase decimals as follows:
1.01 = 1 + .01.
.99 = 1 - .01.

Notice that (1 + .01) and (1 - .01) are CONJUGATES:
= (1 + .01)(1 - .01)
= 1² - (.01)²
= 1 - .0001
= .9999.
Notice also that the product of these conjugates (.9999) is ALMOST IDENTICAL to one of the numerators in the problem above (.99999999).

The two DENOMINATORS in the problem above can be rephrased as follows:
1.0001 = 1 + .0001
1.0003 = 1 + .0003.

In order for these two denominators to CANCEL OUT, the two NUMERATORS are almost certainly composed of the following sets of CONJUGATES:
(1 + .0001)(1 - .0001)
(1 + .0003)(1 - .0003).

Thus:
0.99999999/1.0001 - .99999991/1.0003

= [(1 + .0001)(1 - .0001) / (1 + .0001)] - [(1 + .0003)(1 - .0003) / (1 + .0003)]

= (1 - .0001) - (1 - .0003)

= .0002

= 2 * 10^(-4).

The correct answer is D.

The full solution below is taken from the GMATFix App.



-Patrick