(1.0002)(0.9999) – (1.0001)(0.9998) =

Question

A. 0.00000001
B. 0.00000002
C. 0.00000004
D. 0.0001
E. 0.0002

GreginChicago · Accepted Answer

Method #1: 
 Don’t actually multiply these numbers out! The key here is to substitute a variable for a tiny value that shows up in several places.

Specifically, let x = 0.0001. Now you can rewrite all the numbers as 1 plus or minus x or 2x.

(1.0002)(0.9999) – (1.0001)(0.9998)
= (1 + 2x)(1 – x) – (1 + x)(1 – 2x)

Now, distribute each product in the expression separately.

First product: (1 + 2x)(1 – x) = 1 + 2x – x – = 1 + x –

Second product: (1 + x)(1 – 2x) = 1 + x – 2x – = 1 – x –

Subtract the two products:

1 + x – – (1 – x – )
= 1 + x – – 1 + x + 
= 2x

Finally, substitute back in for x. The difference is 2(0.0001) = 0.0002.

It might seem odd to solve an arithmetic problem by turning it into algebra! But in this case, doing so saves you a ton of work.

Method #2: 
Another way to tackle the problem is to estimate judiciously. Of course, if you round every number to 1 in the original expression, you get 0. But consider rounding the numbers in this way:

(1.0002)(0.9999) – (1.0001)(0.9998)
≈ (1.0002)(1.0000) – (1.0001)(0.9999) — round both of the second numbers up by 0.0001, and because the first numbers in each product are approximately equal, you’ll only have a truly small rounding error
= 1.0002 – (1.0001)(0.9999). Now, if this were 1.0002 – 1.0001, you’d get 0.0001. But you’re subtracting something even smaller (since the 1.0001 is being multiplied by a number less than 1), so the difference must be larger than 0.0001.

The correct answer is E.

chetan2u · Answer

remove the decimal before calculations and get that back in the final answer..

so we reomove 8 decimal points from .0002 and .9999 and similarly from second term..

10002*9999-10001*9998 = (10000+2)(10000-1)-(10000+1)(10000-2)=  
 = 10000^2-2*10000-10000*1-10000^2+2*10000-1*10000 = 40000-20000=20000

now get 8 decimal points back in the answer
 \frac{20000}{10^8} = 0.0002

E

EMPOWERgmatRichC · Answer

Hi All,

This question actually has some nice shortcuts built into it that can help you avoid doing math the "long way":

The first part of the calculation...

(1.0002)(0.9999)

…will have 8 decimal places (4 decimal points x 4 decimal points = 8 total decimal points) and the last digit will be an 8 (since 2 x 9 = 18)

The second part of the calculation….

(1.0001)(0.9998)

….will also also have 8 decimal places (for the same reason that the first part has 8 decimal points) and the last digit will also be an 8 (1 x 8 = 8)

When subtracting the second value from the first value, the resulting number will have a '0' in the 8th decimal 'spot.' That clearly doesn't happen in the first 3 answer choices, so we can eliminate Answers A, B and C.

To get the exact correct answer, we have to do a little more work. This time, I'll start by breaking the second calculation into two pieces:

(1.0001)(0.9998) = 
(1)(0.9998) + (.0001)(0.9998) = 
0.9998 + 
0.00009998
---------
0.99989998

With the first calculation, we have to pay a bit more attention to the number of digits involved (remember though - there's still only 8 total decimal points):

(1.0002)(0.9999) +
(1)(0.9999) + (.0002)(0.9999) = 
0.9999 + 
0.00019998
---------
1.00009998

1.00009998 -
0.99989998

You should notice the 5th through 8th decimal points 'cancel out':

1.0000 -
0.9998

....leaving us with a 4th decimal point that must be a 2....

Final Answer:  E/spoiler]

GMAT assassins aren't born, they're made, 
Rich

pushpitkc · Answer

(1.0002)(0.9999) – (1.0001)(0.9998)
= (1+2*10^-4)(1-10^-4) - (1+10^-4)(1-2*10^-4)

Consider
1= a
10^-4= b

(a+2b)(a-b)-(a+b)(a-2b)
= (a^2)-(ab)+(2ab)-(2b^2)-(a^2)+(2ab)-(ab)+(2b^2)
= 2ab
= 2(1)(10^-4)
= 0.0002 (E)

Antreev · Answer

You changed the question. You have 8 decimal places both sides plus you changed the actual numbers.

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GyMrAT · Answer

(1.0002)(0.9999) – (1.0001)(0.9998) = (1 + 0.0002)(1 - 0.0001) - (1 + 0.0001)(1 - 0.0002)  
 
  = (1 + 2*10^{-4})(1 - 1*10^{-4}) - (1 + 1*10^{-4})(1 - 2*10^{-4})

= 1 - 1*10^{-4} + 2*10^{-4} - 2*10^{-8} - 1 + 2*10^{-4} - 1*10^{-4} + 2*10^{-8}  
 
  = 10^{-4} * (-1 + 2 + 2 - 1) = 0.0002

Answer E.

Thanks,
GyM

EMPOWERgmatRichC · Answer

Hi Antreev,

Thank you for catching that error. I accidentally copied over an explanation for a similar OG question (in the OG2018, it's PS question #216 on pg. 178). I've edited my original post accordingly.

GMAT assassins aren't born, they're made,
Rich

GMATGuruNY · Answer

One more approach:

(1.0002)(0.9999) – (1.0001)(0.9998)

=    (1 + 0.0001    +   0.0001)   (1 - 0.0001)    -    (1 + 0.0001)(1 - 0.0001    -   0.0001)

=    (1 + 0.0001)(1 - 0.0001)    +   (0.0001)(1 - 0.0001)   -    (1 + 0.0001)(1 - 0.0001)    +   (0.0001)(1 + 0.0001)

=   0.0001 - (0.0001)^2 + 0.0001 + (0.0001)^2

= 0.0002

E

Antreev · Answer

OR after the last third step you could have just cancelled first and third items as they have opposite signs and solved the rest like this.
0.0001(1-0.0001+1+0.0001)
0.0001(2)
0.0002 so E

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GMATGuruNY · Answer

Yes, indeed.
I considered this approach but suspected that most test-takers would intuitively distribute 0.0001 in the second and fourth terms rather than factor it out.

loneRanger1050 · Answer

We can write decimals as..

\frac{10002}{10^4}  *  \frac{9999}{10^4}  ] -  \frac{10001}{10^4}  * \frac{9998}{10^4}  ]

Let say, x = 10000 =  10^4

Solving for numerator, since the denominator is  10^8

(x+2)(x-1) - (x+1)(x-2)

(x^2 + x - 2) - (x^2 - x - 2)  =  2x

Final solution =  \frac{2x}{10^8}  =  (2*10^4)/10^8  =  \frac{2}{10^4} = 0.0002

lincy93 · Answer

(1.0002)(0.9999)-(1.0001)(0.9998)
=(1.0001+0.0001)(0.9998+0.0001)-(1.0001)(0.9998)
1.0001=a
0.0001=x
0.9998=b
=(a+x)(b+x)-ab
=ab+x(a+b)+x^2-ab
=x(a+b+x)
=0.0001*2
=0.0002

GyMrAT · Answer

Thats a very efficient solution. Smart!!

kdgupta · Answer

(1.0002)*(0.9999)-(1.0001)*(0.9998) Can also be expressed as 
(1+2/10^4)*(1-1/10^4)-(1+1/10^4)*(1-2/10^4)

if 1/10^4 = a , then 
(1+2a)*(1-a)-(1+a)*(1-2a) 
= 1+2a-a-2a^2-1-a+2a+2a^2
=4a-2a
=2a 
=2*1/10^4
=0.0002

GmatPoint · Answer

(1.0002)(0.9999) – (1.0001)(0.9998) =.
﻿Consider a = 1, b = 1, x = 0.0002, y = 0.0001.
﻿The product can be written as :
﻿(a+x)*(b -y) - (a+y)*(b-x).
﻿This when expanded we have :
﻿ab-ay+bx-xy - (ab-ax+by-xy).
﻿ax-ay +bx - by.
﻿(x-y)*(a+b).
﻿(0.0001)*(1+1) = 0.0002.
﻿

webarebears · Answer

(1.0002)(0.9999) – (1.0001)(0.9998)
= (1+2*10^-4)(1-10^-4) - (1+10^-4)(1-2*10^-4)

Consider
1= a
10^-4= b

(a+2b)(a-b)-(a+b)(a-2b) 
= (a^2)-(ab)+(2ab)-(2b^2)-(a^2)+(2ab)-(ab)+(2b^2)
= 2ab
= 2(1)(10^-4)
= 0.0002 (E)

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(1.0002)(0.9999) – (1.0001)(0.9998) =

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(1.0002)(0.9999) – (1.0001)(0.9998) =

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