Q. 199 on OG 13
(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
The right answer is D. Is there any shortcut for this question? The solution in the OG is tedious.
Complex decimal problem - any shortcuts?
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- rachitakapoor
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(0.99999999 ÷ 1.0001) - (0.99999991 ÷ 1.0003)rachitakapoor wrote:Q. 199 on OG 13
(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
The right answer is D. Is there any shortcut for this question? The solution in the OG is tedious.
(9999.9999/10001) - (9999.9991/10003)
(9999 * 1.0001/10001) - (9997 * 1.0003/10003)
(9999 * 10^-4) - (9997 * 10^-4)
0.9999 - 0.9997
0.0002
Option D
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Approach 1:rachitakapoor wrote:Q. 199 on OG 13
(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
The right answer is D. Is there any shortcut for this question? The solution in the OG is tedious.
(9999.9999)/(10001) - (9999.99991)(10003)
= (10^-4)(99999999/10001) - (10^-4)(99999991/10003)
= (10^-4) (99999999/10001 - 99999991/10003)
Only C, D and E include 10^-4.
Thus, the correct answer must be C, D or E, implying three possibilities:
99999999/10001 - 99999991/10003 = 3.
99999999/10001 - 99999991/10003 = 2.
99999999/10001 - 99999991/10003 = 1.
Since their difference is an integer, the quotients here are almost certainly integer values themselves.
Focus on the UNITS DIGITS:
The units digit of 99999999/10001 must be 9, since 10001 must be multiplied by a units digit of 9 to yield 99999999.
The units digit of 99999991/10003 must be 7, since 10003 must be multiplied by a units digit of 7 to yield 99999991.
Since 9-7=2:
99999999/10001 - 99999991/10003 = 2.
The correct answer is D.
Approach 2:
(9999.9999)/10001 - (9999.9991)/10003
= [10003(9999.9999) - 10001(9999.9991)] / (10001)(10003)
≈ [10003(10^4) - 10001(10^4)] / (10^4)(10^4)
≈ (10003-10001)/(10^4)
≈ 2(10^-4).
The correct answer is D.
Approach 3:
Try a simpler version of the problem.
.99/1.1 - .91/1.3
= 99/110 - 91/130
= ( 99/11 - 91/13) (1/10)
= (9-7)(10^-1)
= 2(10^-1).
This result suggests that the correct answer almost certainly is D.
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- rachitakapoor
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Many thanks, rijul007!rijul007 wrote:(0.99999999 ÷ 1.0001) - (0.99999991 ÷ 1.0003)rachitakapoor wrote:Q. 199 on OG 13
(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
The right answer is D. Is there any shortcut for this question? The solution in the OG is tedious.
(9999.9999/10001) - (9999.9991/10003)
(9999 * 1.0001/10001) - (9997 * 1.0003/10003)
(9999 * 10^-4) - (9997 * 10^-4)
0.9999 - 0.9997
0.0002
Option D
- rachitakapoor
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Thanks a tonne for all the approaches! Looking at various approaches does help a lot and I'm gonna try finding atleast 2 of them for the tough problems.GMATGuruNY wrote:Approach 1:rachitakapoor wrote:Q. 199 on OG 13
(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
The right answer is D. Is there any shortcut for this question? The solution in the OG is tedious.
(9999.9999)/(10001) - (9999.99991)(10003)
= (10^-4)(99999999/10001) - (10^-4)(99999991/10003)
= (10^-4) (99999999/10001 - 99999991/10003)
Only C, D and E include 10^-4.
Thus, the correct answer must be C, D or E, implying three possibilities:
99999999/10001 - 99999991/10003 = 3.
99999999/10001 - 99999991/10003 = 2.
99999999/10001 - 99999991/10003 = 1.
Since their difference is an integer, the quotients here are almost certainly integer values themselves.
Focus on the UNITS DIGITS:
The units digit of 99999999/10001 must be 9, since 10001 must be multiplied by a units digit of 9 to yield 99999999.
The units digit of 99999991/10003 must be 7, since 10003 must be multiplied by a units digit of 7 to yield 99999991.
Since 9-7=2:
99999999/10001 - 99999991/10003 = 2.
The correct answer is D.
Approach 2:
(9999.9999)/10001 - (9999.9991)/10003
= [10003(9999.9999) - 10001(9999.9991)] / (10001)(10003)
≈ [10003(10^4) - 10001(10^4)] / (10^4)(10^4)
≈ (10003-10001)/(10^4)
≈ 2(10^-4).
The correct answer is D.
Approach 3:
Try a simpler version of the problem.
.99/1.1 - .91/1.3
= 99/110 - 91/130
= ( 99/11 - 91/13) (1/10)
= (9-7)(10^-1)
= 2(10^-1).
This result suggests that the correct answer almost certainly is D.
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Yes!! This is an awesome strategy that all students should practice while preparing for the GMAT (for all math questions).rachitakapoor wrote: Looking at various approaches does help a lot and I'm gonna try finding at least 2 of them for the tough problems.
Cheers,
Brent
- KateG1302
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Hi,rijul007 wrote:(0.99999999 ÷ 1.0001) - (0.99999991 ÷ 1.0003)rachitakapoor wrote:Q. 199 on OG 13
(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
The right answer is D. Is there any shortcut for this question? The solution in the OG is tedious.
(9999.9999/10001) - (9999.9991/10003)
(9999 * 1.0001/10001) - (9997 * 1.0003/10003)
(9999 * 10^-4) - (9997 * 10^-4)
0.9999 - 0.9997
0.0002
Option D
sorry but I just don't understand how to get from:
(9999.9999/10001) - (9999.9991/10003)
to
(9999 * 1.0001/10001) - (9997 * 1.0003/10003)
I see how the third approach works (i.e. using "simplified" numbers) but I am worried that I will not recognise this pattern during the real test, so I'd like to understand the calculations properly.
Thanks in advance !
Kate
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Hi Kate,KateG1302 wrote: Hi,
sorry but I just don't understand how to get from:
(9999.9999/10001) - (9999.9991/10003)
to
(9999 * 1.0001/10001) - (9997 * 1.0003/10003)
I see how the third approach works (i.e. using "simplified" numbers) but I am worried that I will not recognise this pattern during the real test, so I'd like to understand the calculations properly.
Thanks in advance !
Kate
The step from (9999.9991/10003)
to
(9997 * 1.0003/10003)
is a gigantic step. Don't worry, the vast majority of students won't see it either.
I believe that the test maker wants us to recognize that both 9999.9999 and 9999.9991 can be rewritten as differences of squares.
First, 0.99999999 = 1 - 0.00000001
= (1 - 0.0001)(1 + 0.0001)
Similarly, 9999.9991 = 1 - 0.00000009
= (1 - 0.0003)(1 + 0.0003)
Original question: 0.99999999/1.0001 - 0.99999991/1.0003
= (1 - 0.0001)(1 + 0.0001)/(1.0001) - (1 - 0.0003)(1 + 0.0003)/(1.0003)
= (1 - 0.0001)(1.0001)/(1.0001) - (1 - 0.0003)(1.0003)/(1.0003)
= (1 - 0.0001) - (1 - 0.0003)
= 0.0002
= 2 x 10^(-4) = D
Cheers,
Brent