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(1-1/10)(1-1/11)(1-1/12)…(1-1/100) =?

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by Max@Math Revolution » Mon Oct 15, 2018 11:24 pm

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E

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[Math Revolution GMAT math practice question]

(1-1/10)(1-1/11)(1-1/12)...(1-1/100) =?

A. 1/10
B. 9/10
C. 9/100
D. 1/100
E. 9/50
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by GMATGuruNY » Tue Oct 16, 2018 3:46 am
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

(1-1/10)(1-1/11)(1-1/12)...(1-1/100) =?

A. 1/10
B. 9/10
C. 9/100
D. 1/100
E. 9/50
(1-1/10)(1-1/11)(1-1/12)...(1-1/99)(1-1/100)

= (9/10)(10/11)(11/12)....(98/99)(99/100).

The values in blue all cancel out, leaving only the values in red:
9/100.

The correct answer is C.
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1-1/10)(1-1/11)(1-1/12)…(1-1/100) =?

by fskilnik@GMATH » Tue Oct 16, 2018 11:41 am
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

(1-1/10)(1-1/11)(1-1/12)...(1-1/100) =?

A. 1/10
B. 9/10
C. 9/100
D. 1/100
E. 9/50
\[? = \left( {1 - \frac{1}{{10}}} \right)\left( {1 - \frac{1}{{11}}} \right)\left( {1 - \frac{1}{{12}}} \right) \ldots \left( {1 - \frac{1}{{100}}} \right)\]
Just look for a PATTERN:
\[1 - \frac{1}{{\boxed{10}}} = \frac{9}{{\boxed{10}}}\,\,\]
\[\left( {1 - \frac{1}{{10}}} \right)\left( {1 - \frac{1}{{\boxed{11}}}} \right) = \frac{9}{{10}} \cdot \frac{{10}}{{11}} = \frac{9}{{\boxed{11}}}\]
\[\left( {1 - \frac{1}{{10}}} \right)\left( {1 - \frac{1}{{11}}} \right)\left( {1 - \frac{1}{{\boxed{12}}}} \right) = \frac{9}{{10}} \cdot \frac{{10}}{{11}} \cdot \frac{{11}}{{12}} = \frac{9}{{\boxed{12}}}\,\,\]
\[\left( {1 - \frac{1}{{10}}} \right)\left( {1 - \frac{1}{{11}}} \right)\left( {1 - \frac{1}{{12}}} \right)\left( {1 - \frac{1}{{\boxed{13}}}} \right) = \frac{9}{{10}} \cdot \frac{{10}}{{11}} \cdot \frac{{11}}{{12}} \cdot \frac{{12}}{{13}} = \frac{9}{{\boxed{13}}}\]
\[ \ldots \]
\[? = \left( {1 - \frac{1}{{10}}} \right)\left( {1 - \frac{1}{{11}}} \right)\left( {1 - \frac{1}{{12}}} \right) \ldots \left( {1 - \frac{1}{{\boxed{100}}}} \right) = \frac{9}{{\boxed{100}}}\]

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
English-speakers :: https://www.gmath.net
Portuguese-speakers :: https://www.gmath.com.br
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by Scott@TargetTestPrep » Wed Oct 17, 2018 6:04 pm
Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

(1-1/10)(1-1/11)(1-1/12)...(1-1/100) =?

A. 1/10
B. 9/10
C. 9/100
D. 1/100
E. 9/50
Starting with the first 4 factors we have:

(1-1/10)(1-1/11)(1-1/12)(1 - 1/13)

(9/10)(10/11)(11/12)(12/13) = 9/13

So we see only the first numerator and last denominator remain; thus, the product is 9/100.

Answer: C

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by Max@Math Revolution » Thu Oct 18, 2018 12:03 am
=>

(1-1/10)(1-1/11)(1-1/12)...(1-1/100) = (9/10)(10/11)(11/12)...(99/100) = 9/100.

Therefore, C is the answer.
Answer: C
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