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A function V(a, b) is defined for positive integers a, b and

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A function V(a, b) is defined for positive integers a, b and [#permalink] New post 10 Dec 2012, 12:25
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A function V(a, b) is defined for positive integers a, b and satisfies V(a, a) = a, V(a, b) = V(b, a), V(a, a+b) = (1 + a/b) V(a, b). The value represented by V(66, 14) is ?

(A) 364
(B) 231
(C) 455
(D) 472
(E) None of the foregoing

Bunuel,
I know it shouldn't be here but could you explain the solution of this one?
Thanks in advance!
[Reveal] Spoiler: OA

Last edited by Bunuel on 11 Dec 2012, 01:14, edited 1 time in total.
Edited the question and added the OA.
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Re: A function V(a, b) is defined for positive integers a, b and [#permalink] New post 11 Dec 2012, 01:57
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felixjkz wrote:
A function V(a, b) is defined for positive integers a, b and satisfies V(a, a) = a, V(a, b) = V(b, a), V(a, a+b) = (1 + a/b) V(a, b). The value represented by V(66, 14) is ?

(A) 364
(B) 231
(C) 455
(D) 472
(E) None of the foregoing

Bunuel,
I know it shouldn't be here but could you explain the solution of this one?
Thanks in advance!


Given that:
V(a, a) = a;
V(a, b) = V(b, a);
V(a, a+b) = (1 + \frac{a}{b}) V(a, b).

Question asks to find the value of V(66, 14).

Notice that only the first function gives answer as a simple value rather than another function, thus we should manipulate with V(66, 14) so that to get V(a, a) = a in the end.

V(66,14 ) = V(14,66) = V(14, 14+52);

V(14, 14+52)=(1+\frac{14}{52})V(14,52)=\frac{33}{26}*V(14,14+38);

\frac{33}{26}*V(14,14+38)=\frac{33}{26}*(1+\frac{14}{38})V(14,38)=\frac{33}{19}*V(14, 14+24);

\frac{33}{19}*V(14,14+24)=\frac{33}{19}*(1+\frac{14}{24})V(14,24)=\frac{33}{12}*V(14,14+10);

\frac{33}{12}V(14,14+10)=\frac{33}{12}*(1+\frac{14}{10})V(14,10)=\frac{33}{5}*V(10,14)=\frac{33}{5}*V(10, 10+4);

\frac{33}{5}V(10,10+4)=\frac{33}{5}*(1+\frac{10}{4})V(10,4)=\frac{33*7}{5*2}*V(4,10)=\frac{33*7}{5*2}*V(4, 4+6);

\frac{33*7}{5*2}*V(4, 4+6)=\frac{33*7}{5*2}*(1+\frac{4}{6})V(4,6)=\frac{33*7}{2*3}*V(4,4+2);

\frac{33*7}{2*3}*V(4,4+2)=\frac{33*7}{2*3}*(1+\frac{4}{2})V(4,2)=\frac{33*7}{2}*V(2,4)=\frac{33*7}{2}*V(2, 2+2);

\frac{33*7}{2}*V(2, 2+2)=\frac{33*7}{2}*(1+\frac{2}{2})V(2,2)=\frac{33*7}{2}*2*2=462.

Answer: E.
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Re: A function V(a, b) is defined for positive integers a, b and [#permalink] New post 13 Dec 2012, 02:12
Bunuel wrote:
felixjkz wrote:
A function V(a, b) is defined for positive integers a, b and satisfies V(a, a) = a, V(a, b) = V(b, a), V(a, a+b) = (1 + a/b) V(a, b). The value represented by V(66, 14) is ?

(A) 364
(B) 231
(C) 455
(D) 472
(E) None of the foregoing

Bunuel,
I know it shouldn't be here but could you explain the solution of this one?
Thanks in advance!


Given that:
V(a, a) = a;
V(a, b) = V(b, a);
V(a, a+b) = (1 + \frac{a}{b}) V(a, b).

Question asks to find the value of V(66, 14).

Notice that only the first function gives answer as a simple value rather than another function, thus we should manipulate with V(66, 14) so that to get V(a, a) = a in the end.

V(66,14 ) = V(14,66) = V(14, 14+52);

V(14, 14+52)=(1+\frac{14}{52})V(14,52)=\frac{33}{26}*V(14,14+38);

\frac{33}{26}*V(14,14+38)=\frac{33}{26}*(1+\frac{14}{38})V(14,38)=\frac{33}{19}*V(14, 14+24);

\frac{33}{19}*V(14,14+24)=\frac{33}{19}*(1+\frac{14}{24})V(14,24)=\frac{33}{12}*V(14,14+10);

\frac{33}{12}V(14,14+10)=\frac{33}{12}*(1+\frac{14}{10})V(14,10)=\frac{33}{5}*V(10,14)=\frac{33}{5}*V(10, 10+4);

\frac{33}{5}V(10,10+4)=\frac{33}{5}*(1+\frac{10}{4})V(10,4)=\frac{33*7}{5*2}*V(4,10)=\frac{33*7}{5*2}*V(4, 4+6);

\frac{33*7}{5*2}*V(4, 4+6)=\frac{33*7}{5*2}*(1+\frac{4}{6})V(4,6)=\frac{33*7}{2*3}*V(4,4+2);

\frac{33*7}{2*3}*V(4,4+2)=\frac{33*7}{2*3}*(1+\frac{4}{2})V(4,2)=\frac{33*7}{2}*V(2,4)=\frac{33*7}{2}*V(2, 2+2);

\frac{33*7}{2}*V(2, 2+2)=\frac{33*7}{2}*(1+\frac{2}{2})V(2,2)=\frac{33*7}{2}*2*2=462.

Answer: E.


Dear Bunuel,

Your explanation is brilliant. But do you think this is a kind of question that i will face in GMAT because i think the sollution is quite time consuming or there is quiker way?
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Re: A function V(a, b) is defined for positive integers a, b and [#permalink] New post 13 Dec 2012, 02:15
Expert's post
ziko wrote:
Bunuel wrote:
felixjkz wrote:
A function V(a, b) is defined for positive integers a, b and satisfies V(a, a) = a, V(a, b) = V(b, a), V(a, a+b) = (1 + a/b) V(a, b). The value represented by V(66, 14) is ?

(A) 364
(B) 231
(C) 455
(D) 472
(E) None of the foregoing

Bunuel,
I know it shouldn't be here but could you explain the solution of this one?
Thanks in advance!


Given that:
V(a, a) = a;
V(a, b) = V(b, a);
V(a, a+b) = (1 + \frac{a}{b}) V(a, b).

Question asks to find the value of V(66, 14).

Notice that only the first function gives answer as a simple value rather than another function, thus we should manipulate with V(66, 14) so that to get V(a, a) = a in the end.

V(66,14 ) = V(14,66) = V(14, 14+52);

V(14, 14+52)=(1+\frac{14}{52})V(14,52)=\frac{33}{26}*V(14,14+38);

\frac{33}{26}*V(14,14+38)=\frac{33}{26}*(1+\frac{14}{38})V(14,38)=\frac{33}{19}*V(14, 14+24);

\frac{33}{19}*V(14,14+24)=\frac{33}{19}*(1+\frac{14}{24})V(14,24)=\frac{33}{12}*V(14,14+10);

\frac{33}{12}V(14,14+10)=\frac{33}{12}*(1+\frac{14}{10})V(14,10)=\frac{33}{5}*V(10,14)=\frac{33}{5}*V(10, 10+4);

\frac{33}{5}V(10,10+4)=\frac{33}{5}*(1+\frac{10}{4})V(10,4)=\frac{33*7}{5*2}*V(4,10)=\frac{33*7}{5*2}*V(4, 4+6);

\frac{33*7}{5*2}*V(4, 4+6)=\frac{33*7}{5*2}*(1+\frac{4}{6})V(4,6)=\frac{33*7}{2*3}*V(4,4+2);

\frac{33*7}{2*3}*V(4,4+2)=\frac{33*7}{2*3}*(1+\frac{4}{2})V(4,2)=\frac{33*7}{2}*V(2,4)=\frac{33*7}{2}*V(2, 2+2);

\frac{33*7}{2}*V(2, 2+2)=\frac{33*7}{2}*(1+\frac{2}{2})V(2,2)=\frac{33*7}{2}*2*2=462.

Answer: E.


Dear Bunuel,

Your explanation is brilliant. But do you think this is a kind of question that i will face in GMAT because i think the sollution is quite time consuming or there is quiker way?


I doubt that this is a GMAT question. So, I wouldn't worry about it at all.
_________________

NEW TO MATH FORUM? PLEASE READ THIS: ALL YOU NEED FOR QUANT!!!

PLEASE READ AND FOLLOW: 11 Rules for Posting!!!

RESOURCES: [GMAT MATH BOOK]; 1. Triangles; 2. Polygons; 3. Coordinate Geometry; 4. Factorials; 5. Circles; 6. Number Theory; 7. Remainders; 8. Overlapping Sets; 9. PDF of Math Book; 10. Remainders; 11. GMAT Prep Software Analysis NEW!!!; 12. SEVEN SAMURAI OF 2012 (BEST DISCUSSIONS) NEW!!!; 12. Tricky questions from previous years. NEW!!!;

COLLECTION OF QUESTIONS:
PS: 1. Tough and Tricky questions; 2. Hard questions; 3. Hard questions part 2; 4. Standard deviation; 5. Tough Problem Solving Questions With Solutions; 6. Probability and Combinations Questions With Solutions; 7 Tough and tricky exponents and roots questions; 8 12 Easy Pieces (or not?); 9 Bakers' Dozen; 10 Algebra set. ,11 Mixed Questions, 12 Fresh Meat

DS: 1. DS tough questions; 2. DS tough questions part 2; 3. DS tough questions part 3; 4. DS Standard deviation; 5. Inequalities; 6. 700+ GMAT Data Sufficiency Questions With Explanations; 7 Tough and tricky exponents and roots questions; 8 The Discreet Charm of the DS ; 9 Devil's Dozen!!!; 10 Number Properties set., 11 New DS set.


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Re: A function V(a, b) is defined for positive integers a, b and [#permalink] New post 31 Jan 2014, 13:43
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Re: A function V(a, b) is defined for positive integers a, b and   [#permalink] 31 Jan 2014, 13:43
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