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# If a = 1 + 1/4 + 1/16 + 1/64 and b = 1 + 1/4a, then what is the value

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Math Expert
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Posts: 58335
If a = 1 + 1/4 + 1/16 + 1/64 and b = 1 + 1/4a, then what is the value  [#permalink]

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30 Jul 2018, 21:56
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75% (02:11) correct 25% (02:49) wrong based on 1032 sessions

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If $$a = 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64}$$ and $$b = 1 + \frac{1}{4}a$$, then what is the value of a – b ?

A. -85/256

B. -1/256

C. -1/4

D. 125/256

E. 169/256

NEW question from GMAT® Quantitative Review 2019

(PS14293)

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Posts: 7953
Re: If a = 1 + 1/4 + 1/16 + 1/64 and b = 1 + 1/4a, then what is the value  [#permalink]

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30 Jul 2018, 23:12
3
1
Bunuel wrote:
If $$a = 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64}$$ and $$b = 1 + \frac{1}{4a}$$, then what is the value of a – b ?

A. -85/256

B. -1/256

C. -1/4

D. 125/256

E. 169/256

NEW question from GMAT® Quantitative Review 2019

(PS14293)

Bunuel, pl relook ..$$b = 1 + \frac{1}{4a}$$ should be $$b = 1 + \frac{1}{4}*a$$
otherwise you will not get the denominator as a multiple of 4

$$a = 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64}=1+\frac{1}{2^2}+\frac{1}{2^4}+\frac{1}{2^6}$$...
it is a geometric progression
sum = $$\frac{a(1-r^n)}{1-r}=1*(1-\frac{1}{4}^4)/(1-\frac{1}{4})=\frac{255}{256}/\frac{3}{4}=\frac{255*4}{256*3}=\frac{85}{64}$$...

$$b = 1 + \frac{a}{4}=1+\frac{85}{64*4}=\frac{341}{256}$$....

$$a-b=\frac{85}{64}-\frac{341}{256}=\frac{340-341}{256}=\frac{-1}{256}$$

B
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Re: If a = 1 + 1/4 + 1/16 + 1/64 and b = 1 + 1/4a, then what is the value  [#permalink]

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31 Jul 2018, 02:47
6
Bunuel wrote:
If $$a = 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64}$$ and $$b = 1 + \frac{1}{4}a$$, then what is the value of a – b ?

a = 1 + 1/4 + 1/16 + 1/64
a = ( 64 + 16 + 4 + 1 ) / 64
a = 85 / 64

Now,
b = 1 + 1/4 * 85/64
b = 1 + 85/256
b = 341/256

Therefore,
a - b = 85/64 - 341/256
a - b = ( 85*4 - 341 ) / 256
a - b = (340 - 341)/256
a - b = - 1 / 256

Hence, B.
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##### General Discussion
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Joined: 11 Jul 2018
Posts: 19
Re: If a = 1 + 1/4 + 1/16 + 1/64 and b = 1 + 1/4a, then what is the value  [#permalink]

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30 Jul 2018, 22:49
2
Value of a = 1 + 1/4 = 1.25 (ignoring other bit which will make it slightly bigger than 1.25)
Value of b = 1 + 1/(4*1.25) = 1 + 1/5 = 1.20

Roughly a - b = 1.25 - 1.20 = .05 => Actual value of a will be slightly more than 1.25 and therefore, value of b will be slightly less than what is present. This will mean this difference will slightly bigger but not drastically big.

I will go with smaller option of D.

This is not a proper way to solve this query as it can be solve using equations or GP series etc in a proper manner but that will be too time consuming in exam.
Math Expert
Joined: 02 Sep 2009
Posts: 58335
Re: If a = 1 + 1/4 + 1/16 + 1/64 and b = 1 + 1/4a, then what is the value  [#permalink]

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30 Jul 2018, 23:15
chetan2u wrote:
Bunuel wrote:
If $$a = 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64}$$ and $$b = 1 + \frac{1}{4a}$$, then what is the value of a – b ?

A. -85/256

B. -1/256

C. -1/4

D. 125/256

E. 169/256

NEW question from GMAT® Quantitative Review 2019

(PS14293)

Bunuel, pl relook ..$$b = 1 + \frac{1}{4a}$$ should be $$b = 1 + \frac{1}{4}*a$$
otherwise you will not get the denominator as a multiple of 4

$$a = 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64}=1+\frac{1}{2^2}+\frac{1}{2^4}+\frac{1}{2^6}$$...
it is a geometric progression
sum = $$\frac{a(1-r^n)}{1-r}=1*(1-\frac{1}{4}^4)/(1-\frac{1}{4})=\frac{255}{256}/\frac{3}{4}=\frac{255*4}{256*3}=\frac{85}{64}$$...

$$b = 1 + \frac{a}{4}=1+\frac{85}{64*4}=\frac{341}{256}$$....

$$a-b=\frac{85}{64}-\frac{341}{256}=\frac{340-341}{256}=\frac{-1}{256}$$

B

_______________
Edited. Thank you.
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Re: If a = 1 + 1/4 + 1/16 + 1/64 and b = 1 + 1/4a, then what is the value  [#permalink]

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31 Jul 2018, 09:49
1
Bunuel wrote:
If $$a = 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64}$$ and $$b = 1 + \frac{1}{4}a$$, then what is the value of a – b ?

A. -85/256

B. -1/256

C. -1/4

D. 125/256

E. 169/256

NEW question from GMAT® Quantitative Review 2019

(PS14293)

Easier way to do rather than using Geo series formula is using substitution method.

$$b=1+\frac{1}{4} a$$
$$b=a+\frac{1}{256}$$
so, $$b-a = -\frac{1}{256}$$
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Re: If a = 1 + 1/4 + 1/16 + 1/64 and b = 1 + 1/4a, then what is the value  [#permalink]

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31 Jul 2018, 10:06
can this be done in 2 minutes?
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Re: If a = 1 + 1/4 + 1/16 + 1/64 and b = 1 + 1/4a, then what is the value  [#permalink]

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06 Aug 2018, 02:42
1
Bunuel wrote:
If $$a = 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64}$$ and $$b = 1 + \frac{1}{4}a$$, then what is the value of a – b ?

A. -85/256

B. -1/256

C. -1/4

D. 125/256

E. 169/256

NEW question from GMAT® Quantitative Review 2019

(PS14293)

Bunuel

I think this question is already appeared in OG Quantitative review 2018. Kindly verify
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Re: If a = 1 + 1/4 + 1/16 + 1/64 and b = 1 + 1/4a, then what is the value  [#permalink]

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10 Aug 2018, 19:06
3
Bunuel wrote:
If $$a = 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64}$$ and $$b = 1 + \frac{1}{4}a$$, then what is the value of a – b ?

A. -85/256

B. -1/256

C. -1/4

D. 125/256

E. 169/256

a - b

a - (1 + ¼a)

¾a - 1

¾(1 + ¼ + 1/16 + 1/64) - 1

¾(64/64 + 16/64 + 4/64 + 1/64) - 1

¾(85/64) - 1

255/256 - 256/256

-1/256

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Re: If a = 1 + 1/4 + 1/16 + 1/64 and b = 1 + 1/4a, then what is the value  [#permalink]

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13 Aug 2018, 04:52
whatfielddoido wrote:
can this be done in 2 minutes?

Apparently yes "73% (01:44) correct"
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Re: If a = 1 + 1/4 + 1/16 + 1/64 and b = 1 + 1/4a, then what is the value  [#permalink]

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14 Aug 2018, 01:32
2
How I solved it in 30s.

a= 1+0.25+(small numbers)=almost 1.3
b= 1+0,25*1.3=1+ slightly more than 0.3.
a - b= (less than) 1.3 - (more than) 1.3 = negative, but close to 0.

Answer options allow for B as best answer choice. B is correct.
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Re: If a = 1 + 1/4 + 1/16 + 1/64 and b = 1 + 1/4a, then what is the value  [#permalink]

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26 Aug 2018, 20:16
Bunuel wrote:
If $$a = 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64}$$ and $$b = 1 + \frac{1}{4}a$$, then what is the value of a – b ?

A. -85/256

B. -1/256

C. -1/4

D. 125/256

E. 169/256

NEW question from GMAT® Quantitative Review 2019

(PS14293)

Hi Bunuel,

I took the LCM of b=1+1/4a

Then b became 5/4a

I got the answer as -85/256

Why can't we take the LCM and do it this way?

And if we can then either way, we must get the same answer..

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Re: If a = 1 + 1/4 + 1/16 + 1/64 and b = 1 + 1/4a, then what is the value  [#permalink]

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26 Aug 2018, 21:11
Hi Bunuel,

I took the LCM of b=1+1/4a

Then b became 5/4a

I got the answer as -85/256

Why can't we take the LCM and do it this way?

And if we can then either way, we must get the same answer..

Hi zanaik89,
Refer the highlighted portion, It should be :
$$b=1+\frac{1}{4}a$$-------------(2)
=$$\frac{4+a}{4}$$ (Though we don't require this step)
a=1+$$\frac{1}{4} + \frac{1}{16} + \frac{1}{64}$$=$$\frac{64+16+4+1}{64}$$=$$\frac{85}{64}$$------------(1)

Hence $$a-b=\frac{85}{64}-(1+\frac{1}{4}*\frac{85}{64})=\frac{85}{64}-1-\frac{85}{4*64}=\frac{(4*85)-(4*64)-85}{4*64}$$=-$$\frac{1}{256}$$

Another Method:-
$$a=1+\frac{1}{4}+\frac{1}{16}+\frac{1}{64}$$
So, $$\frac{1}{4}a=\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\frac{1}{256}$$ (Dividing both sides by 4)
So, $$b=1+\frac{1}{4}a=1+\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\frac{1}{256}$$

Hence, $$a-b=(1+\frac{1}{4}+\frac{1}{16}+\frac{1}{64})-(1+\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\frac{1}{256})$$
Or, $$a−b=−\frac{1}{256}$$
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Re: If a = 1 + 1/4 + 1/16 + 1/64 and b = 1 + 1/4a, then what is the value  [#permalink]

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09 Dec 2018, 12:11
Hello to everyone!
Is there any way to solve this task in 1 minute, please? Any tricks or magic?

The task is not complicated, but the problem takes more than 1.5 to be solved out.

Thanks.
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Re: If a = 1 + 1/4 + 1/16 + 1/64 and b = 1 + 1/4a, then what is the value  [#permalink]

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09 Jan 2019, 11:10
get value of A and B then substitute, you will easily get the answer.
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Re: If a = 1 + 1/4 + 1/16 + 1/64 and b = 1 + 1/4a, then what is the value  [#permalink]

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12 Jan 2019, 07:24
3
Bunuel wrote:
If $$a = 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64}$$ and $$b = 1 + \frac{1}{4}a$$, then what is the value of a – b ?

A. -85/256

B. -1/256

C. -1/4

D. 125/256

E. 169/256

NEW question from GMAT® Quantitative Review 2019

(PS14293)

Hi,

this is my first post. Hope I did it correctly.

$$a=1+\frac{1}{4}+\frac{1}{16}+\frac{1}{64}$$
$$b=1+\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\frac{1}{256}$$

$$a-b=1+\frac{1}{4}+\frac{1}{16}+\frac{1}{64}-1-\frac{1}{4}-\frac{1}{16}-\frac{1}{64}-\frac{1}{256}$$
or
$$a-b=1-1+\frac{1}{4}-\frac{1}{4}+\frac{1}{16}-\frac{1}{16}+\frac{1}{64}-\frac{1}{64}-\frac{1}{256}$$

Just simplify:

$$=-\frac{1}{256}$$
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If a = 1 + 1/4 + 1/16 + 1/64 and b = 1 + 1/4a, then what is the value  [#permalink]

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26 Apr 2019, 21:56
a=1+1/4+1/4^2 +1/4^3

B=1+1/4a , substitue vaule a in to b

b= 1+1/4(1+1/4+1/4^2 +1/4^3 )
= 1+1/4+1/4^2 +1/4^3 +1/4^4

a-b= 1+1/4+1/4^2 +1/4^3 - (1+1/4+1/4^2 +1/4^3 +1/4^4 )

=-1/4^4
=-1/256
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Re: If a = 1 + 1/4 + 1/16 + 1/64 and b = 1 + 1/4a, then what is the value  [#permalink]

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04 Aug 2019, 19:19
Tried both geometric progression method and substitution, and substitution was faster for me.

Geometric progression just had too much calculation with all the fractions; substitution reduced the calculations.

Given:
$$a = 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64} = \frac{85}{64}$$
$$b = 1 + (\frac{1}{4})*a$$

Substitution Method:
a - b
$$= \frac{85}{64} - (1 + (\frac{1}{4})*(\frac{85}{64}))$$
$$= \frac{85}{64} - \frac{64}{64} - (\frac{1}{4})*(\frac{85}{64})$$
$$= \frac{21}{64} - (\frac{1}{4})*(\frac{85}{64})$$
$$= \frac{84}{256} - \frac{85}{256}$$
$$= \frac{-1}{256}$$

Answer is B $$\frac{-1}{256}$$

Bunuel wrote:
If $$a = 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64}$$ and $$b = 1 + \frac{1}{4}a$$, then what is the value of a – b ?

A. -85/256

B. -1/256

C. -1/4

D. 125/256

E. 169/256

NEW question from GMAT® Quantitative Review 2019

(PS14293)
Intern
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Posts: 1
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Re: If a = 1 + 1/4 + 1/16 + 1/64 and b = 1 + 1/4a, then what is the value  [#permalink]

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17 Aug 2019, 01:43
Simpler method -> no LCM, No GP

a-b = (1 + 1/4 + 1/16 + 1/64) - (1 + (1/4) * a)

a-b = 1/4 + 1/16 + 1/64 - (1/4)*a

a-b = 1/4 (1 + 1/4 + 1/16 - a)

From the value of 'a' given in the question, we can derive -> 1 + 1/4 + 1/16 = a - 1/64

So, a-b = 1/4 (a - 1/64 - a)

a-b = 1/4 * -1/64 = -1/256
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If a = 1 + 1/4 + 1/16 + 1/64 and b = 1 + 1/4a, then what is the value  [#permalink]

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17 Aug 2019, 01:58
Bunuel wrote:
If $$a = 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64}$$ and $$b = 1 + \frac{1}{4}a$$, then what is the value of a – b ?

A. -85/256

B. -1/256

C. -1/4

D. 125/256

E. 169/256

NEW question from GMAT® Quantitative Review 2019

(PS14293)

Given: $$a = 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64}$$ and $$b = 1 + \frac{1}{4}a$$

Asked: What is the value of a – b ?

$$a - b = a - (1 + \frac{1}{4}a) = \frac{3}{4} a -1$$
$$a = 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64}$$

$$a = \frac{64+16+4+1}{64}$$

$$a = \frac{64+16+4+1}{64}$$

$$a = \frac{85}{64}$$

$$a - b = a - (1 + \frac{1}{4}a) = \frac{3}{4} a -1$$

$$a - b = \frac{3}{4} *\frac{85}{64} -1$$

$$a - b = \frac{255}{256} -1$$

$$a - b = \frac{-1}{256}$$

IMO B

Another method:

$$a = 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64}$$
$$\frac{a}{4} = \frac{1}{4} + \frac{1}{16} + \frac{1}{64}+ \frac{1}{256}$$
$$b = 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64}+ \frac{1}{256}$$
$$a - b = (1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64}) - (1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64}+ \frac{1}{256})$$
$$a - b = -\frac{1}{256}$$

IMO B
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If a = 1 + 1/4 + 1/16 + 1/64 and b = 1 + 1/4a, then what is the value   [#permalink] 17 Aug 2019, 01:58

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