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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
Joined: 02 Sep 2009
Posts: 48044
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
1
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35% (medium)

Question Stats:

76% (01:43) correct 24% (02:09) wrong based on 164 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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Joined: 11 Jul 2018
Posts: 20
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
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.
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Posts: 6556
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
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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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

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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, 02:47
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 = 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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Posts: 13
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
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
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
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

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

Answer: B
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Joined: 16 Mar 2017
Posts: 13
GMAT 1: 640 Q38 V40
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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Concentration: Finance, Statistics
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
1
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.
Re: If a = 1 + 1/4 + 1/16 + 1/64 and b = 1 + 1/4a, then what is the value &nbs [#permalink] 14 Aug 2018, 01:32
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