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

Question

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

(PS14293)

ScottTargetTestPrep · Accepted Answer

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

mt7777 · Answer

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}

chetan2u · Answer

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

bangu · Answer

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.

sudarshan22 · Answer

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  .

Drb89 · Answer

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}

ErikLewe · Answer

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.

Kinshook · Answer

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{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

Jsound996 · Answer

Simple solution to this problem if you apply logic:
We know that 
 a = 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64}  and 
 b = 1 + \frac{1}{4}a 
We're solving for a - b
Plug a in for b so you get the following:

( 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64} ) - ( 1 + \frac{1}{4} ]

Distrubute  \frac{1}{4}

( 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64} ) - ( 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64} + \frac{1}{256} ]

The values in Red cross out, leaving you with  \frac{-1}{256} 
 ( 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64} )  -  ( 1 + \frac{1}{4} + \frac{1}{16} + \frac{1}{64}   +  \frac{1}{256} ]

Answer B

Kinshook · Answer

Asked: 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
b = 1+1/4+1/16+1/64+1/256

a-b=-1/256

IMO B

Posted from my mobile device

StudiosTom · Answer

Hi  chetan2u , Can you please elaborate the geometric progression formula aind its application.

chetan2u · Answer

HI,

A GP is a series where each successive number is SOME times the preceding term and this SOME could be any number. This is also referred as the RATIO or simply r. so if a, b, c,d ois the series in GP, b/a=r...
If 2, 4, 6, 8..r=4/2=2
If 4, 2, 1, 1/2....r=2/4=1/2.

There can be various applications which meets this requirement. Say a certain species doubles itself every year, or a certain amount of money increases by 10% every 4 months and so on.

MHIKER · Answer

Multiplying  a  by  \frac{1}{4} 
 \frac{1}{4}a  =  \frac{1}{4}  +  \frac{1}{16}  +  \frac{1}{64}  +  \frac{1}{256}

Now  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}

= -  \frac{1}{256}

The answer is B

samarth1222 · Answer

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,  
I also followed this same method and it took more than 2 minutes, so I ended up guessing and moving on. and I was doing it correctly. so if you were able to solve this question in under 2 mins so could you give me some advice on how you do that question under 2 mins?

hkalabdu · Answer

a - b = 1 + 1/4 + 1/16 + 1/64 - 1 - 1/4 - 1/16 - 1/64 - 1/256

The trick here to pay attention that since a -b >> it means all b numbers will be negative.

From the above we subtract the same numbers with opposite signs and we will -1/256 remaining

Which is answer B

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

Prep Toolkit

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