Marge received a certain amount of money as a gift. She spent half of

Question

Marge received a certain amount of money as a gift. She spent half of the amount in the first week. In each subsequent week for the next 12 weeks, she spent half of the amount remaining from the previous week. What was the first week in which the total amount of the gift that Marge had spent was greater than $\frac{19}{20}$ of the amount of the gift?

A. 4th

B. 5th

C. 6th

D. 9th

E. 10th

Runjhun26 · Accepted Answer

Can somebody confirm if this is the right way to do this question-

let the amount be 1000, and 19/20 of 1000 is 950
We have to check by which week the total amount spent is greater than 950

spent in week 1- 500 (given half of the entire amount)
remaining 500, out of which half is spent in week 2= 250
remaining 250, out of which half is spent in week 3= 125
remaining 125, out of which half is spent in week 4= 62 (approx.)
remaining 62, out of which half is spent in week 5= 31

We stop to check for week 5, because given in the options- 
Amount spent by week 5= 500+250+125+62+31= 968 > 950

Therefore answer is WEEK 5

rakman123 · Answer

Simple Approach

Always think of a number easy to breakdown. Money received as a gift = $200.

w1: Spent half of 200 --> $100 left.  Now, 100 is a very nice number to break down easily. 
w2: 1/2 of 100 = $50
w3: 1/2 of 50 = $25
w4: 1/2 of 25 = $12.5
w5: 1/2 of 12.5 = $6.25

Add them all up, (100+50+25+12.5+6.25) (without even needing to add the decimals for quick & simple math), you've spent $193 on w5, which is the first week you'd spend  more  than 19/20 of the $200 (which is $190 btw).

Voila.

gmatophobia · Answer

Let's assume that Marge received $ 2x  as a gift

The amount she spends the first week =  x

The amount she spends the second week =  \frac{x}{2}

The amount she spends the third week =  \frac{x}{4}

The amount she spends the fourth week =  \frac{x}{8} 
.
.
so on

Question:   What was the first week in which the total amount of the gift that Marge had spent was greater than  \frac{19}{20}  of the amount of the gift?

Expenses by Marge = { x ,  \frac{x}{2} ,  \frac{x}{4} ,  \frac{x}{8} , ...}

The expenses by Marge represent a Geometric progression with a common ratio of  \frac{1}{2}

\frac{19}{20} * 2x < x * \frac{(1-(\frac{1}{2})^n)}{1-\frac{1}{2}}

\frac{19}{20} * 2x < 2x(1-(\frac{1}{2})^n)

Dividing both sides of the equation by  2x

\frac{19}{20} < 1-(\frac{1}{2})^n

(\frac{1}{2})^n < 1 - \frac{19}{20}

(\frac{1}{2})^n < \frac{1}{20}

Hence, the minimum value of n for which   (\frac{1}{2})^n < \frac{1}{20}  is  5

Option B

uCS2020 · Answer

To solve this question:
Let’s take the amount of money as $X
Now let’s list first 4 weeks of how much she spent

1st week : X/2
2nd week : X/4
3rd week : X/8
4th week : X/16

Before we move on,let’s find out how much 19/20 of X is
19X/20

Next step:Find the total money she spent in 4 weeks
15X/16 and 19X/20

OH NO… 
15X/16 < 19X/20
This is not what we were looking for
How about we check for 5 weeks.

15X/16 + X/32 = 31X/32
Now next ste-
Wait…
31X/32>19X/20
This is exactly what we wanted 
Therefore the answer is
Answer =B

Posted from my mobile device

GMACOG · Answer

Hello  gmatophobia

There seems to be a typo in this answer posted by you.
Could you please check it once?

Instead of common difference it should be common ratio
Thanks

915amit525 · Answer

* If more than 19/20th part of gift has already been spent on starting of some week then must have left with less than or equal to              1/20th part on the same weak.
   Therefore find out in which weak gift amount comes below 1/20th part. i.e for what smallest value of 'n'(weak) 
    (1/2)^n <=1/20
      n=5  (Ans)

Vordhosbn · Answer

Total amount spent greater than 19/20 means total amount remaining is less than 1/20
(1/2)^n < 1/20
n=4 gives us 1/16 which is greater than 1/20. So, n=5.

KarishmaB · Answer

Say Marge received $x.

Every week, she spent 1/2 of what she had.

First week = 1/2 spent, 1/2 remaining
Second week = 1/4 spent (1/2 of 1/2 spent, 1/2 of 1/2 remaining)
Third week = 1/8 spent (1/2 of 1/4 spent, 1/2 of 1/4 remaining)

Do we see the pattern? In every nth week, she spent 1/2^n of what she got and was left with 1/2^n of what she got.

We want to find the week when she had spent more than 19/20 of what she got i.e. she was left with less than 1/20 th of what she got.

In the 4th week, she was left with 1/2^4 = 1/16th of what she got.

In the 5th week, she was left with 1/2^5 = 1/32 of what she got.

Hence, she had spent more than 19/20th of x for the first time in 5th week

Answer (B)

hemanthPA · Answer

Let 1000 be the total amount of money

1000*19/20 = 19*50 = 950
Find the week where the total exceeds $950

Given 1st week = 1000/2 spent = 500$ spent

Remaining
2nd week spent = 500/2 = $250 spent

Remaining – 250, total spent so far (500+250) = $750

3rd week spent = 250/2 = $125
Remaining = 125, total spent so far as of 3rd week = $750 + $125 = $875

4th week spent = 125/2 = $62.5
Remaining = $62.5, total spent so far as of 4th week = $875 + $62.5 = $937.5

5th week spent = 62.5/2 = $31.5
Remaining = $31.5, total spent so far as of 5th week = $937.5 + $31.5 = $969 > 950

bumpbot · Answer

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Marge received a certain amount of money as a gift. She spent half of

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