12 Days of Christmas GMAT Competition - Day 7: An investor has two

Let C be the maximum capacity of both wallets.

Tokens in main wallet initially: m/100 * C
Tokens in backup wallet initially: b/100 * C

Tokens needed to fill the main wallet = C - m/100 * C = (1 - m/100) * C

Remaining tokens in backup wallet post transfer = b/2 * C

Tokens transferred = b/100 * C - b/200 * C = b/200 * C

b/200 * C = (1 - m/100) * C
b = 200 - 2*m

Now testing answer choices,

if m = 30, then b > 100 => Not Possible
if m = 45, then b > 100 => Not Possible
if m = 50, then b = 100; but we don't have this answer choice
if m = 55, then b = 90; this is CORRECT
if m = 60, then b = 80; but we don't have this answer choice

Answer 55 and 90

Let's say the capacity of each is c,

So as per current policy max limit for holding .. mc/100, Backup bc/100.

So now as per new law the main has to be full, m so to do that m-mc/100 worth cryptos need to be transferred from Backup to main.

So crypots left in backup is bc/100 - m + mc/100.

Given that this is equal to 1/2(bc/100).


So if we equate both we will get b/2 + m =100.

So among the options b has to be 90, m = 55, (45+55=100).


Hence IMO b=90, m=55

It is given that the capacity of both wallets is identical
Let the capacity be 100

Main wallet hold m% of its capacity = m/100 * 100 = m
Backup wallet hold b% of its capacity = b/100 * 100 = b

After transferring from the backup to the main wallet,
Main wallet new balance = m + transferred money = 100
Backup wallet new balance = b/2

It is given that the backup wallet had a higher percentage before transferring, b > m

Looking at the given options, and starting with the highest for b,

b = 90
After transferring, b = 90/2 = 45

45 is transferred to m

m + 45 = 100
m = 55

m = 55
b = 90

IMO
m: 55
b: 90

To solve this problem, we need to determine the values of m and b that satisfy the given conditions. Let's break down the conditions
Given Conditions:
1. The Main wallet holds mm of its maximum capacity.
2. The Backup wallet holds bb of its maximum capacity.
3. Both wallets have identical maximum capacities.
4. After transferring tokens from the Backup wallet to the Main wallet until the Main wallet reaches its maximum capacity, the Backup wallet contains exactly half of its previous percentage.
5. The Backup wallet initially held a higher percentage than the Main wallet.


Now the best way to do this by Options

Checking the Option we see only m=45 and b=90 satisfy the conditions

2m+b= 200 and b>m
Now we substitue values from choices. m cannot be lesser than 50% as b would have to be more than 100% which is not possible so we can eliminate a and b and substitute the rest.
2(55)+90=200

Hence m=55, b=90

IMO Answer m=55, b=90

m=55, b=90
After the transfer, b=90/2=45, m=55+45=100

total capacities be M & B:
M=B given

transferred= 0.5*B*(b/2)

mM/100 + 0.5*bB/100 = M

b=200-2m

m=55, b =90

Answer 55 & 90

By looking at the answers we can eliminate any amount that ends with a 5 for b and use those that end with zero so that we do not get a decimal point. With this information b could be 50, 60 or 90 but of the three only 50 yields a figure of m that is more than the original b hence the answer is m=30 and b=50

Main = m%
Backup = b%
Total is the same = Assume that it is 100% for both the wallets as it is given in percentage

Half of whatever was in Backup wallet has been transferred to the Main wallet.
New Main = m% + (b/2)%
New Backup = (b/2)%

Now, the Main wallet has to add up to 100% as it reaches its maximum capacity after the transfer.
If m = 55% and b = 90%, then b/2 = 45%

Main Wallet = m% + (b/2)% = 55% + 45% = 100%
Old Wallet = (b/2)% = 45%

Hence, m = 55 and b = 90

1. Let's say y tokens are transferred to the Main wallet
2. Initial conditions:
   
   • Main: m%
   • Backup: b%
   • b > m (given)
3. After transfer:
   
   • Main: 100% (reached maximum)
   • Backup: b/2%
4. The transfer amount y must satisfy:
   
   • m + y = 100 (Main reaches max)
   • b - y = b/2 (Backup halves)
5. From these equations:
   
   • y = 100 - m
   • y = b/2
6. Therefore:
   
   • 100 - m = b/2
   • b = 2(100 - m)
7. Testing pairs where b > m: For m = 50:
   
   • b = 2(100 - 50) = 100 (not in options)
   For m = 55:
   
   • b = 2(100 - 55) = 90

Answer: m = 55, b = 90

Answer is m = 55 and b = 90

We know 100 - m = b/2 --> 100 = m + b/2, so m = 55 and b = 90 are the only values that satisfy this condition

Given m<b; m+b/2 = 100;
Observations -> b/2 complete the m to 100, such that m<b;

first, find out those values from list that will provide m+b/2 = 100;

and you will only find values for that 55,90;

and also m=55; b=90; m<b; bingo

m% -> M=m/100
b% -> B=b/100

c=maximum capacity

c-cM=c(1-M): free space in Main

cB-c(1-M)=cB/2
2B-2+2M=B
B=2-2M

If m=55, M=0.55 and B=0.90, b=90

m=55 and b=90

Please find the attached solution

Here's how to approach the problem:

Given:

1. The Main wallet initially holds m% of its capacity.


2. The Backup wallet initially holds b% of its capacity.


3. After a transfer, the Main wallet reaches 100% , and the Backup wallet ends up with b/2% of value .



Constraints:

1. Backup initially held a higher percentage than the Main: b>m


2. After the transfer:

Main wallet's increase = Backup wallet's decrease:

b/2 (Transferred amount)+m=100
b=2(100-m)


Steps:

1. Plug in the possible values of (30, 45, 50, 55, 60) and calculate b.


2. Ensure b>m to satisfy the initial condition.



Calculation:

For m=30 :

b=2(100-30)=140
This is not one of the options provided even though b>m & hence can be eliminated


For m=45:


b = 2(100 - 45) = 2(55) = 110
This is not one of the options provided even though b>m & hence can be eliminated

For m=50:


b = 2(100 - 50) = 2(50) = 100
This is not one of the options provided even though b>m & hence can be eliminated


For m=55:


b = 2(100 - 55) = 2(45) = 90
This is one of the options provided & hence is a valid answer as it also satifies the constraint of b>m

For m=60:


b = 2(100 - 60) = 2(40) = 80.
This is not one of the options provided even though b>m & hence can be eliminated


Answer:

Hence the only possible pair of options & henceforth the answer is:
b=90
m=55



.

Let maximum capacity be 100

Then Main wallet initially has Mi = m
Backup wallet initially has Bi = b

After transferring,
Backup wallet finally has Bf = b/2
Main wallet finally has Mf = 100 = m + b/2

Also note that b>m (given)

Starting with the highest option for b ie b=90
Then b/2 = 45 and m = 55