At the start of a year, Elena deposited $800 into Bank A and $800 into

Question

At the start of a year, Elena deposited $800 into Bank A and $800 into Bank B. Each deposit earned interest at its own fixed annual rate, compounded annually, and no additional deposits or withdrawals were made. What was the annual interest rate at Bank B?

(1) During the first year, the deposit at Bank B earned $40 more interest than the deposit at Bank A.
(2) During the second year, the deposit at Bank B earned $50 more interest than the deposit at Bank A.

M47-163

Bunuel · Accepted Answer

GMAT Club Official Solution:

At the start of a year, Elena deposited $800 into Bank A and $800 into Bank B. Each deposit earned interest at its own fixed annual rate, compounded annually, and no additional deposits or withdrawals were made. What was the annual interest rate at Bank B?

Let a be the annual interest rate at Bank A, and let b be the annual interest rate at Bank B.

(1) During the first year, the deposit at Bank B earned $40 more interest than the deposit at Bank A.

So:

800b - 800a = 40

800(b - a) = 40

b - a = 0.05

This gives only the difference between the two interest rates, not the value of b.

Not sufficient.

(2) During the second year, the deposit at Bank B earned $50 more interest than the deposit at Bank A.

At the beginning of the second year, the deposit at Bank A was:

800(1 + a)

So the interest earned at Bank A during the second year was:

800(1 + a) * a

At the beginning of the second year, the deposit at Bank B was:

800(1 + b)

So the interest earned at Bank B during the second year was:

800(1 + b) *b

So, the (2) gives:

800b(1 + b) - 800a(1 + a) = 50

This is one equation with two unknowns, so b cannot be determined.

Not sufficient.

(1)+(2) From statement (1):

b - a = 0.05

From statement (2):

800b(1 + b) - 800a(1 + a) = 50

Divide by 800:

b(1 + b) - a(1 + a) = 50/800

b + b^2 - a - a^2 = 1/16

(b - a) + (b^2 - a^2) = 1/16

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

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

Since b - a = 0.05 = 1/20:

1/20 * (1 + a + b) = 1/16

1 + a + b = 20/16 = 5/4

a + b = 1/4

Now we have:

b - a = 0.05

a + b = 0.25

So b is determined.

Sufficient.

Answer: C.

ankushsambare · Answer

Let:
 
 Bank A rate = a 
 Bank B rate = b  
Both deposits = 800

Statement (1):
First-year interest difference:
800b − 800a = 40
⇒ b − a = 0.05
Only a relationship between two variables → not sufficient.

Statement (2):
Second-year interest difference depends on balances after compounding:
800(1+b)b − 800(1+a)a = 50
One equation with two variables (a, b) → not sufficient.

Together:
From (1), substitute b = a + 0.05 into (2), giving one equation in one variable → solvable uniquely.
So together they are sufficient.

Answer: C

SS990 · Answer

Given: The Principal invested in accounts A and B each is 800 , compounded annnually, no changes in between to the account
To find: Interest rate of account B = Rb

1) (800Rb/100) = (8000Ra/100) +40 Not suffiecient to find Rb
 we have relationship between Ra and Rb here

2) Consider seconder as any other year with additional amounts added just for the sake of this choice. It is still same as above coice. This is not sufficient

((800Rb/100)+800)Rb/100 = ((8000Ra/100)+800)Ra/100

Considering together, we know the relationship between rate and rate if fixed. Hence we can solve for Rb. Hence C

At the start of a year, Elena deposited $800 into Bank A and $800 into

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