Data Sufficiency

Tamir deposits $100,000 in a basic savings account that yields b% simple annual interest. At the same time, he deposits $100,000 in a special account that yields s% interest, compounded annually. At the end of two years, is the amount of interest earned by the special account greater than the amount of interest earned by the basic account?

1. b < 1.05s
2. The amount of interest Tamir earns from the basic account during the first year is greater than $11,000.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are not sufficient.

Give it a shot, and I'll post the official explanation next Monday!

Jessie@Knewton wrote:Tamir deposits $100,000 in a basic savings account that yields b% simple annual interest. At the same time, he deposits $100,000 in a special account that yields s% interest, compounded annually. At the end of two years, is the amount of interest earned by the special account greater than the amount of interest earned by the basic account?

1. b < 1.05s
2. The amount of interest Tamir earns from the basic account during the first year is greater than $11,000.

statement 1:INSUFFICIENT

Principal Amount = $100,000
time period = 2 years

Generalized Way:

Special Account:

A=P(1 + s/100)^t
A= 100000((100+s)/100)^2
A= 10(100+s)^2 = 100000 +2000s + 10s^2
Is = 100000 +2000s + 10s^2 - 100000
=> Is = 10s(200 + s)

Basic Saving Account :(max rate b=1.05s)

Ib = Pbt/100
=>100000*b*2/100
=> 1000*1.05s*2
Ib = 2100s

10s(200+s)>2100s => 2000s +10s^2 -2100s>0
10s^2 - 100s > 0
10s(s-10)>0
=>s-10>0 => s>10%

Hence Is can be greater than Ib ,if s>10%, or less than Ib,if s<10%.

Statement 2:INSUFFICIENT

From this statement we can calculate the interest earned by the basic savings account at the end of 2 year
But there is no information about the interest earned by the special account or the % interest it yields.

Combining both the statements

We know from the first statement that s>10% for Is to be greater than Ib

let s =10% ,
from first statement, b< 1.05*10 => b<10.5 . But we know from second statement that b>11%. Therefore s has to be greater than 10% to satisfy the inequality.

Hence the correct answer is C

Jessie@Knewton wrote:Tamir deposits $100,000 in a basic savings account that yields b% simple annual interest. At the same time, he deposits $100,000 in a special account that yields s% interest, compounded annually. At the end of two years, is the amount of interest earned by the special account greater than the amount of interest earned by the basic account?

1. b < 1.05s
2. The amount of interest Tamir earns from the basic account during the first year is greater than $11,000.

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are not sufficient.

Give it a shot, and I'll post the official explanation next Monday!

We are told that basic account with rate b% yields simple interest.
Also, special account with rate s% yields annual compound interest.

We need to find whether the interest for special account is greater than that of the special account.
For ease of calculation, let principal = 100,000$ = x. Then if the interest of one differs from the other by a certain amount, the amount accumulated must also differ by the same amount.
So, if we can determine whether the total amount after 2 years for special account is greater than that of the basic account, we'd have sufficient data.

Now Amount for simple interest after 2 years = x*(1+0.01*2*b) = x*(1+0.02b)
Amount for compound interest after 2 years = x*(1+0.01s)^2 = x*(1+(0.01s)^2 + 0.02s)
Second - First = x*(1+(0.01s)^2+0.02s - 1 - 0.02b) = x*( (0.01s)^2 + 0.02(s-b))
= 0.01*x*[0.01s^2+2(s-b)].
So if we can definitely determine whether 0.01*s^2 +2(s-b) >0, we'll be fine.

With that in mind let's look at the statements:

1. b < 1.05s
=> b < s+0.05s => s-b > -0.05s

Now for the smallest value of (s-b), if the expression is always positive, we'll be fine. Otherwise, the data would be insufficient.

So for s = -0.05s, 0.01*s^2 -2*(0.05s) = 0.01*s^2 -0.1s = 0.01s*(s-10). We find two things. One, the expression with this condition depends solely on value of s. Second, this expression is positive only when s>10. Hence the data is insufficient.

2. The amount of interest Tamir earns from the basic account during the first year is greater than $11,000. Since x=100,000$, this translates to the rate b being greater than 11/(100) = 11%.
So we get that b> 11. We know nothing about s. can't compare. Insufficient.

Together, s> b/1.05 . b > 11, hence s> 11/1.05 >10.
This is the condition we required in our analysis of statement 1. Sufficient.

Hence C is the correct answer.

hi eagleeye,

Can you please elaborate on the below statement?

"Then if the interest of one differs from the other by a certain amount, the amount accumulated must also differ by the same amount."

Firstly with these questions, take the thousands out. This saves time and ink, and since it's data sufficiency, so you don't even need to put them back in.

SIMPLE interest:
100*(b/100) + 100*(b/100) = 2b

COMPOUND interest:
100 -> 100 + s -> 100 + s + (100+s)*(s/100)=100 + 2s + s^2/100
Therefore interest = 2s + s^2/100

(If you like, the difference between simple and compound interest is the s^2/100 term, which for typical interest rates will be quite small, because we are squaring a small decimal).

So the question is: 2b < 2s + s^2/100?

Statement 1
If b=1.05s, we'd have 2.1s < 2s + s^2/100
0.1s < s^2/100
10s < s^2
10 < s (or s is negative which it can't be).

So at the limit (b=1.05s), s would have to be greater than 10% for the compound account to "win". We don't know that it is though. Neither do we know that b isn't very small, in which case the compound account would definitely "win". So, INSUFFICIENT.

Statement 2
b > 11
Clearly on it's own this is insufficient, because it tells us nothing about s. (Trivially, if s=b, s wins, irrespective of b). INSUFFICIENT.

Together, we have:
Statement 2: b > 11
Therefore, substituting into statement 1
s > 11 * 20/21
s > 220/21 (about 10.5)

So using both statements, s>~10.5. Recalling statement 1, we found that knowing s>10 was sufficient. We now know this and hence the answer is C.

kullayappayenugula wrote:hi eagleeye,

Can you please elaborate on the below statement?

"Then if the interest of one differs from the other by a certain amount, the amount accumulated must also differ by the same amount."

Since the starting principal amounts are the same in both cases,
(Principal + interest in case b ) - (Principal + interest in case s ) = interest in case b- interest in case s
=> amount in case b - amount in case s = interest in case b- interest in case s
So difference of amounts equals the difference in interests.

OA is C

This is the OE:
This question asks whether the amount of compound interest earned in a special account is greater than the amount earned by the same amount of money in an account earning simple interest.

The basic account has an interest rate of b%, so for ease of notation, let the rate of interest earned be named B (where B = b%). Then the amount of interest the basic account earns in two years will be:

Tbasic = 2 × B × $100,000 = 2B(100,000).

Similarly, let the rate of interest earned by the special account be called S (where S = s%). Then the amount of interest earned in the first year is S(100,000). During the second year, the principal has increased to the original principal plus the interest earned in the first year, or (100,000 + S(100,000)) = (1 + S)(100,000). The amount of interest earned on this principal during the second year is S(1 + S)(100,000) = (S2 + S)(100,000), so the total interest for the two years is:

Tspecial = (S2 + S)(100,000) + S(100,000) = (S2 + 2S)(100,000).

We could also calculate the total interest earned by using the formula for compound interest, Pt = (1 + r)t(Pi), where t = 2 and r = S. For the new principal after 2 years, the formula gives P2 = (1 + S)2 (100,000). The difference is: total interest = P2 - P i= (1 + S)2(100,000) - 100,000 = (1 + 2S + S2 - 1)(100,000) = (2S + S2)(100,000). Note that since this expression is the same as the one we found, it can be derived as shown above without memorizing the actual formula.

We wish to know whether Tspecial > Tbasic, or:

Is (S2 + 2S)(100,000) > 2B(100,000)?

Since 100,000 is positive, we can divide both sides of the inequality by 100,000, leaving us with the question: Is S2 + 2S > 2B?

Statement 1 alone tells us that b < 1.05s. Since B = b × 100% and S = s × 100%, we also have B < 1.05S.

Thus, if S2 + 2S > 2(1.05S), then since 2(1.05S) must be greater than 2B by statement 1, S2 + 2S > 2B, making the answer "Yes." We can rearrange the first inequality to get:

S2 + 2S - 2(1.05)S > 0 → S2 + 2S - 2.1S > 0 →

S2 - 0.1S > 0 → S(S - 0.1) > 0

Since S is positive, we can divide both sides by S to get:

S - 0.1 > 0 → S > 0.1

Converting 0.1 into a percentage gives:

S > 10%

So if S > 10%, then the answer to the question is "Yes."

However, we do not know what S is. For high values of S, such as S = 40% and B = 41% (which is less than 1.05 × 40% = 42%), then the interest earned by the basic account is Tbasic = 2 × 0.41 × 100,000 = 82,000. The interest earned by the special account is Tspecial = (0.402 + 2(0.40))(100,000) = (0.16 + 0.8)(100,000) = 96,000, which is greater than the interest earned by the basic account.

But for a low value of S, the interest earned by the basic account may be greater. For instance, if S = 1% and B = 1.04% (which is less than 1.05 × 1% = 1.05%), the interest earned by the basic account is Tbasic = 2 × 0.0104 × 100,000 = $2,080. The interest earned by the special account is Tspecial = (0.012 + 2(0.01))(100,000) = $2,010, which is less than the interest earned by the basic account.

Statement 1 is not sufficient to answer the question definitively. Eliminate answer choices A and D.

Statement 2 alone tells us that the amount of interest earned from the basic account in the first year is greater than $11,000. In other words, b > 11, or B > 11%.

But this does not tell us anything about the special account. For example, if B = 12% and S = 1%, then the amount earned by the basic account is Tbasic = 2 × 0.12 × 100,000 = 24,000 dollars, while the amount earned by the special account would be 2,010 dollars, as we found above. But if B = 12% and S = 40%, then the special account earns 96,000 dollars, as we found above, which is more than the basic account.

Statement 2 alone is insufficient. Eliminate choice B.

Now let's take the two statements together. We know from Statement 1 that if S > 10%, then the special account earned more interest. We know from Statement 2 that B > 11%. We also know that b < 1.05s, or, B < 1.05S. Since B > 11%, if we plug B = 11 into the inequality, we can determine the smallest possible value of S:

B < 1.05S → 11 < 1.05S

We could divide both sides by 1.05, but that would be cumbersome arithmetic. It is enough to notice that if S = 10, the inequality would read 11 < 10.5, which is false. Thus, S must be greater than 10 to make the inequality true. Since S > 10%, the special account must have earned more interest than the basic account. The two statements together are sufficient to answer the question.

Answer choice C is correct.

Hey everyone,

Thanks for the great participation. Karthikpandian19 - that is indeed the full OA, which makes this job a lot faster for me. What do you make of this one:

If 4^w = n, what is the units digit of n?

1. w is an even integer
2. w > 0

(A) Statement (1) ALONE is sufficient, but statement (2) alone is not sufficient.
(B) Statement (2) ALONE is sufficient, but statement (1) alone is not sufficient.
(C) BOTH statements TOGETHER are sufficient, but NEITHER statement ALONE is sufficient.
(D) EACH statement ALONE is sufficient.
(E) Statements (1) and (2) TOGETHER are NOT sufficient.

Give it your best shot!

4^0 = 1
4^1 = 4
4^2 = 16
4^3 = 64
4^4 = 256

So we see a pattern:
All even positive powers of 4 give a number that ends in 6
All odd positive powers of 4 give a number that ends in 4

Statement 1:
We don't know if w=0 in which case '6' may not be the unit's digit.
We don't know if w<0 in which case n will be a fraction
INSUFFICIENT, POE A and D

Statement 2:
We only know that w>0, but what if w = 2.5?
INSUFFICIENT, POE B

Statement 1&2:
w is integer
w is positive and NOT zero
w is even
Therefore a definite answer to the question stem is possible (6 is the unit's digit).

Therefore C is correct.

Kartikshah, great job. Here's the full OA:

Initially, it might seem like we do not have enough information to solve this problem; certainly, we are unable to determine the value ofw, and hence n, on the basis of the prompt alone (or even the statements, for that matter). Still, let's take a look at a table of values of the powers of 4.

4^2 = 16
4^3 = 64
4^4 = 256
4^5 = 1024
4^6 = 4096
4^8 = 65536
4^10 = 1048576

Notice that the units digit of all the even powers of 4 is the same: 6. This is because all even powers of 4 are also powers of 4^2 = 16.

Remember that-for the purposes of this problem-when we look at the units digit of a product, we need only consider the units digits of its factors; thus, the units digit of 4^4 = (4^2)2 = 16 × 16 will be 6, because the units digits of each factor is 6, and 6 × 6 = 36 has a units digit of 6. Similarly, the units digit of 4^6 = (4^4) × 4^2 = (4^4) × 16 will be 6, because the units digit of both (4^4) and 16 is 6, and 6 × 6 has a units digit of 6. (Do not be confused by all the 6's; to take a contrasting example, 7 × 7 = 49 has a units digit of 9.)

Having examined the prompt, we turn to the statements.

Statement 1 tells us that w is an even integer. Be careful, though. Although we found above that positive even integer exponents of 4 have a units digit of 6, the number zero is an even integer for which 4w does not have a units digit of 6; rather 40 = 1 has a units digit of 1. Thus, the fact that w is an even integer is not sufficient to determine the units digit of n. Eliminate answer choices A and D.

The correct answer choice is B, C, or E.

Statement 2 tells us that w > 0. We have already seen that positive odd integer exponents yield a units digit of 4, while positive even integer exponents yield a units digit of 6. Thus, the fact that w is positive is not sufficient to solve for the units digit of n. (In fact, Statement 2 does not even specify whether w is an integer.) Eliminate answer choice B.

The correct answer choice is either C or E.

Combined, Statements 1 and 2 tell us that w is a positive even integer. Thus, n must have a units digit of 6, as we found above. Both statements together are sufficient.

Answer choice C is correct.