Samantha invests i1 dollars in bond X, which pays r1

Question

Samantha invests i1 dollars in bond X, which pays r1 percent simple interest annually, and she invests i2 dollars in bond Y, which pays r2 percent simple interest annually. After one year, will she have earned more interest, in dollars, from bond X than from bond Y ?

(1)  (r1)^2 > (r2)^2 
(2) The ratio of i1 to i2 is larger than the ratio of r1 to r2 .

Why cant 1 be sufficient .
We know that interest cannot be negative , therefore (1) eventually leads to r1> r2 .
The values of i1 and I2 don’t matter , because even when i2 is greater than i1 , the interest would be less .Eg. i1 : 100 r1: 5% , i2: 150 , r2: 3% , i1r1 > i2r2 giving the required result .

Bunuel · Accepted Answer

First of all: "bond X pays r1 percent simple interest annually" --> so r1 and r2 can not be negative.

After 1 year bond X will earn  i_1*\frac{r_1}{100}  and bond Y  i_2*\frac{r_2}{100} .

So the question is  i_1*\frac{r_1}{100}>i_2*\frac{r_2}{100} ? --> is  i_1*r_1>i_2*r_2 ?

(1)  (r_1)^2>(r_2)^2  --> not sufficient as no info about  i_1  and  i_2 .
(2) The ratio of i1 to i2 is larger than the ratio of r1 to r2 -->  \frac{i_1}{i_2}>\frac{r_1}{r_2}  -->  i_1*r_2>i_2*r_1 . Not sufficient.

(1)+(2) From (1)  (r_1)^2>(r_2)^2  -->  r_1>r_2  -->  \frac{r_1}{r_2}>1  --> so  \frac{i_1}{i_2}>\frac{r_1}{r_2}>1  -->  \frac{i_1}{i_2}>1  -->  i_1>i_2 . So as both  r_1>r_2  and  i_1>i_2  then  i_1*r_1>i_2*r_2 . Sufficient.

Or: as both part in both inequalities ( (r_1)^2>(r_2)^2  and  i_1*r_2>i_2*r_1 ) are positive then we can multiply them:  (r_1)^2*i_1*r_2>(r_2)^2*i_2*r_1  --> reduce by  r_1r_2  -->  i_1*r_1>i_2*r_2 . Sufficient.

Answer: C.

shrouded1 · Answer

Interest from bond X : i1*r1/100
Interest from bond Y : i2*r2/100
we need to know if i1*r1 > i2*r2 ?

(1) r1^2 > r2^2 ... Not sufficient as it doesn't tell us anything about the actual amounts invested
(2) i1/i2 > r1/r2 which implies i1*r2 > i2*r1 ... doesnt help us solve our inequality
(1+2) r1^2 > r2^2 
 So r1 > r2 as interest rate on bond cannot be negative
 i1*r1 > i1*r2
 i2*r1 > i2*r2
 Combining with condition known from Statement 2, we get i1*r1 > i2*r2

Answer is (C)

Bonds can't have negative interest. Variable is irrelevant here as we are given it is fixed

sibanmishra · Answer

Well..No one till now commented on the possibility of negative interest rate??

gurpreetsingh · Answer

-ve interest rates are not possible for such investment. Do not over generalize the things on gmat.

Bunuel · Answer

There are comments on this in two posts above: we are told that "bond X  pays  r1 percent simple interest annually" so it can't be negative.

nandini11 · Answer

Hello Bunuel , Shrouded1 , 
What abt the following consideration .
from (1) we know r1 > r2 so irrespective of i1 being greater or less than i2 , we would know that i1*r1 yields high value .
Example 
Consider i1 > i2 (and we know r1 >r2 and i1 and i2 have to be positive values ).
say i1 : 150 , i2 : 100 
Value of Inv on X : i1*r1 : 150*5% : 7.5
Value of Inv on Y : i2*r2 : 100*3% : 3
Therefore X > Y
Take the other case of i1< i2 ( and still r1 >r2 and i1 and i2 have to be positive values ).
Say i1 : 100 , i2 : 150 
Value of Inv on X : i1*r1 : 100*5% : 5
Value of Inv on Y : i2*r2 : 150*3% : 4.5 
So it does not matter what is the principle amount hence (A) or stmnt 1 is sufficient .
Where am I going wrong ?

PS : Gurpreet I have corrected the OA .Sorry abt that .

Bunuel · Answer

That's not correct. Are you saying that interest earned depends only on interest rate?

So if I ask you what would you prefer 1% annual interest from $1,000,000,000 or 100% annual interest from $1?

adishail · Answer

Bunuel, I will be more than happy to accept the former with only 0.25% :lol:

nandini11 · Answer

I get it now .My choice of plugin numbers was not comprehensive enuf .
Thanks a lot Bunuel .

adishail · Answer

Nandini, the numbers you have taken in this particular case are playing the trick. In your example, the ratio of the R's and I's are peculiar. Take extreme example. If you think i1 & i2 do not matter. Do this: Case 1 i1=$10000 and i2=$100 r1=5% and r2=4.99% .. as a result r1>r2 Value of return ......... X = 10000*5% = $500 Y = 100*4.99% = $4.99 X >> Y Case 2 i1=$100 and i2=$10000 r1=5% and r2=4.99% .. as a result r1>r2 Value of return ......... X = 100*5% = $5 Y = 10000*4.99% = $499 X << Y

Rohan_Kanungo · Answer

Can the rates R1 and R2 be fractions? In that case R1^2 > R2^2 would not necessarily mean R1>R2. :shock:

Bunuel · Answer

Actually it would. For non-negative x and y if x^2>y^2 , then x>y : since both sides of x^2>y^2 are non-negative we can take the square root from both sides and get x>y . Adding/subtracting/multiplying/dividing inequalities: help-with-add-subtract-mult-divid-multiple-inequalities-155290.html Hope it helps.

jlgdr · Answer

Is i1r1>i2r2?

Statement 1

ri>r2

This is clearly insufficient

Statement 2

i1/i2>r1/r2

i1r2>r1i2.

Insufficient on its own

Now both together

i1r2>r1i2
r1>r2

Multiplying inequalities since we know both sides are positive then simplifying we get

i2>i2

Since we know that r1>r2 as well then

i1r1>i2r2

Sufficient

C

Hope this helps
Cheers
J

PiyushK · Answer

Read to eat stuff: it will refresh your concept and keep you away from any confusion during exam. Kudos me if it is helpful. Ratio properties.jpg

MathRevolution · Answer

Forget conventional ways of solving math questions. For DS problems, the VA (Variable Approach) method is the quickest and easiest way to find the answer without actually solving the problem. Remember that equal numbers of variables and independent equations ensure a solution.

Since we have 4 variables (i1, i2, r1 and r2) and 0 equations, E is most likely to be the answer. So, we should consider conditions 1) & 2) together first. After comparing the number of variables and the number of equations, we can save time by considering conditions 1) & 2) together first.

Conditions 1) and 2):
Since r1 and r2 are positive, (r1)^2 > (r2)^2 implies r1 > r2 or r1/r2 > 1.
Since i1/i2 > r1/r2 from the condition 2), i1/i2 > r1/r2 > 1 implies i1/i2 > 1 or i1 > i2.
i1 * r1 > i2 * r2.
Thus the interest from the bond X is more than the interest from the bond Y.

Both conditions together are sufficient.

Therefore, C is the answer.

In cases where 3 or more additional equations are required, such as for original conditions with “3 variables”, or “4 variables and 1 equation”, or “5 variables and 2 equations”, conditions 1) and 2) usually supply only one additional equation. Therefore, there is an 80% chance that E is the answer, a 15% chance that C is the answer, and a 5% chance that the answer is A, B or D. Since E (i.e. conditions 1) & 2) are NOT sufficient, when taken together) is most likely to be the answer, it is generally most efficient to begin by checking the sufficiency of conditions 1) and 2), when taken together. Obviously, there may be occasions on which the answer is A, B, C or D.

bumpbot · Answer

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Samantha invests i1 dollars in bond X, which pays r1

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