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OG 13: DS Q#77

by ProGMAT » Fri Aug 16, 2013 8:34 pm

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A total of $60,000 was invested for 1 year. Part of this amount earned simple annual interest at the rate of x percent per year, and the rest earned simple annual interest at the rate of y percent per year. If the total interest earned by the $60,000 for that year was $4080, what is the value of x?

(1) x=3y/4
(2) The ratio of the amount that earned interest at the rate of x percent per year to the amount that earned interest at the rate of y percent per year was 3 to 2.

OA: C
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Source: — Data Sufficiency |
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by GMATGuruNY » Sat Aug 17, 2013 4:32 am

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A total of $60,000 was invested for one year. Part of this amount earned simple annual interest at the rate of x percent per year, and the rest earned simple annual interest at the rate of y percent per year. If the total interest earned by the $60,000 for that year was $4,080, what is the value of x?

(1) x = 3y / 4
(2) The ratio of the amount that earned interest at the rate of x percent per year to the amount that earned interest at the rate of y percent per year was 3 to 2.
Since 4080 is between 5% and 10% of 60,000, it is likely that x and y are between 5 and 10.

Statement 1: x = (3/4)y
It's possible that x=6 and y=8 or that x=6.6 and that y=8.8.
To accommodate the desired combination of percentages, we could simply adjust the amount invested at each percentage so that the total amount of interest earned = 4080.
Since x can be different values, INSUFFICIENT.

Statement 2: The ratio of the amount that earned interest at the rate of x percent per year to the amount that earned interest at the rate of y percent per year was 3/2.
Sum of the parts of the ratio = 3+2 = 5.
Since the actual amount invested = 60,000, and 60,000/5 = 12000, the two parts of the ratio must be multiplied by 12,000:
12,000(3:2) = 36,000:24,000.
Thus, 36,000 earns x% interest and 24,000 earns y% interest.
It's possible that x=6 or that x=6.6.
To accommodate each value for x, we could simply adjust the value of y so that the total amount of interest earned = 4080.
Since x can be different values, INSUFFICIENT.

Statements 1 and 2 combined:
32,000 earns x% interest and 24,000 earns y% interest.
Since x=(3/4)y, for every 4% of interest earned by the $24,000, 3% interest is earned by the $36,000.
If x=3 and y=4:
(.03)(36,000) + (.04)(24,000) = 1080+960 = 2040.
Since 2040 is half the amount of interest needed, the percentages must be doubled to 6% and 8%.
Thus, x=6.
SUFFICIENT.

The correct answer is C.
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by faraz_jeddah » Wed Sep 25, 2013 4:43 am

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Is there a shortcut to this problem?
Maybe structuring it in the "x Equations x variables" format?
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by theCodeToGMAT » Wed Sep 25, 2013 5:08 am

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faraz_jeddah wrote:Is there a shortcut to this problem?
Maybe structuring it in the "x Equations x variables" format?
We don't have to solve for values.. remember it's Data Sufficiency..

My steps:
Total Interest = Interest in First Slot + Interest in Second Slot
4080 = (A*x*1 + (60000-A)*y*1)/100


Statement 1: We have no information about "A" hence we will not be able to deduce a proper numerical answer
INSUFFICIENT

Statement 2:
First Slot is countable
Second Slot is 2/5*60000
here, we don't have info about the X & y relation.. hence we would be able to deduce numerical answer.
INSUFFICIENT

Combining..
we have all values required..

Answer [spoiler]{C}[/spoiler]
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by [email protected] » Wed Sep 25, 2013 11:31 pm

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Hi faraz_jeddah,

Yes, there is a way to use a "system math" rule to answer this question, but you have to be careful about how you deal with the variables in the initial prompt.

You can clearly see the variables X and Y, which represent the 2 unknown interest rates. You only need 1 more variable to account for the invested money.

M = amount invested at X%
(60,000 - M) = amount invested at Y%

We're told that M(X/100) + (60,000 - M)(Y/100) = $4080. This is 1 equation with 3 variables

We're asked to solve for X.

Fact 1: X = 3Y/4

Here we have another equation, but this 1 equation isn't enough for us to figure out what X is.
Fact 1 is INSUFFICIENT

Fact 2: This sentence translates into this: M/(60,000 - M) = 3/2

Here we can figure out the value of M, but we don't have enough to figure out what X is.
Fact 2 is INSUFFICIENT

Combining Facts, we have 3 variables and 3 unique equations. You CAN solve this system and get the value of X.

Final Answer: C

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by Scott@TargetTestPrep » Fri Nov 09, 2018 12:24 pm

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ProGMAT wrote:A total of $60,000 was invested for 1 year. Part of this amount earned simple annual interest at the rate of x percent per year, and the rest earned simple annual interest at the rate of y percent per year. If the total interest earned by the $60,000 for that year was $4080, what is the value of x?

(1) x=3y/4
(2) The ratio of the amount that earned interest at the rate of x percent per year to the amount that earned interest at the rate of y percent per year was 3 to 2.
We are given that $60,000 was invested for 1 year. We are also given that part of the investment earned x percent simple annual interest and the rest earned y percent simple annual interest. We are also given that the total interest earned was $4,080. Let's start by defining a variable.

b = the amount that earned x percent simple interest

Using variable b, we can also say:

60,000 - b = the amount that earned y percent simple annual interest

Since we know that the total interest earned was $4,080, we can create the following equation:

b(x/100) + (60,000 - b)(y/100) = 4,080

Note that in the equation above, we express "x percent" as x/100 and "y percent" as y/100 in the same way that we would express, say, 24 percent as 24/100.

Statement One Alone:

x = 3y/4

Although we have an equation with x and y, we still need a third equation to be able to determine the value of x because our equation from the given information has three variables. Statement one alone is not sufficient to answer the question.

Statement Two Alone:

The ratio of the amount that earned interest at the rate of x percent per year to the amount that earned interest at the rate of y percent per year was 3 to 2.

From our given information we know that b is the amount that earned interest at the rate of x percent per year and that 60,000 - b is the amount that earned interest at the rate of y percent per year. Thus, we can create the following equation:

b/(60,000 - b) = 3/2

Without a third equation, statement two alone is not sufficient to determine the value of x.

Statements One and Two Together:

From the given information and statements one and two we have the following 3 equations:

1) b(x/100) + (60,000 - b)(y/100) = 4,080

2) x = 3y/4

3) b/(60,000 - b) = 3/2

Since we have 3 independent equations with variables x, y, and b, we are able to determine the value of x.

Answer: C

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