A total of $60,000 was invested for one year. Part of this

Application of weighted average -
Some amount at x% and some at y%. Total interest earned is $4080. So we can find the average interest earned (4080/60,000).
So weighted average of x and y is (4080/60,000). What is x?
We do not know the ratio of principal (i.e. the weights) and we have two variables x and y.
(1) x = (3/4) y
A variable reduced but the ratio of weights is still not known.

(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.

Got the ratio of weights but still have 2 variables.

Both together, now I have the ratio of weights and just one variable x, so definitely I will get the value of x using
w1/w2 = (A2 - Aavg)/(Aavg - A1)
3/2 = (4/3x% - 4080/60,000)/(4080/60,000 - x%)

Of course, it is a DS question so I will not try to find it but I could so sufficient. Answer (C).

Hi bunuel,

Your expert thoughts on a clarification I have got ,

when you have a equation in some problems there is only one solution and we can arrive on a unique value.
such that no other values of x or y can satisy that equation.

In the stmt above i spent 30 secs thinking if a unique solution would be available!
how do we coem to a conclusion tat there is not unique solution and we can say its not sufficeint like stmt 2?

One question... Why are we not considering the time for which each investment was made? isnt the formula for Interest earned = P x R X T/100 ??

Apologise if its a dumb question

Welcome to GMAT Club. Below is an answer for your question.

We are told that "A total of $60,000 was invested for one year", so we can omit multiplying by 1.

For more on this kind of problems please check: math-number-theory-percents-91708.html

Hope it helps.

But when we are breaking it into 2 parts 'a' and '60,000 - a' , wouldn't it matter if i invested 'a' for 2 months or 10 months? The interest amount of 4080 can come for a specific duration only, right?

It follows from the stem that both \(a\) and \(60,000-a\) were invested for one year.

Hope it's clear.

Hi guys.

I want to explain my more-abstract solution for this question. A kind of solution one require for answering a DS question in an efficient way.

First lets model what the question put in our table. Consider the part of investment that have x percent interest A and the other part (60k-A). Now what we have is this: Ax+(60k-A)y = 4080

Now lets begin from statement 1: It says that x=(3/4)y. If we put this equation into the original model then we got nothing. Why? because what we got is 1 equation with 2 unknown variables. so cross off statement 1.

Now statement 2: we got A/(60k-A)=3/2. Like the first statement if we plug this equation into original one we still have 1 equation with 2 unknown variables. Now statement 2 is out too.

1+2: from statement 2 we can draw that the 60k investment has 5 part (because the ration of two parts was 3:2) then we can calculate that each part is 12k (60k/5). Now if we plug statement 1 into the original model we got 1 equation with 1 unknown parameter. Problem solved. The answer is C.

My main point here was to stress that for solving DS question try to avoid doing the problem with math concepts only. If you see that you reach to the point that the equation has an unique answer then pick the right answer choice and go ahead!

Hi Karishma,

I have been following your solutions. They are an eye opener for me. Now I am stuck in this problem with my approach. Request you to kindly help reason out why I can't take this approach.

From statement 1 x=3y/4
so x/y=3/4
hence I can take x=3 and y=4.
since I know the interest which is 4080 and the ratio of interest 3 and 4
can't I apply these 4080/7*3 to calculate x.

If I can't take this approach in this question then under what question can this approach be taken

Of the 60,000, say I invest 30,000 at x% and 30,000 at y%.
Now if x/y = 3/4, the actual interest earned by two investments will be in the ratio 3:4 so you can calculate interest earned ta x% using 4080/7*3 and hence can get the value of x.

But say, I invest 10,000 at x% and 50,000 at y% (I don't know how the principal is split), will the actual interest earned at x% and at y% be in the ratio 3:4? No. Actual Interest earned at x% will be much less than actual interest earned at y% because amount invested at x% is very little.

Hence, we need the way the principal was split to get the ratio of interest earned.

Hi Bunuel,

This makes complete sense in retrospec but I tried to solve using the weighted avg formula:

W1/W2 = A2-Avg/Avg-A1

I realize that the Avg is 6.8% but I was thrown off by the values of W1/W2 and A2, A1. Is w1 and w2 supposed to be the amount that accrues interest rate 1 and 2, or in this case, x and y?

Thanks.

\(4x = 3y\)

\(\frac{y-6.8}{6.8-x}=\frac{3}{2}\).

For more check:
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/03 ... -averages/
and
https://www.gmatclub.com/forum/veritas-prep-resource-links-no-longer-available-399979.html#/2011/04 ... ge-brutes/

Let the amounts be A and B.
Stem:
A + B = 60,000
Ax + By = 4,080

We have 4 variables, so we need 4 different equations.

Statement 1, x = 3y/4, gives us the relation between x and y but not x or y themselves

Statement 2, A=3B/2 , in combination with A + B = 60,000, gives us A and B but not x or y

With statements 1 and 2, we have the 4 equations needed.

Answer C

VeritasPrepKarishma - I was trying to use weighted average with statement 1, can you pls help.

We are given that x/y = 3/4 and we can calculate the overall rate by using the 60,000 and the interest earned 4,080. using this won't we be able to get the ratio in which 60,000 would be invested at X and Y%?

Won't we enough data with equation 1 itself to solve it for X%?

TIA

Answer: option C

Check solution as attached

Using weighted averages,

4080/60,000 = 6.8%

Using stmnt 1:
w1/w2 = (4x/3 - 6.8)/(6.8 - x)
Now here is the problem: We don't know w1/w2 using stmnt 1 alone. So we cannot calculate the value of x.

Using stmnt 2, we get the value of w1/w2 which is 3/2. But here we don't have the ratio of x and y.

So you need both statements to get

3/2 = (4x/3 - 6.8)/(6.8 - x)

And now you can get a unique value of x.

Answer (C)

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

VeritasKarishma

I mistakenly read (2) as saying "the amount of interest earned at x" rather than "the amount that earned interest." If it was said in the former way (2) would be sufficient on its own, right? You could find the value of each out of 4080, so you could also determine x% and y% from just (2). Would the GMAT do something tricky like this?

4080/5 = 816
So 2448 at x and 1632 at y.
2448/4080 = .6 and 1632/4080 = .4