David used part of $100,000 to purchase a house. Of the remaining port

David used part of $100,000 to purchase a house. Of the remaining portion, he invested 1/3 of it at 4 percent simple annual interest and 2/3 of it at 6 percent simple annual interest. If after a year the income from the two investments totaled $320, what was the purchase price of the house?
Let X be the invested amount
Sol : 1x/3 * 0.04 + 2x/3 * 0.06 = 320 (total investment)
or 0.04x/3 + 0.12x/3 = 320
or 0.16x/3=320
or x = 320 * 3 * 100 /16 = 6000
David used part of $1,00,000(given) , hence = 100000-6000 = $ 94,000 QA

Of every 3 dollars invested, $1 earns 4% interest and $2 earns 6% interest.
Average interest earned by every $3 = (1*4 + 2*6)/3 = 16/3% = 16/300.

From here we can plug in the answers, which represent the purchase price of the house.

Answer choice C: 88,000
Amount invested = 100,000 - 88,000 = 12000.
Interest earned = (16/300)(12000) = 640.
Since the interest earned is too great, the purchase price of the house must INCREASE, so that the amount invested DECREASES.

Answer choice B: 94,000
Amount invested = 100,000 - 94,000 = 6000.
Interest earned = (16/300)(6000) = 320.
Success!

The correct answer is B.

Let the Price of house was x.

Then

\(\frac{1}{3}\)(10000-x) * \(\frac{4}{100}\) --> 1st investment

\(\frac{2}{3}\)(10000-x) * \(\frac{6}{100}\) --> 2nd investment


\(\frac{4}{300}\)(10000-x) *(1+3) = 320


(10000-x) = 20 * 300 = 6000

x = 94000

Option B is correct

Let \(x\) be the amount of money left after purchase of the house. So \(\frac{1}{3} * x\) was invested under \(\frac{104}{100}\) percent and \(\frac{2}{3} * x\) was invested under \(\frac{106}{100}\). and \(x + 320\) we will get our investment back with the interests

\(\frac{x}{3} * \frac{104}{100} + \frac{2}{3}x * \frac{106}{100}\) =\(x + 320\)
\(\frac{104x + 2 * 106x}{300} = x + 320\)
\(316x = 300x + 320 * 300\)
\(16x = 32 * 10 * 300\)
\(x = 6000\)

To get the price of the house we should substract the money left after after purchasing from 100,000 :

\(100000 - 6000 = 94000\)

Hence, the answer is B

dave13 - important steps were not spelled out.

This one is a bear to explain.
I tacked (*1) to the first expression in the equation
because (*1) changes nothing but makes the factoring out easier to see.
I have broken the steps down below.

That said, I would not and did not use much algebra to solve.

\(x\) = price of house
\((100,000 - x)\)= amount invested
End game: Find a common factor

Original equation (not in solution above)
(\(\frac{1}{3}*($100,000-x)*\frac{4}{100}) + (\frac{2}{3}*($100,000-x)*\frac{6}{100})= $320\)

\((\frac{$100,000-x}{3} * \frac{4}{100} * 1) + ((\frac{1}{3}*\frac{2}{1})*($100,000-x)*\frac{6}{100})= $320\)

\((\frac{$100,000-x}{3} * \frac{4}{100} * 1) = (\frac{1}{3}*($100,000-x)*\frac{2}{1}*\frac{6}{100})\\
= $320\)

\((\frac{$100,000-x}{3} * \frac{4}{100} * 1) + (\frac{$100,000-x}{3}*\frac{12}{100}) = $320\)

\((\frac{$100,000-x}{3} * \frac{4}{100} * 1) + (\frac{$100,000-x}{3}*\frac{4*3}{100}) = $320\)

\((\frac{$100,000-x}{3} * \frac{4}{100} * 1) + (\frac{$100,000-x}{3}*\frac{4}{100} * 3) = $320\)

COMMON FACTOR in both terms: \(\frac{$100,000-x}{3}*\frac{4}{100}\)

COMMON FACTOR, factored OUT of blue equation above
\((\frac{$100,000-x}{3} * \frac{4}{100} * 1) + (\frac{$100,000-x}{3}*\frac{4}{100} * 3) = $320\)
\((\frac{$100,000-x}{3} *\frac{4}{100})\) * \((1 + 3) = $320\)

When you factor it out of the first expression, you "leave" 1; out of the second expression, you "leave" 3.
(Try multiplying it back the other way, e.g. A(b+c) = Ab + Ac)

The rest:
\((\frac{$100,000-x}{3} *\frac{4}{100})* (1 + 3) = $320\)

\((\frac{$100,000-x}{3} *\frac{4}{100})* (4) = $320\)

\(\frac{$100,000-x}{3} * (\frac{4}{100} * 4) = $320\)

\((\frac{$100,000-x}{3} * \frac{16}{100})= $320\)

Clear the 3 in the denominator. Multiply both sides by 3

\($100,000-x * \frac{16}{100}= $960\)

\($100,000 - x = $960 * \frac{100}{16}\)

\($100,000 - x = \frac{$960}{16}*100\)

\($100,000 - x = $60 * 100\)

\($100,000 - x = $6,000\)

\($100,000 - $6,000 = x\)

\($94,000 = x\)


If you cannot follow that algebra, do not worry.
It is not necessary to solve the problem with that particular version of algebra.
You can use another kind of algebra. Or very little.

I backsolved.

Start with C) $88,000

If the house cost $88,000, then (100,000 - 88,000) = $12,000 is left over

1/3 of that $12,000 is invested at 4 percent
$4,000 earns .04
2/3 of that $12,000 is invested at 6 percent
$8,000 earns .06

Combined, do they earn $320 in a year?

\(.04(4,000) + .06(8,000) =\)
\(160 + 480 = $600\)

NO. If the house costs $88,000, the leftover money is $12,000.
It earns $600 in one year. Too much.
The base principal (the leftover money ) is too great.
The house needs to be more expensive.

Try B) $94,000

If the house cost $94,000, then (100,000 - 94,000) = $6,000 is left over

1/3 of that $6,000 is invested at 4 percent
$2,000 earns .04
2/3 of that $6,000 is invested at 6 percent
$4,000 earns .06

Combined, do they earn $320 in a year?

\(.04(2,000) + .06(4,000) =\)
\(80 + 240 = $320\)

BINGO

Answer B

Hope that helps, dave13
_________________

An alternative solution:

"Of the remaining portion" is split two ways - 1/3 & 2/3. The interest earned is an integer amount. So, we can pick some smart numbers for the remaining amount and calculate the interest rate. Use answer choices to pick a number that is divisible by 3.

Of all the answer choices, (A) and (D) do not offer a number divisible by 3. (E) can be eliminated easily because when you invest 1/3 of $60,000 ($20,000) @ 4% interest rate, you earn $800 interest, which is definitely more than $320. But, if you calculate 2/3 of $60,000 ($40,000) @ 6% interest rate, you earn $2400 interest. Total interest when $60,000 is the remaining amount: $800 + $2400 = $3200. We need $320. It means the remaining amount should be $6000. Therefore, (B) is the answer.

It may seem like a little bit lengthy approach, but with enough practice you can save yourself some trouble solving algebraic equations and save some time

Happy solving!

STRATEGY: Upon reading any GMAT Problem Solving question, we should always ask, Can I use the answer choices to my advantage?
In this case, we can easily test the answer choices.
Now let's give ourselves up to 20 seconds to identify a faster approach.
In this case, we can also solve the question algebraically. However, since I can already see we can immediately eliminate two answer choices, I know that I need only test ONE answer choice.
So I'm going to start testing answer choices

First recognize that the correct answer can't be A or D. Here's why:
If the correct answer were choice A ($96,000), then David invested the remaining $4000, which means he invested 1/3 of $4000 at an interest rate of 4% and 2/3 of $4000 at an interest rate of 6%.
The problem is that $4000 isn't divisible by 3.
So, for example, 1/3 of $4000 = $1333.33333...., which means David invested $1333 and one-third of a penny at 4%.
Since we can't invest one-third of a penny, answer choice A is disqualified.
We can apply the same logic to eliminate answer choice D.

Of the remaining three answer choices (B, C and E), I'll test the middle answer...
(C) $88,000
If David spent $88,000 of his $100,000 on his house, then he invested the remaining $12,000
1/3 of $12,000 is $4000, which means David invested $4000 at a 4% simple interest rate, which means the interest after 1 year = 4% of $4000 = $160
2/3 of $12,000 is $8000, which means David invested $8000 at a 6% simple interest rate, which means the interest after 1 year = 6% of $8000 = $480
So, the TOTAL income from the two investments = $160 + $480 = $640

The question tells us that the total income from the two investments is $320, which means we can eliminate choice C.
Also, since we want the total income to be LESS THAN $640, we need David to invest less than $12,000.
In other words, we need the house price to be greater than $88,000, which means we can also eliminate answer choice E.

By the process of elimination, the correct answer must be B.

Answer: B

100,000 = 100 K
h = house

[100k - h]/3 * (4/100) + [100k - h]/3* (12/100) = 320

Taking [100k - h]/3 * (4/100) as a factor

[100k - h]/3 * (4/100) *[ 1+3] = 320

[100k - h]/3 * (4/100)*4 = 320

[100k - h]*16/100 =320 *3

100k - h = (320*3*100)/16

h= 96000

Hi adiagr,
Could you please clarify why there is no (1+%) in your base formula. As I remember simple interest formula should be interest*(1+years*%/100).
What am I doing wrong?
Thanks you in advance

lor12345 , I think adiagr might not be around anymore. It looks as if you just got the formulas slightly mixed up.

There is no (1 + %) in his formula because $320 is interest only. See his RHS: it's $320. That's interest only. (We weren't given $6,320, which in this case is principal + interest.)

Simple interest with principal PLUS accrued interest ("A") formula is indeed

A = P(1 + rt)

But interest only for simple interest formula is

I = (P)(r)(t)

adiagr used this formula, and defined P as ($100,000 - x)*(1/3) in one case, and ($100,000 - x)*(2/3) in the other.

True, both are multiplied by the respective interest rates, but without the 1 because his x is derived from subtraction from $100,000. If you use a multiplier with a 1, you include interest earned on $100,000. That is not how it worked. David earned interest on some small part of $100,000.

If you want to include 1 in the multiplier, it seems easiest if you separate the principal (which earned interest) from the $100,000 (most of which did not). It's plenty hard either way, IMO.

See Vorovski , above, who used a (1+%) multiplier by considering ONLY x as principal (and not $100,000 - x, until the end).

(Whew. I backsolved (used the answer choices) in about 90 seconds. See how I used the answer choices, here.)

Hope that helps.
_________________

rocko911 , yes, exactly.
The word "income" tells us we are solving for interest only.

Interest is defined in one way as the income earned on a principal amount of money.

The original principal amount of money is NOT defined as "income."

In addition, here, we are NOT told something such as:
"after a year, the total amount of money from David's investment is $_____"

So the word "income" lets us know that we need to use the "interest only" formula.

The income earned from the money invested
= interest = $320

Interest only = (Principal Amt) * \(\frac{rate}{100}\) * time

OR, Interest only = (Principal Amt * rate * time),
if the interest rate is in decimal form

Your first equation is correct.

Hope that helps.
_________________

Hi generis how is life ?:)

well explained:) i actually had the same question but after reading your post got it,

so in general 1+% means a principal account +acrrued percent right ?

but we are interested only in finding accrued percent so that's why we use only 0.04 and 0.06 without "+1" , correct ?

Hi dave13 - life is good. You?

Yes, you are 100% correct.

If you see a 1.xx for an interest rate, it almost always means the principal has been included...

If you see a 0.xx for an interest rate, it means the principal has not been included.

Nice work.

Below is an explanation for WHY the "1" usually means that we're talking about principal, too.
The 1 means 100% of Principal= \(\frac{100}{100} P= 1P\)
For SIMPLE INTEREST (not compound)

• interest only formula is
\(i = P * r * t\)
\(i\) = interest
\(P\) = principal
\(r\) = rate in decimal form
\(t\) = time

• Total Amount(principal plus interest) has a different formula. But it borrows from the interest only formula.

The "1" stands for 100% of original Principal, or \(\frac{100}{100}P\) or \(1P\)

Total amount (principal plus interest) earned formula is:
\(A = P(1 + rt)\)
Breaking it down, Total Amount, A
A = Principal + Interest
A = Principal + (P*r*t) (from above for interest only)
A = P + (P*r*t)
Now factor out P
A = P (1 + rt)
If this makes no sense to you, ignore it.
You have the basic concept!

_________________

Let h be the cost of the house, remaining money for investment --> (100,000-h)
(100,000-h)(1/3 * 4/100) + (100,000-h)(2/3 * 6/100) = 320
(100,000-h)(4/300 + 12/300) = 320
(100,000-h) = 320 * 300/16
100,000-h = 6,000
100,000 - 6,000 = h
h = 94,000

generis, how de got this \(\frac{4}{300}\)(10000-x) *(1+3) = 320[/m]

Number plugging, starting with the middle option:

C) --> 12000 left, 8000 at 6 % = 80*6 = 480, too much.

B --> 6000 left, 4000 at 6 % = 40*6 = 240. Looks promising. 2000 at 4 % = 20*4 = 80 = correct answer.


Im not sure which would have been faster for me but I actually took the algebraic approach here:

x/3 * 4/100 + 2x/3 * 6/100 = 320

16x = 96000, x = 6000. Answer must be B.

I Didn't understand why (1+3)?

Posted from my mobile device

It seems the below question has misnomer wording. It asks us to find the income after 1 year, so it should not be simply addition of simple interests for both the investments, but addition of the total amount received for both the investments after 1 year.

Isn't it?

1/3 interest rate 4% and 2/3 interest rate 6% -> Together the rate must be closer to 6% (aprox 5.3%)

Then test the choices to see which matches with this rate -> B is the only that works 320/6000 = 5.3%