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John invested a total of $100,000 in two accounts. The fir

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by Max@Math Revolution » Mon May 28, 2018 12:01 am

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[GMAT math practice question]

John invested a total of $100,000 in two accounts. The first account had an annual interest rate of 4%, and the second account had an annual interest rate of 6%. If his investment earnings at the end of the first year were at least $5,000, what was the smallest possible amount that he could have invested at an annual interest rate of 6%?

A. $40,000
B. $42,000
C. $45,000
D. $50,000
E. $54,000
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by Max@Math Revolution » Wed May 30, 2018 12:16 am
=>

Let x be the principal invested at the 6% interest rate.
Then we must have x*0.06 + ( 100,000 - x )*0.04 ≥ 5,000.
Now,
x*0.06 + ( 100,000 - x )*0.04 ≥ 5,000
=> x*0.06 + ( 4,000 - x *0.04 ) ≥ 5,000
=> x*0.02 + 4,000 ≥ 5,000
=> x*0.02 ≥ 1,000
=> x ≥ 50,000

Therefore, the smallest amount he could have invested at 6% was $50,000.
Therefore, D is the answer.

Answer: D
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by Jake@ThePrincetonReview » Wed May 30, 2018 2:40 pm
A conceptual explanation: John's making at least 5% (5,000 is 5% of 100,000) interest. An equal amount invested at 4% and 6% would result in an average interest rate of 5%. Any more than half that 100k invested at 4% would bring that average down to below 5% so his interest would be less than 5,000. Thus, John must be investing at least half his money at the 6% rate, so D.

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