Donald plans to invest x dollars in a savings account that

Question

Donald plans to invest x dollars in a savings account that pays interest at an annual rate of 8% compounded quarterly. Approximately what amount is the minimum that Donald will need to invest to earn over $100 in interest within 6 months?

A. 1500
B. 1750
C. 2000
D. 2500
E. 3000

A. 1500
B. 1750
C. 2000
D. 2500
E. 3000

mustdoit · Accepted Answer

Lets solve it under 30 seconds...

8% compounded quarterly = 2% per quarter = 4% for half year

if 4% is 100 then 100% would be 2500.....Answer D. What say....

Hope this works !!!

Bunuel · Answer

Algebra: x(1+0.02)^2 -x > 100 x(1.02^2-1)>100 x*0.0404>100 x>\frac{100}{0.0404}\approx{2475.25} Donald plans to invest x dollars in a savings account that pays interest at an annual rate of 8% compounded quarterly. Approximately what amount is the minimum that Donald will need to invest to earn over $100 in interest within 6 months? A. 1500 B. 1750 C. 2000 D. 2500 E. 3000 Annual rate of 8% compounded quarterly is approximately 4% in 6 months (a bit more). x*0.04=100 --> x=2500. Answer: D. Similar questions to practice: john-deposited-10-000-to-open-a-new-savings-account-that-135825.html on-the-first-of-the-year-james-invested-x-dollars-at-128825.html marcus-deposited-8-000-to-open-a-new-savings-account-that-128395.html jolene-entered-an-18-month-investment-contract-that-127308.html alex-deposited-x-dollars-into-a-new-account-126459.html michelle-deposited-a-certain-sum-of-money-in-a-savings-138273.html leona-bought-a-1-year-10-000-certificate-of-deposit-that-143742.html Theory: math-number-theory-percents-91708.html Hope it helps.

alwynjoseph · Answer

Compound interest formula

A = P ( 1+r/n)power nt

given, n= 4 (quaterly);r =.08

the approach is substitution,

our interest requirement is 100$ after 6 months, 2 compounding period. interest per compounding period is 2%

lets take 1500, after 3 months interest accumulated is 30$, total amount is 1530
after 6 months, interest is 30.6$ and total is 1560.6$, so not 1500

1500 & 1750 have a difference of 250$ only , but the expected interest different is around 40$ hence you can straightaway rule out 1750

2000 is again can be ruled out as approx 4% interest yeilds only 80$

2500$ is a good bet, first 3 months it earns 50$ as interest, next 3 months it will earn 51$ as interest.
hence answer is D

Economist · Answer

E.

For CI:
Final amount = Principal(1+ r/n)^nt
Let principal = x, then final amount = x+100

x+100 = x ^2
x+100 = 1.04x
0.04x=100
x=2500

so if 2500 is invested CI will be exactly 100. To earn more interest more principal should be invested. So E.

TheRob · Answer

Thank you veyr much here is the book explanation

The formula for calculating compound interest is A = P(1 + r/n)nt where the variables represent the following:
A = amount of money accumulated after t years (principal + interest)
P = principal investment
r = interest rate (annual) 
n = number of times per year interest is compounded
t = number of years
In this case, x represents the unknown principal, r = 8%, n = 4 since the compounding is done quarterly, and t = .5 since the time frame in question is half a year (6 months).
You can solve this problem without using compound interest. 8% interest over half a year, however that interest is compounded, is approximately 4% interest. So, to compute the principal, it's actually a very simple calculation:
100 = .04x
2500 = x
The correct answer is D.

rohitgoel15 · Answer

Donald plans to invest x dollars in a savings account that pays interest at an annual rate of 8% compounded quarterly. Approximately what amount is the minimum that Donald will need to invest to earn over $100 in interest within 6 months?

A. 1500
B. 1750
C. 2000
D. 2500
E. 3000

A. $1500
B. $1750
C. $2000
D. $2500
E. $3000

The formula for calculating compound interest is A = P(1 + r/n)nt where the variables represent the following:
A = amount of money accumulated after t years (principal + interest)
P = principal investment
r = interest rate (annual) 
n = number of times per year interest is compounded
t = number of years

temp33 · Answer

IMO D

Solution:
r = Rate = 8% compounded quarterly.. = 2% per quarter
t = 6 motnhs = 2 quarters
So A = P (1+0.02)^2 
=> A = P * 1.0404

Subsitute, P for 1500, 1700, 2000.. The A-P should be greater than 100 (since Interest = Amount - principal).
For 2500, interest is 101...

Hence, D

gurpreetsingh · Answer

A = P(1 + r/n)nt here t =1/2 and n=4
=> A = P(1 + 8 %/4)4*(1/2)
=> A = P(1 + 2%)^2
=> A = P ( 1+ (2%)^2 + 4%) ,(2%)^2 is very small and can be ignored

=> A = P ( 1+4%) => A-P = P*4%
Also A-P = 100 => P = 100/4% = 100*100/4 = 2500

nikhil007 · Answer

what wrong am I doing, can someone point out?

{x(1+.02)^2 } -x > 100

taking x common on LHS
(x)(1.02)^2 -1>100

(x)(1.02)^2 >101

(x) 1.04 >101

x= 101\1.04

I think i am doing some silly calculation mistake here, can someone point me out pl.

vivekvijayan · Answer

Hi Bunnel,

Would it be correct to do the following.

I used the simple interest formula.
Since CI will give a greater amount than SI. an the interest is compounded quaterly. The rate for a quater would be 2%
Since the duration is 6 months there would be two compounding periods. $50 per period.

so... 50= (X. 2.1)/ 100

X would be 2500.... This is SI so CI would be obviously greater.

IS this method correct.?

Bunuel · Answer

Yes, 50= x*0.02 is correct.

Check other  Compound Interest Problems  in our  Special Questions Directory .

Hope it helps.

mikemurawski93 · Answer

Hi  Bunuel ,
 
In your equation above, x(1+.02)^2 - X> 100... where did you get the -X from? The equation for compound interest is A = P (1+r/n) ^nt. I was unaware we needed to subtract an x for the correct answer or where we even got the -X from. Could you please help?

Bunuel · Answer

Approximately what amount is the minimum that Donald will need to invest to earn over $100  in interest  within 6 months?

x(1+.02)^2 is the final amount, while x(1+.02)^2 - x is the interest earned.

KarishmaB · Answer

An annual rate of 8% means a quarterly rate of 2%. 
In 6 months, we have two quarters i.e. two time periods. So in the first time period, 2% interest will be earned. In the second time period, 2% will be earned along with 2% of the previous interest (overall, a very small value).
So we can assume approx 4% interest earned in 6 months.

1% of 1500 is 15 so 4% is 60 - Reject
...
1% of 2000 is 20 so 4% is 80. To reach 100, we need almost another $20 which is 50% of the value of previous interest. Not correct. 
1% of 2500 is 25 so 4% is 100 - This is it.

Answer (D)

sayan640 · Answer

Bunuel  , please check the highlighted part. 
Can you please explain why you say so ? just 8 / 2 = 4 % ?

Bunuel · Answer

An annual rate of 8% compounded quarterly means an interest rate of 2% every three months. In three months, you earn 2% interest on the investment. In six months, you earn 2% on the investment plus 2% on the interest earned in the previous three months, which is a small amount. Therefore, after six months, the interest is slightly more than 4%.

Chayan_C · Answer

The question asks for the minimum amount to be invested to earn an interest of OVER $100. Why have we chosen an option that yields exactly $100?
Shouldn't E be the answer?

Bunuel · Answer

Have you paid attention to the highlighted part in the stem?

ashKing12 · Answer

Quarterly rate = 8 / 4 = 2%
1 quarter = 3 months
so, 6 months = 2 quarters
let principal = x
CI for 1st quarter = x * (2%)
CI for 2nd quarter = x*(2%) +   *2%

total CI = Ci for 1st quarter + CI for 2nd quarter = (404x) / 10000 = 100 (According to question)

X comes out to be slightly less than 2500 
D

Donald plans to invest x dollars in a savings account that

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