On a certain day, Tim invested $1,000 at 10 percent annual interest

Question

On a certain day, Tim invested $1,000 at 10 percent annual interest, compounded annually, and Lana invested 2,000 at 5 percent annual interest, compounded annually.The total amount of interest earned by Tim’s investment in the first 2 years was how much greater than the total amount of interest earned by Lana’s investment in the first 2 years?

A. $5
B. $15
C. $50
D. $100
E. $105

My Doubts :
 1)What does the statement : 10 percent annual interest, compounded annually mean? 
Does compounded annually mean 5% for first 6 months and 5% for next 6 months ??

2)Can someone solve this using detailed approach?

Bunuel · Accepted Answer

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Shoumik · Answer

There are two types of interest rates that GMAT handles with.
1) Simple interest
2) Compound Interest

simple interest is basically when you simply multiply your investment amount * interest rate * number of years.

Compound interest formula is  Interest Earned = A * (1+i)^n - A , where A is your investment amount.

The difference between the two can be conveyed with the following example:
Suppose you invest 5000 dollars in a bank with an interest rate of 10% for 3 years.

Compounded Annually:
After 1 year, your bank total would be:   + $5,000 = $5,500. (Total interest earned = $5,500 - $5000 = $500)
After 2 years, your bank total would be:  $5,500   * 0.10] + $5,000 = $6,050. (Total interest earned = $6,050 - $5000 = $1050)
After 3 years, your bank total would be:   $6,050   * 0.10] + $5,000 = $6655. (Total interest earned = $6,655 - $5000 = $1,655)

* Using the compounded interest formula above will give you the same answer. *

Whereas, simple interest for the same investment would be:
After 1 year, your bank total would be: $5,000 * 0.10 * (1) + $5,000 = $5,500. (Total interest earned = $5,500 - $5000 = $500)
After 2 years, your bank total would be:   +$5,000 = $6,000. (Total interest earned = $6,000 - $5000 = $1000)
After 3 years, your bank total would be:   + $5,000 = $6,500 (Total interest earned = $6,500 - $5000 = $1,500)

As you can see, "Compounded annually" charges interest on the bank balance which includes both the initial investment amount + the cumulative interest earned from that investment whereas simple interest is based solely on the interest earned from just the initial investment amount.

siddhans · Answer

What does the term compounded annually mean ? interest received twice in a year? i.e once every 6 months ?

BenchPrepGURU · Answer

Compounded annually means that the interest is applied once per year. One can have 10% annual interest compounded monthly - in this case 10%/12 would be applied each month, or 10% annual interest compounded daily etc.

With respect to the problem at hand, at the end of two years, Tim would have
1,000(1.10)^2 = 1,000(1.21) = 1,210
and Lana would have
2,000(1.05)^2 = 2,000(1.1025) = 2,205
Thus, Tim earned 210 dollars, while Lana earned 205 dollars
The difference is $5 and the answer is A.

BenchPrepGURU

Shoumik · Answer

Compounded annually means your bank amount is being interested once per year (yearly). If it was semi-annually, it would mean your amount is being interested once every six months.

abid1986 · Answer

calculate simple interest for both lana and tim

Tim interest = 20% X 1000 = 200+ Additional interest due to compounding
Lana interest= 10%x2000 = 200+ additional interest due to compouding

but the difference between two compounded interest would be very less 
Hence Ans A)

Had the option was 7$ as well ,then calculation would be required here.

irda · Answer

After first year both person would have an interest of 100 (10 % of 1000 = 100 & 5 % of 2000 is 100)

The interest for next year is also 100 for both person as above

The difference in the net interest is between the interest added to both account.

Therefore for the account with 10 % interest the additional interest is 10 % of 100 = 10

and for the other account it is 5% of 100 = 5

Hence net difference = 10 -5 = 5

PareshGmat · Answer

Tim: 
Int for 1st Yr =100
Int for 2nd Yr = 110 (compounded)

Lana:
Int for 1st Yr =100
Int for 2nd Yr = 105 (compounded)

Difference:
= 100 + 110 - (100 + 105)
=5

jlgdr · Answer

I agree one should notice that after the first year interest is the same, they both earn 100 bucks
Now, in second year

Tim gains 1100/10 = 110
Lana gains 2100*5/10 = 105

So difference is 5

A is the correct answer

Hope it helps
Cheers
J

Salvetor · Answer

tim :

1st year = 1000+ 10% of 1000 = 1100
2nd year : 1100+ 10% 1100 = 1210

interest earned = 1210-1000 = 210

lana : 1st year = 2000+15% of 2000 = 2100
2nd year = 2100+ 15% of 2100 = 2205

interest earned = 2205-2000 = 205

so ans is 210-205 = 5

CrushTheQuant · Answer

Here's an explanation on YouTube. (This GMAT question, by the way, is a great introduction to questions about compound interest.) https://www.youtube.com/watch?v=DFcQrgM ... aNJHoNeFi0

RadheyaKulkarni · Answer

Can someone solve this and explain? What is a fast way to solve the multiplier raised to an exponent ? (1.1^3 & 1.05^3) Thank you

Adarsh_24 · Answer

Can we say compound interest if compounded annually is as below,

Interest from year 1 = principal * r/100
Interest from year 2 alone = (interest from year1) * ( 1 + r/100)
Interest from nth year alone = (interest from year1) * (1 + r/100)^(n-1)

Is this is a valid formula?

yalhaddad3d · Answer

Tim ====> 1000*10/100 = 100 so first year he has a total of 1000 + 100 which will be compounded again =====> 1100*10/100 = 110
Total compound earning = 100+110 = 210

Lana ====> 2000*5/100 = 100 so first year she has a total of 2000 + 100 which will be compounded again =====> 2100*5/100 = 105
Total compound earning = 100+105 = 205

Difference = 210-205 = 5

Answer is A.

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On a certain day, Tim invested $1,000 at 10 percent annual interest

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