Aptitude - Compound Interest - Discussion

1. 

A bank offers 5% compound interest calculated on half-yearly basis. A customer deposits Rs. 1600 each on 1st January and 1st July of a year. At the end of the year, the amount he would have gained by way of interest is:

[A]. Rs. 120
[B]. Rs. 121
[C]. Rs. 122
[D]. Rs. 123

Answer: Option B

Explanation:

Amount
= Rs. 1600 x 1 + 5 2 + 1600 x 1 + 5
2 x 100 2 x 100
= Rs. 1600 x 41 x 41 + 1600 x 41
40 40 40
= Rs. 1600 x 41 41 + 1
40 40
= Rs. 1600 x 41 x 81
40 x 40
= Rs. 3321.

C.I. = Rs. (3321 - 3200) = Rs. 121


Akhila said: (Aug 31, 2010)  
How did you get 41/40 am not able to get that value please help me.

Sundar said: (Aug 31, 2010)  
Hi Akhila,

= 1 + (5/(2 x 100)

= 1 + (5/200)

= 1 + (1/40)

= (40 + 1)/40

= 41/40

Next step (41/40)^2

= 41/40 x 41/40
...

Hope you understand. Have a nice day!

Sipu said: (Nov 5, 2010)  
YA SUNDER IS CORRECT...

Bala said: (Nov 13, 2010)  
Its a basic thing. Where did you get 3200 in this problem?

Manju said: (Nov 17, 2010)  
Hai, where did you get from this value 3200.

Dinesh Guptha said: (Nov 17, 2010)  
I think customer deposits per year 3200.

So they subtracted 3200 with that final amount it seems.

Dinesh said: (Nov 17, 2010)  
Can anyone clear me the first step.

Venkat said: (Nov 19, 2010)  
venkst:simple customer deposited twice
1600*2=3200

Boney said: (Nov 22, 2010)  
I dont understand the 2nd term in the expression
Amount=1600[1+(1/2)*(5/100)]^2+ 1600[1+(5/2*100)]
Amount compounded half yearly is A=P[1+r/2*100)]^2
Don't understand what this term doing 1600[1+(5/2*100)]
Someone tell me

Lavanya said: (Dec 5, 2010)  
I too dint understand d first step

Kumar said: (Dec 16, 2010)  
I didn't understand someone help me.

Anusha said: (Jan 13, 2011)  
Customer has deposited twice in a year. Once in the begining i.e on Jan 1st n 2nd time on july 1st i.e after 6 months.

So 1st case n=1(1 yr) & 2nd case n=1/2(half yr).

So you get first and 2nd terms coresponding 2 first and 2nd case.

Akanksha said: (Jan 23, 2011)  
@ Anusha: There's nothing. See clearly, 1st january and 1st July are dates of depositions!

And please someone explain why 2nd term is not squared while 1st is?

Malkhan Meena said: (Jan 26, 2011)  
On 1st january customer deposit amount and receive end of year.

But in 1st july he deposit same amount (1600 Rs. ) and receive it after the same date (after 6 month).

It means first he deposit for 1 year and second he deposit for 6 month. And get all amount at the end of year.

Vishal said: (Feb 23, 2011)  
Yes, its right , so in first case n (no. of years) is 1 and in second case (for july) n is 1/2 years, so now you can directly put that values in simple formula.

Vikram said: (Feb 25, 2011)  
Give the mathemetical formula for compund interest.

Rajesh said: (Feb 26, 2011)  
@Anusha:

You told 2 cases one is half yearly(july) and other is yearly(jan). In yearly case the formula should be p(1+r/100)^n but they used half yearly formula only.

Sukumar said: (Feb 28, 2011)  
I can't understand

[1600*41/40(41/40+1)]

Harry said: (Mar 13, 2011)  
Will someone explain the first step?

Amit said: (Apr 18, 2011)  
On 1st january customer deposit amount and receive end of year. It means first he deposit for 1 year.

When interest is compounded Half-yearly:

Amount = P * [ { 1 + (1/2) * (R/100) } ^ 2(n) ]
Amount = 1600 * [ { 1 + (1/2) * (5/100) } ^2(1) ] = 1681


But in 1st july he deposit same amount (1600 Rs. ) and receive it after the same date (after 6 month). It means second he deposit for 6 month(1/2 year).

When interest is compounded Annually but time is in fraction, say 1/2 years.

Amount = P * [ 1 + { (1/2)* R/100}]
Amount = 1600 * [ 1 + { (1/2)* 5/100}] = 1640

So Total = 1681 + 1640 = 3321

Cherry said: (Apr 26, 2011)  
Hi amit , c.I formula is A=p*(1+(r/100)^n) , so i understood the first step , but what s that i am expecting in the 2nd step is Amount = P * [ 1 + { (1/2)* R/100}^1/2 } , why isn't any sqrareroot included there. i mean to say u should mention the "n" i.e no. of years ass "half" right , as we are calculating only for 1/2 year , from july 1st to jan 1 st

Leejo said: (May 14, 2011)  
@Amit:gud answer

Siri said: (Jun 1, 2011)  
What is formula for simple interest?

Priya said: (Jun 9, 2011)  
Can you tell me how come it's 5 months it should be 6 month written, 1 jan to 1 july the time period is 7 month in this question they ask half yearly??

Priya said: (Jun 13, 2011)  
He deposit second time in july know. Beforre that tere are 6 months. So it is termed as half yearly. Is it clear.

Niveditha said: (Jul 2, 2011)  
Can any explain this sum fully?

Mrudhula said: (Jul 3, 2011)  
Let me explain in my point of view...

Firstly... i have calculated the compound interest of 1600/- for 1year..i.e,
[1600*[1+(5/100)]]=80

Then at 7th month the total amount=3200/-

So the compound interest of 3200 for (1/2)year is..
[3200*[1+(5/200)]^2]=162

Now the average of 80 and 162 is..
(80+162)/2 = 121

Prasad Shetty said: (Jul 6, 2011)  
Hi Amit, you have explained the problem correctly and good thinking.

Balaji said: (Jul 23, 2011)  
Very nice amit and mrudhula.

Satheesh said: (Jul 23, 2011)  
Hi Mrudhula i think u have some logical mistakes

here i will provide you the simple procedure

first calculate the C.I for first 1600 in 6 months

A=1600(1+5/200)^2*1/2
=1640

Now another 1600 is invested so now the total amount is

1640+1600=3240 now the C.I for this amount should be calculated

i,e A=3240(1+5/200)^2*1/2
A=3321 SO HE INVESTED 3200 AND THE GAIN IS 121

Pratik said: (Jul 26, 2011)  
At the end of the page, formula is mentioned.

Jyoti said: (Jul 27, 2011)  
Hello Friends
it is mentioned that bank give interest on half yearly Basis
jan to june thr is 6 months
cal of jan to june:1600(5)/200=40.so total is 1640. On july money become 3240 because man deposite on july.
Now cal july to Dec:(3240)5/200=81.
Now u can see total gain is 81+40=121.

Anu said: (Aug 1, 2011)  
Hi jyoti that was really good and simple explanation.

Karthi said: (Aug 22, 2011)  
Why putting 100*2 in denominator?

Dinesh said: (Aug 31, 2011)  
How did you get 41/40 am not able to get that value please help me.

Ram said: (Sep 14, 2011)  
First calculate CI for first deposit Jan 1st.

Here interest is compounded half early

So Formula is
Amount = P [1 + (R/2)/ 100]^2n here n is 1 year(12mnths)
so, amount is 1600[1 + 5/200]^2

and now calculate amount for money deposited on july

Formula is Amount = P [1 + (R/2)/ 100]^2n here n=1/2yrs(6mnths)
so, amount is 1600[1 + 5/200]^1

Add both amounts
and subtract 3200(ie., 1600+1600) we get CI.

Kapil said: (Sep 16, 2011)  
Why is the rate 5% halved while calculating, once it is given that it is calculated half yearly.

Chiranjit said: (Sep 29, 2011)  
I can't understand (1+5/2*10). Please explain it.

Anusha said: (Oct 16, 2011)  
Calculate first deposit jan 1st

amount= p[1+(R/2)/100]^2n here n is 1 year
so amount = 1600[1+(5/2)/100]^2
=1600[1+(5/200)]^2
=1600[1+(1/40)]^2
=1600[41/40]^2
=1681.
now cal amount for money deposited on july
amount=p[1+(R/2)/100]^2n n=1/2 yr
=1600[1+(5/200)]
=1600[41/40]
=1640.
add both amounts
1681+1640=3321
1600 twice the customer deposited 1600*2=3200
3321-3200=121.

Digvijay said: (Nov 7, 2011)  
Thanks to anusha I was confused in second part.

Nandhakumar said: (Nov 19, 2011)  
Now only I understood. Thanks anusha.

Santu said: (Nov 28, 2011)  
Thank you. Anusha.

Annie said: (Dec 7, 2011)  
The question is for one year. What if 5 years? please help me. I've got headache thinking about this topic.

Randheer said: (Dec 16, 2011)  
You people might confused with the formulae and there of three formulaes where n represents exactly year i.e n=1 year, n=1/2 half year so on.,

Then another thing the three formulaes were having a much difference look at it as individual dont miggle those all three formulaes 2gether .

Ex: to calculate half year compund intrest u shoud go with the second one
don't bother about the duration here we are calculating C.I for every half yearly what ever the term it be either 1 year or 2 years u shud use the half year formulae for calculating that... look at it as very abstract.

Amit said: (Jan 10, 2012)  
CI is nothing but interest on interest.
so first find SI for 1600
so SI= (1600 * 5 * 1/2)/100 = 40
now total money is 1640 after 6 month. he again deposit 1600 rs so
now in july beginning total money is 1640 + 1600 = 3240 rs
again find SI = (3240 * 5 * 1/2)/100 = 81 rs

So total interest is = 81 +40 = 121 Rs ans.

Shreya said: (Jan 17, 2012)  
Sateesh and Jyoti gave the nice explaination.

Saurav Karmakar said: (Jan 21, 2012)  
At the time of first deposit i.e on January 1st

Amount= p[1+(R/2)/100]^2(n)
here n is 1 year beginning of the year
so amount = 1600[1+(5/2)/100]^2
=1600[1+(5/200)]^2
=1600[1+(1/40)]^2
=1600[41/40]^2
=1681.

Secondly when he deposited its July i.e after 6 months
so n=half of the year=1/2

Amount=p[1+(R/2)/100]^2n n=1/2 year
=1600[1+(5/200)]
=1600[41/40]
=1640.

now add both amounts
1681+1640=3321
1600 deposited 2 times by the customer in a year therefore 1600*2=3200

gain => 3321-3200=121.

Vishal said: (Jan 28, 2012)  
Best way was shows by Jyoti.... Kudos Jyoti... :)

Mangesh said: (Feb 11, 2012)  
From 1st Jan to next 1st Jan i.e n=1(12 months)
C.I.1= 1600[1+(5/2*100)]^(2*1)

Now from 1st July to 1st Jan i.e n=1/2(6 months)
C.I.2= 1600[1+(5/2*100)]^(2*1/2)

C.I.=C.I.1 + C.I.2

Naveenraj said: (Jun 11, 2012)  
1600
Interest 5% in 6 month (1/2 of a year) = 5/100 * 1600 * 1/2 = 40
1640 + 1600 = 3240(principal + previnterest + new deposit)
5% of 3240 in next 6 month = 5/100 * 3240 * 1/2 = 81
End of year total = 3240 + 81 = 3321

Gain = total with interest - deposit = 3321 - 3200 = 121

Student said: (Aug 2, 2012)  
FV=P(1+r/n)^(nt)

FV = Future value of the deposit.
P = Principal or amount of money deposited.
r = Annual interest rate (in decimal form).
n = Number of times compounded per year (1 or 2 or 3... etc).
t = Time in years for which the money has been deposited.

Use this formula and you will get the answer.

Vivek Kumar said: (Oct 29, 2012)  
Can someone tell me why have they considered 1600 value for the next 6 months (P) should be 1600 + Interest for the next half.

Pankaj Parashar said: (Dec 21, 2012)  
According to me itis given that compounded half yearly
So rate=5/2=2.5%
So first 6 months ci=1600*2.5/100=40
Now amount=1600+40=1640

Now 2nd 6 months ci=3240*2.5/100=81

So total ci = 80+40 = 121.

Zaid Junaid said: (Feb 15, 2013)  
Well according to me it means that he is depositing the amount 2 times first in Jan and than in July.

So for Jan years will be 1 but for July it will be 1/2 years. So that's why the second term has not been squared.

Suresh said: (Feb 27, 2013)  
Well. Is there any shortcuts to solve the problem?

Arvind said: (Mar 23, 2013)  
Hi.

Can someone please let me know why the 2nd time of deposit been multiplied by 2*100 (in denominator).

As per my understanding the period is only 6 months in this case hence the time period is 1 and so is the case with the interest right we don't have to multiply it with 2 as only for 6 months.

Kindly clarify why the denominator is multiplied with 2?

Nil.Dhongde@Gmail.Com said: (Apr 2, 2013)  
My gosh. So many comments. I know it is bit confusing and I too.
Let me try to make you understand.

So here we start. you might have familiar with the formula,

C.I = p[1+(R/100)]^n.

Here p = principal amount.
R = rate.
n = no.of years.

But in the problem we are dealing with half year.

Means we are getting C.I on 6 months
* we have given annual rate of 5%.

So for half year it would be R/2
* As we are calculating C.I over every 6 months, so for a year n become 2 (as two half year is equal to one year).

So here n = 2.
So our formula becomes,

C.I = P[1+(R/2*100)]^2.

Here p is given as 1600 Rs. Now after 6 months, on date 1 July another amount of 1600 Rs got deposited.

So again we have to calculate the C.I for this amount for a 6 months only(upto 31 Dec) so that we can get the C.I from Jan 1 to July 1 and from July 1 to Dec 31.

So as to complete one year. as We are asked about C.I over total one year.

So,
For a second amount formula for C.I becomes,

C.I = P[1+(R/2*100)]^1.

Combining two we have,

C.I = P[1+(R/2*100)]^2 +P[1+(R/2*100)]^1.

Pooja said: (Apr 7, 2013)  
@Amit: Thanks a lot you made me understand the concept. It took me whole day to ponder over the 2nd half of the expression.

Amarnath said: (Jun 5, 2013)  
Let me try to make you understand.

>> First 6 months .

* Deposit is 1600.

* Interest is 5% half yearly.

So, Interest is 5/100 for 6 months,

i.e., one year is represented as 1 (one).

Half yearly means 1/2 (six months), {quarterly 1/4 (4 months each) for example}.

To calculate interest on deposit.

>> deposit * interest * years.
>> 1600 * 5/100 * 1/2.
>> 40.

Interest on 1600 is 40,
So 1600 + 40 = Rs. 1640.

After first six months, the total amount available is 1640.

>>> Next Six months.
>> Deposit is 1600.

So, Total available balance is 1600 + 1640(from first 6 months).

>> 3240 * (5/100) * (1/2).
>> 81.

Finally, So total interest is 40 + 81 = 121.

Vasavi said: (Jul 11, 2013)  
As it is half yearly basis,

The formula is A=p[1+(R/2)/100]^2(n).

But here only time varies i.e n=1y & n=1/2y.

Now substitute.

Sharmi said: (Sep 14, 2013)  
Any shortcut to solve this problem?

Vicky said: (Sep 27, 2013)  
Shortcut guys,

1600*5/100*6/12 = 40 then after 6 months 1600+1600 = 3200 & also 40 will be interest calculate.

3200*5/100*6/12 = 80.

40*5/100*6/12 = 1.

= 40+80+1 = 121 Answer.

Anshika said: (Oct 20, 2013)  
Hey guys can someone guide me. If time period is given 3/4 years in a question of compound interest compounded annually then which formula to use?

Preeti said: (Nov 14, 2013)  
I can't understand why the time is 1st year for deposition on jan and 1/2 on july?

Vishal said: (Dec 13, 2013)  
If compound interest is given and no.of years have to be calculate then?

Priya said: (Feb 26, 2014)  
Hello @Preeti.

In first case it is 1 year because its from 1st Jan to end of the year so its of 12 months = 1 year.

In sec case it is 1/2 year because its from 1st July to end of the year its of 6 months = 1/2 year.

Hope you ll get.

Venkatesan M said: (Apr 29, 2014)  
1st Jan-First Half Yr 1600 x 5/100 x 6/12 = 40.00.
1st Jan-Second Half Yr (1600+40)x 5/100 x 6/12 = 41.00.
1st Jul-second Half Yr 1600 x 5/100 x 6/112 = 40.00.
_______

Total Int = 121.00.

Venkatesan M said: (Apr 29, 2014)  
SI : 625 x 4% x 2yr = ---------------------------------- 50.00.
CI : 625 x 4% x 1st Yr ------------------ 25.00.

(625+25) x 4% x 2nd yr ------------- 26.00.
          ------      -------
Total    51.00    50.00.

Difference between SI & CI (51.00 - 50.00) 1.00.

Shekhar said: (May 27, 2014)  
Why did subtract one in most of cases? please reply.

Shail said: (Jul 17, 2014)  
Hi,

What does "5% compound interest calculated on half-yearly basis" actually mean?

This 5% interest is, interest per annum.
Or interest per 6 months?

Expert said: (Aug 1, 2014)  
1st half 1600. After half half yr @2.5% 40 so total 1640.

2nd half 1640+1600 = 3240 so after 1yr 3321.

Gain 3321-3200 = 121.

Mzcool said: (Aug 2, 2014)  
What is the difference between simple interest and compound interest?

Ravi said: (Aug 9, 2014)  
Hey guys what formula use for this? please help me.

Kasinath @Hyd said: (Sep 5, 2014)  
Since Bank offers compound interest on HALF YEARLY basis: so convert 1 Year into two Half years. so, use only half year formula.

Half Year formula: (1+ R/2*100)^2n.

1 year = Half Year + Half Year.

=>(1+ R/2*100)^2*1 n=1 is bcoz (1/2 +1/2).

Half year = (1+ R/2*100)^1 1 is bcoz (2*1/2) here n=1/2.

Siva Nandi Reddy said: (Sep 18, 2014)  
There is no formula to get the answer.

Simple logic is there that,
For 1 year 80/- is the interest then half of that amt is 40/- for half year.

So next half year for Jan amount 40/- and 1/- (it is obtained on first half interest). So total is 81/-.

and for July amt 40 because it is also half year, So the total is 121/-.

Aramide Akintmehin said: (Sep 25, 2014)  
Please why did you multiply the 100 by 2.

Srikanth said: (Oct 21, 2014)  
He deposited 1600 for 1/2 year but our question he asks yearly so 1600+1600 = 3200.

Giri said: (Oct 27, 2014)  
When interest is compound half quarterly.

Amount = p[(1+(r/2)/100]^2n.

So amount = 1600[1+(5/2)/100]^2.

Simplify above equation, we get,

Amount = 1600[1+5/2*100]^2.

Purnika said: (Nov 8, 2014)  
Why we are supposed to add the compound interest for half year and one year to get the amount and why C.I. = (3321-3200)?

Iovd said: (Nov 23, 2014)  
@Purnika.

They have calculated CI for 1 year not for 1/2 year see carefully.

Roman Pearce said: (Nov 23, 2014)  
First i did for one year then converted into half:

CI = 1600*5/100 = 80 for one year we want for 6 months then its = 40 for next half year.

CI = 1640+1600*5/100 = 164 one year but we want for 6 months then its = 81.

Total interest is 40+81 = 121.

Ranjith said: (Dec 22, 2014)  
I have taken this formula:

A = 1600(1+2.5/100)^2+1600(1+2.5/100).
= 1600(1+41/40)^2+1600(1+41/40).

= 1681+1640.
= 3321.

Therefore less principle amount of 1600*2 = 3200 to get interest earned.

3321-3200 = 121.

Rak said: (Dec 27, 2014)  
Why subtract 3200 from 3321?

Gurpreet said: (Feb 6, 2015)  
Because 3200 is the total amount of one year which is principle.

Thirumurugan Sundaram said: (Feb 20, 2015)  
I think it will be as correct answer for everyone, the first deposit amt is 1600, after six month total amt is 1640 (Including 5% of one year interest for six months). We should assign this as A-case.

Next July 1st day second deposit amt is 1600, It should be as B-case. The end of the year amt is 1640 (Including 5% of one year interest of six months). But A-case deposited at one year, So total Mature amt is 1681 (Including 5% of one year interest of twelve months. Interest of Rs. 40 is Rs-1, So amt Rs-40 became as Rs-41).

Finally A-case + B-case are 1681 + 1640 = 3321. Deposit amt is Rs-3200 and Interest amt is Rs-121.

Prasoon Dawn said: (Mar 17, 2015)  
I am not able to understand the problem in these quick steps. Can anyone please help me understand in detailed process.

Prathik said: (May 1, 2015)  
Guys most of you might be confused with the formula, lets do this way:

1st 6 months deposit = 1600,

Interest earned at 5% upto July = (1600 x 5/100 x 1/2) = 40.

Note: In compound interest interest gets added to the principle.

So P = 1640 now, and next deposit in July = 1600.

Total deposit = (1640+1600), interest = (3240 x 5/100 x 1/2) = 81.

Total profit earned as interest = 40+81 = 121.

Hope this helps.

Anna said: (Jun 19, 2015)  
Your answer are so useful guys. Keep it up.

Swarna said: (Jul 18, 2015)  
Such a brilliant answers I got here. How you people can make this possible?

Please give some suggestion to improve my aptitude problem solving ability.

Dias said: (Aug 9, 2015)  
Is there any other way to solve this problem?

Monalisa said: (Aug 20, 2015)  
I am confused somebody help me a better way to do this problem?

Anu said: (Sep 2, 2015)  
1600+1600 = 3200.

Nidhi said: (Sep 3, 2015)  
Thanks @Anu you explain it nicely.

Here half yearly means every six months so the amount of 1600 is submitted two times so that it become 3200.

Thank you all have a good day.

Pragya said: (Sep 23, 2015)  
Good explanation.

Beaulah Abigail said: (Sep 27, 2015)  
Someone please explain the question.

Rohan said: (Oct 19, 2015)  
Please explain the 1st question I am not able to understand.

Aps said: (Oct 27, 2015)  
I don't understand please explain it.

Nikesh said: (Nov 10, 2015)  
Please solve this question.

In how many years will a sum of Rs 800 at 10% per annum compounded semiannually becomes Rs 926.10.

Is this process wrong?

926.1 = 800(1+5/200))^2n.

Where n is in half yearly. I got 4/5.

Hanmanth said: (Nov 11, 2015)  
How 3200 is coming please explain detail me?

Donalika said: (Nov 29, 2015)  
Easy way to do it quite clearly.

Double the amount of 1600 = 1600+1600 = 3200.

Is this way is correct? I think its correct.

Sirisha said: (Dec 10, 2015)  
How come 81/40? Can any one explain?

Spg said: (Jan 12, 2016)  
5/2 (since half yearly, had it been quarterly 5/4 or monthly 5/12).

= 2.5 => 100 + 2.5 = 102.5 => 102.5/100 = 1.025.

For 1st amount = 1600 * 1.025 * 1.025 = 1681.

For 2nd amount = 1600 * 1.025 = 1640.

Therefore interest earned 81 + 40 = 121.

Kishor said: (Feb 11, 2016)  
For half yearly basis (R) = 5/2 = 2.5%.

(2.5% of 1600)/100 = 1600*2.5. = 40

Customer has rs 1600 + 40 = 1640.

He again adds 1600 on 1st July so 1600 + 1640 = 3240.

(2.5% of 3240)/100 = 3240*2.5 = 81.

Therefore, total interest earned is 81 + 41 = 121.

Deepa said: (Feb 24, 2016)  
Thank you all those helped me to clear the problem.

Kajal said: (Apr 7, 2016)  
I solve this by the simple interest method.

Where p = 1600, r = 5, T = 0.5(6 months) (January amount).
Interest = (1600*5*0.5)/(100*2) = 40 Rs.

Now p =1640(January) + 1600(newly added in July).
So, p = 3240, r = 5, t = 0.5(6 months).
Interest = (3240*5*0.5)/(100*2) = 81 Rs.

Now the total interest = 40 Rs + 81 Rs =121 Rs.

Anandaganesh said: (Apr 20, 2016)  
Thank you @Amit. Your explanation is simple to understand the concept.

Tiger Karim said: (May 17, 2016)  
It is very easy to solve without formula.

Such as 1600 *.05 = 80/2 =40.
So, 1640 *.05 = 82/2 = 41.
So that interest = 81 and 1600*.05 = 40
Then Ans = 81 + 40 = 121.

Abdul Wahab said: (May 18, 2016)  
Where the present value (pv) = 1600.

Rate: 5%/2.

Then, n:2 for 1st jan to dec and 1 for 1st jul to dec.

Therefore fv =1600 (1 + 0.025) ^2 + (1 + 0. 025).

= 1600 * 2. 07563.

= 3321.

Now the amount he would have is 3321 - 3200 = 121Rs.

Mon Doley said: (Jun 7, 2016)  
Still in confusion, Check out these:

According to question Rs. 1600 is invested twice. i.e. from 1st Jan to till the end of the year, and from 1st July to the end of the year.

So we have -.

P= Rs. 1600; R = 5%; N1= 1yr; N2= 1/2yr.

So, Amount->1 = p{1+ (R/2) /100}^2 * N1, ie CI calculated half yearly.

= 1600 (1 + 5/200) ^2 * 1.
= 1600 (205/200) ^2.
= 1600 (41/40) ^2.
= 1600 * 41 * 41/40 * 40.
= 41 * 41 = 1681 --->(i).

And,
Amount->2 = p{1 + (R/2) /100}^2 * N2.

= 1600 (1 + 5/200) ^2 * 1/2.
= 1600 (1 + 5/200).
= 1600 (41/40).
= 40 * 41.
= 1640 ----> (ii).

Therefore Total Amount:

(i) + (ii) = 1681 + 1640 = 3321 Rs.

As the same amount, 1600 is invested twice so Total principal = 1600 * 2 =3200.

Therefore, C.I = Amount - Principal.

= 3321 - 3200.
= Rs. 121.

Abi said: (Jun 15, 2016)  
@Mon Doley.

Thanks, you done a great job.

Kunal said: (Jun 28, 2016)  
Here is a very simple method.

1600 * 5 * 1/2 * 100 = 40.

Now the amount you have on 1st July is 1600 + (1600 + 40) = 3240.

So.
3240 * 5 * 1/2 * 100 = 81.
Now add 40 + 81 = 121.

V!Cky said: (Jul 6, 2016)  
If interest is 5% compounded semi-annually. Then what is the effective interest for 1st half, is it 5/2% or 5%?

Anonymous said: (Jul 17, 2016)  
I don't understand how come it is divided by 200? What is 200?

And isn't the formula P(1+i)^n?

Nishant said: (Jul 25, 2016)  
Thanks @Amit, @Jyoti and @Nil.Dhongde.

Arpita Mandal said: (Aug 1, 2016)  
Why the amount of first six months is not included with 1600 as principal while calculating the amount of next six months?

Anurag Srivastava said: (Aug 5, 2016)  
PRINCIPLE AMT p=1600RS. THIS AMOUNT IS COMPOUNDED HALF YEARLY SO r=5/2%.

Time= 2t this is theory based on the amount calculated half yearly.

For the first six months he received an interest we calculate as follow.

= p (1 + r/100) ^2t.

Here t = 6 month since we have to put it in the formula as a year so 6 month means 1/2 year.

So compound interest for 6 month = 1600 (1 + 5/2 * 100) ^2 * 1/2.

= 1600 (1 + 5/200) =1640 this is the amount he has in his account after six months.

Again bank gives him six-month interest on this amount after the completion of first six months now understand clearly after the six-month interest will be given on 1640 so this is principle amount for the bank again follow the above procedure.

= 1640 (1 + 5/2 * 100) ^2 * 1/2.
= 1640 (1= 5/200) =1681.

The total amount he received at the end of year = 1640 + 1681 = 3321.

Total amount he paid = amount paid for first six month + amount paid for next six month = 1600 + 1600 = 3200.

Interest he received at the end of 12 month = 3321 - 3200 = 121.

Kumar said: (Aug 15, 2016)  
Most helpful for banking. Thanks to all friends.

Sahil said: (Aug 17, 2016)  
I got 123, How that's possible?

Panda said: (Aug 21, 2016)  
From where did 3200 came?

Ranjith said: (Aug 23, 2016)  
Why does the second part not have a square i.e ^2 for it?

Kaviya said: (Aug 25, 2016)  
I don't understand it. Please give me a simple explanation.

Lesner said: (Sep 2, 2016)  
Only see the remaining year from starting date of deposit and multiplication done according to that value. As year remains from 1 Jan is 1 while from 1 July is 1/2. This is the main concept.

Soundarrajan said: (Sep 7, 2016)  
I am confused with the first step could anyone explain?

Shree said: (Sep 9, 2016)  
They have asked the amount, not CI. Why is CI the answer?

Yamini said: (Oct 11, 2016)  
Its is already given that 5% on half yearly basis than when why by the rate of interest calculated further?

Sathish said: (Oct 18, 2016)  
@Yamini. Read the question carefully.

Kotturi said: (Oct 18, 2016)  
@Yamini.

Half year CI was given. But in question, they are asking annual CI.

Chi said: (Nov 13, 2016)  
@Jyoti thanks. Very well explained.

Adithya said: (Dec 3, 2016)  
Very good explanation @ Anusha.

Lomoj said: (Dec 8, 2016)  
Good & fabulous job, Thank you all for the given solution.

Anupam Das said: (Dec 14, 2016)  
1600 * 5/200 = 40(1st half)
1600 + 1600 + 40 = 3240
3240 * 5/200 = 81(2nd half )
So, total interest = 40 + 81 = 121.

Lithin Gowda said: (Dec 28, 2016)  
Thank you for all your answers.

Revant Borawat said: (Dec 29, 2016)  
Annually rate 5%.
for 1st installment 2.5 * 2.5 + 0.625 = 5.0625.

Amount=1681.
For 2nd 40 ÷ 1600 * 41 = 1640,
3321 - 3200 = 121.

S Simadri said: (Jan 6, 2017)  
Easy trick : compounded half yrly so 5%\2 = 2.5%.
1st half year interest : 2.5 % of 1600 = 40.
2nd half year intrest : 2.5% of 3200 + 2.5% of 40 = 81. Then CI is 40 + 81 = 121.

Maneesh said: (Jan 26, 2017)  
Guys we are dealing here with the condition : When interest is compounded Half-yearly having formula.

Amount = P[1+(R/2)/100]^2n .

So in 1st case we have n=1 which means time is 1 year and in 2nd case, we have n=1/2 which mean half a year, simply put these values of 'n' and u will get the desired result.

Brijesh said: (Jan 28, 2017)  
1:- 6 month 1600 * 5 * 1/2 * 100 = 40.
2:- 6 month 1640 + 1600 = 3240.
3240 * 5 * 1/2 * 100 = 81.
Total:-3321 - 3200 = 121.

Harshit said: (Feb 4, 2017)  
I am not able to get why the second term has the rate of interest as 5/2 but time is one year as we can see the second term has power 1.

Please clarify this.

S. Ganesh Babu said: (Mar 17, 2017)  
Very smart explanation, thanks @Brijesh.

Asif Ansari said: (Mar 21, 2017)  
@ALL.

You all are facing problem because you are obsessed with "n" to substitute as "year" always.
"n" is basically how many "times" rate is applied on amount.

Here, it is clearly given that the interest is calculated on "half-yearly" basis. Now because of half yearly basis.

Here "n" will be "1" for 6 months (and r=r/2).
and "n" will be "2" for 1 year ( and r=r/2).

Suppose if it was given quarter-yearly,
"n" would be "1" for 3 months ( and r=r/4).
"n" would be "2" for 6 months (and r=r/4).
"n" would be "3" for 9 months (and r=r/4).
"n" would be "4" for 1 year (and r=r/4).

So, this guy kept his first 1600Rs amount for " two period" time.
that is, from 1st jan to 30th June (1st period) and from 1st July to 31st Dec (2nd Period).
he also deposited another amount on 1st July, this amount is kept for "one period".

That is, from 1st July to 31st Dec.

Therefore there is square(n=2) in a first Formula and no square(n=1) in a second formula.

Hope I helped little bit.

Rana Suresh Varma said: (Apr 14, 2017)  
1st calculate CI on 1600 half yearly for one year so 1600(1+5÷200)^2.

= 1600(205÷200)(205÷200),
= 1681,
= CI = 1681-1600 = 81->eqn1.

Next calculate CI on 1600 which is deposited after 6 months i.e. for 6 months.
= 1600(1+5÷200),
= 1600(205÷200),
= 1640.
= CI is = 1640-1600= 40->eqn2.

Add eqn1 and eqn2 we get;

= 81 + 40 = 121 this is the gain.

Jyoti Ranjan said: (Apr 17, 2017)  
This is very simple.

Here, the compound interest half yearly formula has been applied. But 1st case 1600 deposit time is 1 year and 2nd case 1600 deposit time n=1/2 year. Rest calculation is same.

Vinod said: (May 23, 2017)  
CI CAN BE EXPRESSED IN TERMS OF SI.

IN THIS PROMBLEM,

FOR FIRST 6 MONTHS CALCULATE SI OF PRINCIPAL SINCE CI CALCULATED ON HALFYEARLY BASIS.
SI FOR 6 MONTHS = (1600 * (1/2)*5)/100 = 40.
CI FOR NEXT 6 MONTHS = (1600+40) * (1/2) * 5)/100 = 41,
TOTAL CI = 40 + 41 = 81.

SI FOR 6 MONTHS FROM JULY = (1600*(1/2)*5)/100 = 40,
TOTAL EARNING = 40 + 81 = 121.

Sam said: (Jun 5, 2017)  
As far, I know Compound Interest =p(1+r)^n, I can't understand how n=2?

Can anyone help me? Please.

Anubhab said: (Jun 10, 2017)  
@Sam.

As we know when interest is compounded half-yearly, we divide the rate by 2 and multiple the year(n) by 2.

As he deposits in Jan and bank gives interest half yearly so there so by the next year Jan there will be 2 r% interest which is the reason why n=2 and for the deposit in July he gets the only one which is the reason n=1.

Hope it helped.

Mahesh said: (Jul 2, 2017)  
A person lends certain money from bank at 10% compound interest. If he returns, Rs. 1600, Rs. 1500 and Rs. 4400 at the end of 1st, 2nd and 3rd year respectively to complete his loan. Find how much money he lends from the bank.

Please give solution for this.

Chandan Anand said: (Jul 4, 2017)  
Here answer should be 244, because the total amount at the end of 6 months will be added to 1600. And so, the new P=A+1600.
solution:
for 6 months:

A=1600{1+(5/100)}^1
=1680
So for next 6 months:
P= 1680+1600=3280,
A=3280{1+(5/100)}^1.
=3444.
therefore, the total interest earned= 3444-3200 = 244.

Thavaz@ said: (Jul 13, 2017)  
Helo guys.
Here is a simple method..
Now 1600rs at 5% intrest per annum. So 2.5% for 6 months. its 40rs then at the 2nd investment 1600+40+1600=3240.
Here 2.5% is 81rs, So 40+81=121rs (5% per annum so 2.5% per 6mnths).

Rohit said: (Jul 18, 2017)  
Very nice explanation @Prathik.

Arvind Uike said: (Jul 18, 2017)  
3321-3200 how 3200 can give here?

Please give the clear solution.

Prince said: (Jul 30, 2017)  
1600 he deposits once and 1600 again twice in a year that why there is 3200?

Shashwat Srivastava said: (Aug 10, 2017)  
Shortcut method.

Total ci =5+2.5+(5*2.5/100)
=7.6%.

ci=7.6 of 1600=121.

Rojit said: (Aug 20, 2017)  
Can someone explain me?

If I don't go for formula.. so let's do with normally.

January to july,
Principal-1600
rate-5%
time-1 (i.e half year)

so CI= 1600*0.05*1
= 80.
Hence the amount is 1600+80=1680 which is the principal for the second time.

Now July to December,

His new principal=1680 and he deposit 1600 more so his new principal is 1680+1600=3280.
rate is same = 5%
time=1 (half yearly)

So CI = 3280*0.05*1.
= 164.

So its 164. How 121? Please correct me.

Ashk said: (Aug 27, 2017)  
I think the first step is evaluated as follows:
That square term came from the half-yearly basis.

We have given for half year but we need to find for a full year. So we set n=2 (That is twice of half year) and n=1 for next case (half year).

Umang B. said: (Aug 30, 2017)  
Half yearly formula is used in which for case 1: (of january) n=1;.

Half yearly formula is used in which for case 2: (of july) n=1/2(6 months);

Mr. Mp said: (Sep 1, 2017)  
For 1/2 year formula is =P(1+(R/2)/100)^ 2n.
Now 1600 for 1st 1/2 year or 6 months so n=1/2.
=1600(1+(5/200)^1 = 1640.

This money bank won't returns him on July 1st it is kept in the bank till December. So from July to December 1640 also carries interest again apply formula for this with n=1/2
we get 1681 here 81 is the interest he got.

On July again he deposits 1600 again apply formula we get 1640 40 is the interest so total interest 81+ 40 = 121.

Mahesh said: (Sep 1, 2017)  
1600 initial amount, we get 40 interest. So total money 1st 6 months is 1640 this money will not be returned, it again carries for next 6 months to calculate interest at the end of December. Again 1640 apply formula we get 1681. We can use common sense to calculate interest for 1640 (1600+40).

We know for 1600 40 interest for 40? Which is 1 rs.

Mr.Mp said: (Sep 1, 2017)  
@ALL.

P (1+ R/2 /100) ^ 2n for half yearly.

P (1+R/100) ^n for yearly.

Don't think about the second 1600 deposited on july 1st, forget this completly to avoid confusion.

1600 first deposit becomes 1640 at the June end (use half yearly formula).

Since this is not given back on June end it will remain in the bank.

So what we think is, 1600 is kept for 6 +6 months = 1 year and we use yearly formula. (completely mistaken).

How can we use yearly interest formula when question clearly says interest calculated half yearly.

If we use yearly formula we get 1600 (1+ 5/100) ^1 =1680.

Now see when half yearly formula is used.

1600 (1+ (5/200) ) check half yearly formula.

Which gives 1640.

As I said this 1640 not given back, again apply half year formula.

1640 (1+5/200) = 1681.

So interest is 81.

Compare the interest you got using yearly formula (mistaken formula) and actual formula (correct formula). We got 80 (mistaken) and we got 81 (correct) see the difference.

So it is used p (1+ R/2 /100) ^2n = 1600 (1+ 5/200) ^2 = 1681 correct n= 1/2 +1/2 =1.

Jen said: (Oct 3, 2017)  
Hi, I don't understand how you set up the calculations in that way. The formula is fv=pv (1+r) n. Aren't the calculations suppose to be fv=1600 (1+0.05)1?

Sagar said: (Oct 9, 2017)  
If haf yearly then n=2n and if for 2years then n=2n.

Indhu said: (Oct 9, 2017)  
I did not understand the first calculative statement please help me.

Why [1600 x (1 + 5/2 x 100 )] this step is taken after the half-yearly count?

Henrymc said: (Nov 23, 2017)  
Amount after 1 year on Rs. 1600 (deposited on 1st Jan) at 5% when interest calculated half-yearly.
=P(1+(R/2)100)2T=1600(1+(5/2)100)2*1=1600(1+140)2

Amount after 1/2 year on Rs. 1600 (deposited on 1st Jul) at 5% when interest calculated half-yearly.
=P(1+(R/2)100)2T=1600(1+(5/2)100)2*12=1600(1+140)

Total Amount after 1 year.
=1600(1+140)2+1600(1+140)=1600(4140)2+1600(4140)=1600(4140)[1+4140]=1600(4140)(8140)=41*81=Rs. 3321.

Compound Interest = Rs.3321 - Rs.3200 = Rs.121.

Henrymualchin said: (Nov 23, 2017)  
@Indhu.

1 July, it is 6th month of the year, where there's total 12 months in a year
so, it means half year(6/12=1/2).

Therefore, [1600 x (1+5/2*100)]^2n,
= [1600 x (1+5/2*100)]^2*1/2,
=[1600 x (1+5/200)].

Namrata said: (Dec 15, 2017)  
Why we are using half yearly formula in the first case taking n=1/2, and in the 2nd case taking n=1 but keeping R as R/2?

P Snr said: (Jan 13, 2018)  
A=p(1+R/100)n but why take 200 places of 100?

Kush said: (Feb 8, 2018)  
We use compound interest formula of half-year because the interest rate imposing on 1600 deposited on 1 Jan was only for 6 months.

Siba Nanda said: (Feb 11, 2018)  
Please tell me why you have taken 5/2*100?

Anand said: (Feb 14, 2018)  
The difference between simple interest and compound interest to be compounded annually from the annual interest rate of 16% for 2 years on a fixed amount is 320.

Anu said: (Feb 18, 2018)  
Thank you @Anusha.

Swathi said: (Feb 25, 2018)  
As per formula, S=p*(1+r/200)^2n.Where r=rate of interest per annum. Why we have taken 5/200 there.

As they didn't give rate of interest they gave compound interest.

Suhani said: (Apr 3, 2018)  
I am not at all able to understand the whole sum. Please, someone, help me.

Rahul Singh said: (Apr 5, 2018)  
r=5/2%=1/40.

If p=40 then A=41 ..(bcz t=6 months=1 half yr); CI1=Rs1*40=Rs 40,
again,p=1600 then A=1681...(because t=12 months=2half yr);CI2=Rs 81,
net CI=Rs(81+40)=Rs121.

Shubham said: (Apr 17, 2018)  
I think the answer is not 121. It's 122. Because here 5% in not annually, it is half yearly. It is clearly mentioned in the question. Am I right?

Ashok Raj said: (May 17, 2018)  
@All.

An interest usually is given year base so here calculated compounded half yearly that's why to split interest in two parts first 2.5 on 1600 is 40 and after 6 months he invested again 1600 total 3200 so 2.5 on 3200 is 80 and comp interest means interest on previous year 2.5 on 40 is 1 total interest is 40+80+1=121.

Stupid said: (Jun 26, 2018)  
@All.

In compound interest annually the formula is P(1+R/100)^n. But for half-yearly, it is P[1+2R/100)^2n substitute the values.

Ankur Banerjee said: (Jul 29, 2018)  
Why the value of n is 2?

Yamini Dhanasekar said: (Aug 9, 2018)  
We can also solve it by basic SI formula.
PNR/100.

Customer deposits 1600 twice in a year.
In a first 1/2 yr,p=1600,n=1/2,r=5%,
interest is..1600x1x5 / 2x100 = 40,
In a second 1/2 yr, p becomes 1640,
Interest=1640x1x5 / 2x100 =41,
So interest gained by amount deposited on;
January at the end of yr=81,
And,1600 is deposited on july,again same procedure.
I = 1600 x 1 x 5/2x100 = 40.
So, 81 + 40 = 121.

Shubham said: (Aug 19, 2018)  
Jan= 1600 deposited.
Jan to july =6 month interest= 2.5% of 1600= 40.
July to dec = 2.5% of 40+ 40 = 41.

total interest on amount 1600 deposited on january= 40+41=81.

So, the interest on amount 1600 deposited on july = 40

Then, total interest = 40+81=121.

Prashu said: (Aug 20, 2018)  
How come 2*100?

Please explain me.

Ankush Jaat said: (Nov 5, 2018)  
I am not getting how time be 2? Please explain.

Lince said: (Nov 15, 2018)  
Formula for compound interest: A=P(1+r/m)^m(t).
where:
A=future value.
P=principal.
r=rate.
m=no. of compounding period a yr.
t= no. of years.

In the above problem, we have to solve January and July.

FOR: on January 1st, the customer deposits 1600 which is logically an end of the year.

GIVEN:
P=1600,
r=5% or 0.05,
m=2 (half-yearly basis).
t=1 (one year, as you read a while ago, it's also an end of the year deposit so it's one year).

Using the formula:
A= 1600(1+(0.05/2))^2(1),
= 1681.

FOR: on July 1st he again deposits 1600. 1st of July is the 7th month of the year which is the starting of half of year (1/2).

GIVEN:
P=1600.
r=5% or 0.05,
m=2 (half-yearly basis),
t=1/2 (half of the year),

Using the formula:
A= 1600(1+(0.05/2))^2(1/2).
= 1640.

Now that we're done solving January and July's future value we got:
customer's total deposit is 1600+1600 = 3200.
customer's total deposit WITH interest is 1681+1640 = 3321.

Finally, the question is " what is the amount he would have gained BY WAY OF INTEREST?"

From the formula:
I= Amount-Principal.
= 3321-3200.
= 121 (amount OF INTEREST).

Hope this helps.

Chakri said: (Nov 23, 2018)  
1 year -->5%.
1/2 year -->2.5%.

At jan 1st=Rs.1600 --> 1600+(2.5% of 1600)=1600+40 = 1640.

At july 1st = Rs.1600+Previous amount + (2.5% of previous total) = 1600 + 1640 + (2.5% of(1600 + 1640)).
= 3240+81
= 3321----->At year last.

Total spent is 1600 + 1600 = 3200,
Threfore 3321 - 3200 = 121.

Parv said: (Dec 10, 2018)  
If on calculating the first 6 months then from July it is CI for both. Why only SI is considered in both time.

Narendra Singh said: (Dec 28, 2018)  
Ist half interest =1600*1/40 = 40,
IInd half interest/(1600+1600) * 1/40 = 80,
interest on Ist half intrest = 40 * 1/40 = 1,
Total CI= 40+80+1 = 121.

Ajay said: (Apr 6, 2019)  
Amount from 1st January to 1st July is
Amount= 1600[1+5/(2*100)]^2.
= 1681.

Note: The principal amount becomes 1681 instead of 1600 because the money is deposited in the bank. There is not provided with any information regarding the interest is withdrawn.

So, the new principal amount at 1st July is 1600+1681 = 3281.
The amount at the end of the year becomes;
=3281[1+5/(2*100)]^2,
=3447.1.
The gain is =3447.1-3200.
=247.1 Rs.

Correct me, whether it is wrong.

Anom said: (May 6, 2019)  
I want to ask that, there is a 5% interest on every half a year, why you are calculating 5% interest for a year?

Sanjana said: (May 8, 2019)  
@Anom.

You are right but just once again read the 2nd line of the question.

It is being said that the amount is received in 1st of Jan and 1st of July which are basically half years in simple words - from 1st of Jan to 1st of July it's 6 months which is a half year and same way from 1st of July to 1st Jan is another 6 months which is another half year.

Therefore, the 5% of interest is calculated in Jan and July separately [ p- (1+r/100) ^t + p- (1+r/100) ^t] - p. Which calculated in half year basis only not for a year.

Mahesh S said: (Aug 3, 2019)  
@All.

It's simple, first one is deposited on jan, the year n is 1, second one deposited on July which is half of one year, so year n is 1/2, just put this in the formula for compound interest half yearly you can get the answer.

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