Tag: business mathematics and statistics

Questions Related to business mathematics and statistics

Find the amount(approximately) of an annuity immediate of Rs 2000 per annum for 20 years. The rate of interest is $\dfrac{27}{2}$ % per annum.

business mathematics and statistics insurance and annuity amount of an annuity annuities financial mathematics

  1. 171703

  2. 19000

  3. 21345

  4. 43251

Show answer
Correct Option: A
Explanation:
$n=20,r=(27/2)\%,P=Rs.2000$
$A=P(1+\cfrac{r}{100})^n$
$\implies A=2000[(1+\cfrac{(27/2)}{100})^{20}$$+(1+\cfrac{(27/2)}{100})^{19}......$$+(1+\cfrac{(27/2)}{100})^{1}]\\A=2000\times 85.85\\ A=Rs.171700\approx 171703$

Mr Assem decides to deposit Rs 5000 at the end of year in a bank which pays compound interest the rate of 5% per annum. What will be his total accumulation (approximately) at the end of 15 years?

business mathematics and statistics insurance and annuity amount of an annuity annuities financial mathematics

  1. Rs.$140092$

  2. Rs.$907892$

  3. Rs.$100892$

  4. Rs.$107892$

Show answer
Correct Option: D
Explanation:
$A=P(1+\cfrac{r}{100})^n$
$\implies A=5000[(1+\cfrac{5}{100})$$+(1+\cfrac{5}{100})^{14}......$$+(1+\cfrac{5}{100})^{1}+5000]\\=5000\times 21.5718\\ Rs.107892$

Mr Dev purchased a car paying Rs $90,000$ and promised to pay Rs 5000 every 3 months for the next 10 years. The interest is $6$% p.a. compounded quarterly. If Mr Dev misses first 6 payments , how much should he pay at the time of 7th payment (approximately) to bring himself up to date.

business mathematics and statistics insurance and annuity amount of an annuity annuities financial mathematics

  1. 36333

  2. 46532

  3. 56734

  4. 60322

Show answer
Correct Option: A
Explanation:

$\Rightarrow$  We have $A=Rs.5000,\, n=7$  $I=\dfrac{6}{100}\times \dfrac{1}{4}=0.015$

$\Rightarrow$  At the time of 7th payment, equivalent amount of first 6 missed payment has also to be paid. Thus total payment to be made at the end of 7th period is the amount of annuity of 7 terms. Hence, amount required required to be paid is,
$\Rightarrow$  $M=\dfrac{A}{I}[(1+I)^{n}-1]$


$\Rightarrow$  $M=\dfrac{5000}{0.015}[(1.015)^{7}-1]$           -----  ( 1 )
$\Rightarrow$   Let $x=(1.015)^{7}$
$\Rightarrow$   $log\,x=7\,log\,(1.015)$
$\Rightarrow$   $log\,x=7\,(0.0064)=0.0448$
$\Rightarrow$   $x=antilog\,(0.0448)$
$\Rightarrow$   $x=1.109$
Substitute value of $x$ in ( 1 ) we get,
$\Rightarrow$  $M=\dfrac{5000}{0.015}(1.109-1)$

$\Rightarrow$  $M=\dfrac{5000}{0.015}\times 0.109$

$\Rightarrow$  $M=\dfrac{5000\times 1000}{15}\times \dfrac{109}{1000}$

$\Rightarrow$  $M=Rs.36333.33\approx Rs.36333$

A bank pays interest at the rate of 8 % per annum compounded half yearly. Find how much should be deposited in the bank (approximately) at the beginning of each half year in order to accumulate Rs 8000 for 3 years.

business mathematics and statistics insurance and annuity amount of an annuity annuities financial mathematics

  1. 994

  2. 1161

  3. 4532

  4. 2341

Show answer
Correct Option: B
Explanation:
$A=P(1+\cfrac{r}{100})^n$
$r=8\%/2=4\%, n=2\times 3=6$
$\implies 8000=P[(1+\cfrac{4}{100})^6$$+(1+\cfrac{4}{100})^{5}......$$+(1+\cfrac{4}{100})^{1}]\\ \implies 8000=P\times 21.5718\\ \implies P=Rs.1159\approx Rs.1161$

Find the future value of an ordinary amount of Rs 4000 each six months for 10 years at 8% per annum , compounded semi-annually.

business mathematics and statistics insurance and annuity amount of an annuity annuities financial mathematics

  1. $123877$

  2. $175640$

  3. $213457$

  4. $324156$

Show answer
Correct Option: A
Explanation:

$P=Rs.4000$

Amount payable in half yearly so $,r=8\%/2=4\%,t=10\times 2=20$
$A=\sum _{ n=1 }^{ 20 }{ P{ (1+\cfrac { r }{ 100 } ) }^{ n } } \ =\sum _{ n=1 }^{ 20 }{ 4000{ (1+\cfrac { 4 }{ 100 } ) }^{ n } } \ =4000\times 30.969\ =Rs.123877$

Find the least number of years for which an annuity Rs 1000 must run in order that its amount just exceeds Rs 16000 at 5% pa. compounded annually.

business mathematics and statistics insurance and annuity amount of an annuity annuities financial mathematics

  1. $12$

  2. $9$

  3. $2$

  4. $25$

Show answer
Correct Option: A
Explanation:

$P=4000, r=5\%$

$A=\sum _{ n=1 }^{ n _0 }{ P{ (1+\cfrac { r }{ 100 } ) }^{ n } } \ =\sum _{ n=1 }^{ n _0 }{ 1000{ (1+\cfrac { 5 }{ 100 } ) }^{ n }\  } >1600\ =\sum _{ n=1 }^{ n _0 }{ { (1+\cfrac { 5 }{ 100 } ) }^{ n }\  } >16$
Since it is a sum of a infinite GP. We can write it as a
$\cfrac  {(1.05^{n _0}-1)}{(1.05-1)}\times 1.05>16$
$\implies (1.05)^{n _0}>1.7619$
Taking log on both sides
$\implies \log{(1.05)^{n _0}}>\log 1.7619$
$\implies n _0>\cfrac{\log (1.7619)}{\log (1.05)}$
$\implies n _0>11.6$
$\implies n _0=12$
Hence, least no. of years$=12$.

Find the amount of the annuity of Rs 150 payable in half yearly instalments  for 15 years at 4% per annum interest also payable half yearly

business mathematics and statistics insurance and annuity amount of an annuity annuities financial mathematics

  1. Rs.$9903.45$

  2. Rs.$3003.45$

  3. Rs.$8103.45$

  4. Rs.$3103.45$

Show answer
Correct Option: D
Explanation:

$P=Rs.150$

Amount payable in half yearly so $,r=4\%/2=2\%,t=15\times 2=30$
$A=\cfrac{1}{2}\sum _{ n=1 }^{ 30 }{ P{ (1+\cfrac { r }{ 100 } ) }^{ n } } \ =\cfrac{1}{2}\sum _{ n=1 }^{ 30 }{ 150{ (1+\cfrac { 2 }{ 100 } ) }^{ n } } \ =75\times 41.38\ =Rs.3103.45$

A sum of Rs 2522 is borrowed from a money lender at 5% per annum compounded annually.If this amount is to be paid back in 3 equal instalments , find the annual instalments (approximately).

business mathematics and statistics insurance and annuity amount of an annuity annuities financial mathematics

  1. 925

  2. 800

  3. 875

  4. 567

Show answer
Correct Option: A
Explanation:

$P=\cfrac{A}{(1+\cfrac{R}{100})^n}$

$\implies 2522= \cfrac{A}{(1+\cfrac{5}{100})^1}$$+\cfrac{A}{(1+\cfrac{5}{100})^2}+$$\cfrac{A}{(1+\cfrac{5}{100})^3}$
$ \implies A=926\approx 925$

What is the amount of annuity(approximately) due of Rs 100 yearly payable half yearly for 15 years at 10 % compound interest per annum half yearly.

business mathematics and statistics insurance and annuity amount of an annuity annuities financial mathematics

  1. 3491

  2. 1456

  3. 5434

  4. 2341

Show answer
Correct Option: A
Explanation:

$r=\cfrac{10\%}{2}=5\%,n=2\times 15=30,A=Rs.100$

Amount of annuity due payable half yearly
$=(1/2)\cfrac{A}{r}\times (1+r)\times[(1+r)^n-1]$
$=(1/2)\cfrac{100}{0.05}\times (1+0.05)\times[(1+0.05)^{30}-1]\=Rs.3488\approx  Rs.3491$

Which of the following comes under Annuity due?

business mathematics and statistics insurance and annuity amount of an annuity annuities financial mathematics

  1. Life insurance Premium

  2. Recurring Deposit Payments

  3. Advance Payment of monthly house rent

  4. All of the above

Show answer
Correct Option: D
Explanation:

An annuity is a contract aimed at generating steady income during retirement, where in lump sum payment is made by an individual to obtain certain amounts immediately or at some point of future
all of above comes under annuity.
It includes Life insurance Premium, Recurring Deposit Payments, Advance Payment of monthly house rent.