Tag: amount of an annuity

Questions Related to amount of an annuity

A house is sold for $ Rs \ 30,000$ cash or $ Rs\ 17, 500$ cash down payment and instalments of $ Rs \ 1, 600$ per month for eight months. Determine the approximate rate of interest for instalment.

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

  1. $6.5 \%$

  2. $6 .8 \%$

  3. $ 6. 2 \%$

  4. None of these

  5. $6.3 \%$

Show answer
Correct Option: A
Explanation:

$\Rightarrow$  Cash price = $Rs.30,000$

$\Rightarrow$  Cash down payment = $Rs. 17500$
$\Rightarrow$  Total amount paid in 8 monthly installments = $Rs.(1600\times 8)=Rs.12800$
$\Rightarrow$  Total amount paid under installment paln = $Rs.17500+Rs.12800=Rs.30300$
$\Rightarrow$  Interest charged = $Rs.30300-Rs.30000=Rs.300$
$\Rightarrow$  Principal for 1st month = $Rs.30000-Rs.17500=Rs.12500$
$\Rightarrow$  Principal for 2nd month = $Rs.12500-Rs.1600=Rs.10900$
$\Rightarrow$  Principal for 3rd month = $Rs.10900-Rs.1600=Rs.9300$
$\Rightarrow$  Principal for 4th month = $Rs.9300-Rs.1600=Rs.7700$
$\Rightarrow$  Principal for 5th month = $Rs.7700-Rs.1600=Rs.6100$
$\Rightarrow$  Principal for 6th month = $Rs.6100-Rs.1600=Rs.4500$
$\Rightarrow$  Principal for 7th month = $Rs.4500-Rs.1600=Rs.2900$
$\Rightarrow$  Principal for 8th month = $Rs.2900-Rs.1600=Rs.1300$
$\Rightarrow$  Total principal = $Rs.55200$
$\Rightarrow$  The last installment of Rs.1600 includes Rs.1300 plus Rs.300 interest.
$\Rightarrow$  Time = 1 month = $\dfrac{1}{12}$year, Interest = Rs.$300$
$\Rightarrow$  Interest = $\dfrac{P\times T\times R}{100}$
$\Rightarrow$  $R=\dfrac{I\times 100}{P\times T}$
$\Rightarrow$  $R=\dfrac{300\times 100}{55200\times \dfrac{1}{12}}=\dfrac{300\times 100\times 12}{55200}=\dfrac{150}{23}=6.5$
$\therefore$   $Rate =6.5\%$

Find the present value of an ordinary annuity of $8$ quarterly payments of Rs. $500$ each, the rate of interest being $8\%$ p.a. compounded quarterly.

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

  1. Rs. $3660.20$

  2. Rs. $3662.50$

  3. Rs. $4275$

  4. Rs. $3660$

Show answer
Correct Option: B
Explanation:

Given, $A=$ Rs $500$, $n= 8$

Also, $ r= \dfrac{8}{100} \times \dfrac{1}{4} =0.02 $

$ \therefore V= \dfrac{A}{r} \times \left[1-(1+r)^{(-n)}\right]=\dfrac{500}{0.02} \times \left[1-(1.02)^{(-8)}\right] $

Now, let $ x= (1.02)^{(-8)} $

$\Rightarrow \log{x} = -8\log{1.02}=-8(0.0086) $

$ \Rightarrow \log{x}= -0.0688 $

$ \Rightarrow x= 0.8535 $

$ \Rightarrow  V=\dfrac{500}{0.02} \times [1-0.8535] =$ Rs. $3662.50 $

Thus, the present value of annuity is Rs. $3662.50$.

A man borrowed some money and returned it in $3$ equal quarterly installments of Rs. $4630.50$ each. What sum did he borrow if the rate of interest was $20\%$ p.a. compounded quarterly?

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

  1. Rs. $12000$

  2. Rs. $12100$

  3. Rs. $12160$

  4. Rs. $13000$

Show answer
Correct Option: C
Explanation:

Here, we have to find present value $(V)$ of an ordinary annuity certain.

Given, $A=$ Rs. $4630.50$, $n= 3$

Also $ r= \dfrac{20}{100} \times \dfrac{1}{4}=0.05 $

$ \therefore V=\dfrac{A}{r} \times [1-(1+r)^{(-n)}] $

$=\dfrac{4630.50}{0.05} \times [1-(1.05)^{(-3)}]$

$=$ Rs. $12610 $

Thus, the sum borrowed was Rs. $12160$.

Find the Amount of an ordinary annuity of $8$ quarterly payments of Rs. $500$ each, the rate of interest being $8\%$ p.a. compounded quarterly.

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

  1. Rs. $3660.20$

  2. Rs. $3662.50$

  3. Rs. $4275$

  4. Rs. $3670$

Show answer
Correct Option: C
Explanation:

Here, $A=$ Rs. $500$, $n= 8$

Also $ r= \dfrac{8}{100} \times \dfrac{1}{4} =0.02 $

$ M=\dfrac{A}{r} \times [(1+r)^{(n)}-1]=\dfrac{500}{0.02} \times [(1.02)^{8}-1] $

Let $ x= (1.02)^(8) $

$\Rightarrow \log{x}=8\log{1.02}=0.0688 $

$ \Rightarrow x= 1.171 $

$ \Rightarrow  M= \dfrac{500}{0.02} \times [1.171-1] =$ Rs $4275 $

Thus, the amount is Rs. $4275$.

The present value of an amount ____ its future value.

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

  1. Greater than

  2. Less than

  3. Equal to

  4. Not equal to

Show answer
Correct Option: B

A man borrows Rs $37500$ and agrees to repay in semi-annual installments of Rs $2250$ each, the first due in $6$ months. How many payments must he make if rate of interest is $6\%$ compounded semi-annually?

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

  1. $23$

  2. $24$

  3. $25$

  4. $22$

Show answer
Correct Option: B
Explanation:

$\Rightarrow$  Here, we have to find the number of payments, $n$.

$\Rightarrow$  $V=Rs.37500$  and $A=Rs.2250$
$\Rightarrow$  Rate of interest compounded semi-annually = $\dfrac{1}{2}\times 6\% = \dfrac{1}{2}\times \dfrac{6}{100}=0.03$
$\Rightarrow$  $V=\dfrac{A}{r}\times [1-(1+r)^{-n}]$

$\Rightarrow$  $37500=\dfrac{2250}{0.03}\times [1-(1.03)^{(-n)}]$

$\Rightarrow$  $1-(1.03)^{-n}=\dfrac{37500\times 0.03}{2250}$

$\Rightarrow$  $(1.03)^{(-n)}=0.5$

$\Rightarrow$  $-n\, log(1.03)=log(0.5)$

$\Rightarrow$  $-n(0.0128)=-0.3010$

$\Rightarrow$  $n=\dfrac{-0.3010}{-0.0128}$

$\therefore$    $n=23.51 \approx 24$

Find the Present value of an annuity due of Rs $500$ per quarter for $8$ years and $9$ months at $6\%$ compounded quarterly.

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

  1. Rs $27032.30$

  2. Rs $23137.98$

  3. Rs $13740.86$

  4. Rs $24017.25$

Show answer
Correct Option: C
Explanation:

Here, rate of interest, r =$1.5$% per interest period =$0.015$
Number of interest periods, $n = 4 \times 8 +3 = 35$
Each installment, $A=Rs $ $500$
Present value of annuity due,
$v = \dfrac{A}{r} \times (1+r) \times [1-(1+r)^{-n}]$

= $\dfrac{500}{0.015} \times 1.015 \times [1-(1.015)^{-35}]$
= $13740.86$

A man borrowed some money and returned it in $3$ equal quarterly installments of Rs $4630.50$ each. Find the interest charged (in Rs) on the sum he borrowed, if the rate of interest was $20\%$ p.a. compounded quarterly?

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

  1. $1731.50$

  2. $1200$

  3. $1300$

  4. $1251.80$

Show answer
Correct Option: A
Explanation:

Here, we have to find present value $(V)$ of an ordinary annuity certain.

Given, $A=$ Rs $4630.50$, $n= 3$

Also $ r= \dfrac{20}{100} \times \dfrac{1}{4}=0.05 $

$ \therefore V=\dfrac{A}{r} \times [1-(1+r)^{(-n)}] =\dfrac{4630.50}{0.05} \times [1-(1.05)^{(-3)}]=$ Rs. $ \: 12610 $

Thus, the sum borrowed was Rs. $12160$

Now, total money repaid $ = 3 \times 4630.50 =$ Rs. $ \: 13891.50 $

Therefore, interest paid $ =$ Rs, $ \: 13891.50$ $-$  Rs. $ \: 12160 =$ Rs. $ \: 1731.50 $.

Find the amount of an annuity due of Rs $500$ per quarter for $8$ years and $9$ months at $6\%$ compounded quarterly.

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

  1. Rs $27452.30$

  2. Rs $23137.98$

  3. Rs $13740.86$

  4. Rs $24671.30$

Show answer
Correct Option: B
Explanation:

Number of interest periods, $n = 4 \times 8 +3 = 35$
Each installment, A=Rs $500$
Present value of annuity due,
$v = \frac{A}{r} \times (1+r) \times [1-(1+r)^{-n}]$
= $\frac{500}{0.015} \times 1.015 \times [1-(1.015)^{-35}]$
= $13740.86$
$v = \frac{A}{r} \times (1+r) \times [1-(1+r)^{-n}-1]$
= $\frac{500}{0.015} \times 1.015 \times [1-(1.015)^{-35}-1]$
= $23137.98$

Three equal instalments each of $Rs 200$ were paid at the end of the year for the sum borrowed at $20 \%$ interest compounded annually. Find the sum. 

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

  1. $ 600$

  2. $421.3$

  3. $ 400$

  4. $ 431.1$

Show answer
Correct Option: B
Explanation:
Each Installment = $\dfrac{P{\cdot}r}{100[1-{\{ \dfrac{100}{100+r} \}}^n]}$
Here Installement$ = 200$
Rate of interest $(r) = 20%$
Number of years$ (n) = 3$
Solving the above equation gives $P = 421.3$