Tag: business mathematics and statistics

Questions Related to business mathematics and statistics

A company borrows Rs 10000 on condition to repay it with compound interest at $5$% p.a . by annual instalments at Rs 1000 each. In how many years will the debt be paid off?

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

  1. 14.2

  2. 21.7

  3. 12.67

  4. None of these

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Correct Option: A
Explanation:
$\Rightarrow$  Company borrow Rs.10000 i.e. $pv=Rs.10000$ and $I=5\%=0.05$. $A$ is also given which is $Rs.1000$
$\Rightarrow$  Present value of annuity regular
$\Rightarrow$  $pv=A\times [\dfrac{(1+I)^n-1}{I\times (1+I)^n}]$

$\Rightarrow$  $10000=1000\times [\dfrac{(1+0.05)^n-1}{0.05\times (1+0.05)^n}]$

$\Rightarrow$  $(1.05)^n-0.5\times (0.5)^n=1$
$\Rightarrow$  $(1.05)^n=2$
Taking log both sides
$\Rightarrow$  $n=\dfrac{log\,2}{log\, 1.05}$
$\therefore$   $n=14.2\, years$

Mr Dev purchased a car paying Rs $90,000$ and promising to pay Rs 5000 every 3 months for the next 10 years. The interest is $6$% p.a. compounded quarterly. If at the end of 5th year , he wants to finish his liability by a single payment , how much should he pay?

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

  1. 90100

  2. 80100

  3. 34504

  4. 54345

Show answer
Correct Option: A
Explanation:

$\Rightarrow$  We have $A=Rs.5000,\,I=\dfrac{6}{100}\times \dfrac{1}{4}=0.015$ and $n = 20$

$\Rightarrow$  If at the end of 5th year, i.e., at the time of 20th payment, he wants to finish off the liability, then lump sum payment required is,
$\Rightarrow$  $5000$ + Present value of the remaining 20 installments.
$\Rightarrow$  $5000+V$
$\Rightarrow$  $5000$ + $\dfrac{A}{I}[1-(1+I)^{-n}]$

$\Rightarrow$  $5000+\dfrac{5000}{0.015}[1-(0.015)^{-20}]$    ---- ( 1 )

$\Rightarrow$  Let $x=(1.015)^{-20}$
$\Rightarrow$  $log\,x=-20\,log\,(1.015)$
$\Rightarrow$  $log\,x=-20(0.0064)=-0.128=\bar{1}.8720$
$\Rightarrow$  $x=antilog\,(\bar{1}.8720)=0.7447$
Substitute value of $x$ in ( 1 ),
$\Rightarrow$  $5000+\dfrac{5000}{0.015}(1-0.7447)$

$\Rightarrow$  $5000+\dfrac{5000}{0.015}\times 0.2553$

$\Rightarrow$  $Rs.90100$

Amit buys a house for Rs 500000. The contract is that amit will pay Rs 200000 immediately and the balance in 15 equal instalments with 15 % p.a compound interest . How much has he to pay annually (approximately)?

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

  1. Rs$51,305$

  2. Rs$54,005$

  3. Rs$51,843$

  4. Rs$91,305$

Show answer
Correct Option: A
Explanation:

Present value $=Rs.50,000Rs.20,000=Rs.30,000$

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

$\implies 30,000=\cfrac{A}{1+(\cfrac{15}{100})}$$+\cfrac{A}{1+(\cfrac{15}{100})^2}+.....$4
$ \implies A\times 5,847 $

$\implies A=Rs51,305$

Z invests Rs. $10,000$ every year starting for today for next $10$ years. Suppose interest rate is $8\%$ per annum compounded annually. Calculate future value of the annuity. Given that $(1+0.08)^{10}=2.15892500$.

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

  1. $1,44,865.625$

  2. $1,56,454.875$

  3. $1,54,654.875$

  4. $1,44,568.625$

Show answer
Correct Option: B
Explanation:

Step-$1$: Calculate future value as though it is an ordinary annuity
Future value of the annuity as if it is an ordinary annuity
$=10,000\left[\displaystyle\frac{(1+0.08)^{10}-1}{0.08}\right]$
$=10,000\times 14.4865625$
$=Rs. 1,44,865.625$
Step-$2$: Multiply the result by $(1+i)$
$=1,44,865.625\times (1+0.08)$
$=1,56454.875$.

A machine costs Rs. $98,000$ and its effective life is estimated at $12$ years. If the scrap value is Rs. $3, 000$, what should be cut out of the profit at the end of each year to accumulate at compound rate of $5\%$ per annum so that a new machine can be purchased after $12$ years ?

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

  1. Rs. $6,000$

  2. Rs. $5,968$

  3. Rs. $4,787$

  4. Rs. $4,763$

Show answer
Correct Option: B
Explanation:
Effective cost of the machine is $98000–3000 = 95000$.
We know that Future value of annuity (FV) $=$ annuity $\times$ Compount Value factor of Annuity (CVAF)
That is $\text{FV}= \text{annuity} \times \text{CVAF} _{(5\%, 12)}$
$\text{FV}= \text{annuity} \times \left(\dfrac{(1+r)^n-1}{r}\right)$, where $r=0.05, n=12$
$\Rightarrow 95000 = \text{annuity} \times \dfrac{(1+0.05)^{12}-1}{0.05} $
$\Rightarrow 95000 = \text{annuity} \times \dfrac{0.795856}{0.05} $
$\Rightarrow 95000=\text{annuity} \times 15.917$
$ \therefore \text{annuity} = \dfrac{95000}{15.917} =5968$
Therefore, Rs. $5,968$ should be cut out of the profit at the end of each year to accumulate at compound rate of $5\%$ per annum, so that a new machine can be purchased after $12$ years.