Question

What annual installment will discharge a debt of $Rs.420$ due in $5$ years at $10\mathrm{\%}$ SI?
$A)Rs.60$
$B)Rs.70$
$C)Rs.800$
$D)Rs.900$

Answer 1

Hint: First, assume the annual installment to be $x$ and with the given number of years and rate of interest we can find the amount using a simple interest formula and the sum of the amount after each year gives the debt and solving the $x$ we get the required annual installment.

Complete step-by-step solution:
Since we assume the annual installment to be $x$ . we are given that the debt as $Rs.420$. The number of years is given as $5$
The rate of the given interest is $10\mathrm{\%}$
Hence, we know the simple interest formula that is ${\displaystyle \frac{pnr}{100}}$ where p is the principal amount, n is the time taken and r is the rate of interest.
Hence the amount is given by $p+{\displaystyle \frac{pnr}{100}}$
Hence the sum of the years from one to five can be calculated using the end of each year with the amount $Rs.420$ and annual installment to be $x$
Thus, we have the sum as $(x+{\displaystyle \frac{x\times 1\times 10}{100}})+(x+{\displaystyle \frac{x\times 2\times 10}{100}})+(x+{\displaystyle \frac{x\times 3\times 10}{100}})+(x+{\displaystyle \frac{x\times 4\times 10}{100}})+x=420$
At the end of fifth year we don’t have to pay any interest, that's why we have not added the interest for the fifth year. Further solving we have $(x+{\displaystyle \frac{x}{10}})+(x+{\displaystyle \frac{2x}{10}})+(x+{\displaystyle \frac{3x}{10}})+(x+{\displaystyle \frac{4x}{10}})+x=420$
$({\displaystyle \frac{11x}{10}})+({\displaystyle \frac{12x}{10}})+({\displaystyle \frac{13x}{10}})+({\displaystyle \frac{14x}{10}})+x=420$
Now cross multiplying we have ${\displaystyle \frac{11x+12x+13x+14x+10x}{10}}=420\Rightarrow {\displaystyle \frac{60x}{10}}=420$
Further solving we have $6x=420\Rightarrow x=70$ (by division operation)
Hence the annual installment is $Rs.70$
Therefore, the option $B)Rs.70$ is correct.

Note: Simple interest is the quickest and easy method of calculating the interest charge on a loan. Simple interest can be determined by multiplying the daily interest rate by the principal by the number of days that elapse between payments.
Annual installments mean a series of amounts to be paid annually over a predetermined period of years in the substantially equal periodic payments, except to the extent any increase in the amount reflects reasonable earnings though the date amount is paid.