1. A person borrowed a sum of Rs 6000 at 10% p.a., interest compounded annually. If the money is to be repaid in three equal annual installment, each payable at the end of the year, then what is the value of each installment?
a. Rs.2,000
b. Rs.2,413
c. Rs.2,314
d. Rs.2,662

Answer: B

Explanation:
Sum borrowed = Rs 6000
Interest = 10% at compounded annually and time = 3 years
Let x be the amount paid at the end of each of the 3 years.
$ \Rightarrow 6000\left( {1 + \displaystyle\frac{{10}}{{100}}} \right)^3$ = $x\left( {1 + \displaystyle\frac{{10}}{{100}}} \right)^2$ + $x\left( {1 + \displaystyle\frac{{10}}{{100}}} \right) + x$
$ \Rightarrow 6000(1.1)^3 = 1.21x + 1.1x + x$
$ \Rightarrow x = \displaystyle\frac{{6000(1.331)}}{{3.31}} = 2412.68$
So Installment amount is Rs.2413

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2. Three equal installment, each of Rs 200, were paid at the end of year on a sum borrowed at 20% compound interest compounded annually. Find the sum.
a. Rs. 600
b. Rs. 400
c. Rs. 421.30
d. Rs 432.10

Alternate Method:
we use discounting method. We discount the installments for the present value.
$P = \displaystyle\frac{{200}}{{\left( {1 + \displaystyle\frac{{20}}{{100}}} \right)^1 }}$ + $\displaystyle\frac{{200}}{{\left( {1 + \displaystyle\frac{{20}}{{100}}} \right)^2 }}$ + $\displaystyle\frac{{200}}{{\left( {1 + \displaystyle\frac{{20}}{{100}}} \right)^3 }}$
$P = \displaystyle\frac{{200}}{{1.2}} + \displaystyle\frac{{200}}{{1.44}} + \displaystyle\frac{{200}}{{1.718}}$
P = 421.3

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3. A man borrows a certain sum of money and pays it back in 2 years in two equal installments. If C.I. is reckoned at 5% per annum and he pays back annually Rs. 441, what sum did he borrow?
a. Rs. 820
b. Rs. 800
c. Rs. 882
d. Rs. 850

Answer: A

Explanation:
Money borrowed which becomes Rs. 441 after one year =$\displaystyle\frac{{441}}{{\left( {1 + \displaystyle\frac{5}{{100}}} \right)}}$ = \(441 \times \left( {\dfrac{{20}}{{21}}} \right) = 420\)

Money borrowed which become Rs. 441 after two years = $\displaystyle\frac{{441}}{{\left( {1 + \displaystyle\frac{5}{{100}}} \right)}}$ = \(441 \times {\left( {\dfrac{{20}}{{21}}} \right)^2} = 400\)
Total money borrowed = Rs. 820.

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4. A man borrows Rs. 2100 and undertakes to pay back with compound interest @ 10% p.a. in 2 equal yearly installments at the end of first and second year. What is the amount of each installment?
a. 1200
b. 1210
c. 1300
d. 1310

Answer: B

Explanation:
Let the installment be \(x\).
The value of first installment before 1 year = \(\dfrac{x}{{\left( {1 + \dfrac{{10}}{{100}}} \right)}} = \dfrac{{10x}}{{11}}\)
The value of second installment before 2 years = \(\dfrac{x}{{{{\left( {1 + \dfrac{{10}}{{100}}} \right)}^2}}} = \dfrac{{100x}}{{121}}\)
The sum of the above values should equal to principal.
\( \Rightarrow \dfrac{{10x}}{{11}} + \dfrac{{100x}}{{121}} = 2100\)
\( \Rightarrow \dfrac{{110x + 100x}}{{121}} = 2100\)
\( \Rightarrow \dfrac{{210x}}{{121}} = 2100\)
\( \Rightarrow x = 1210\)

Alternative method:
The value of the principal after 2 years = The value of 1st installment after 1 year + the value of 2nd installment.
\( \Rightarrow \) \(2100{\left( {1 + \dfrac{{10}}{{100}}} \right)^2}\) = \(x\left( {1 + \dfrac{{10}}{{100}}} \right) + x\)
\(2100\left( {\dfrac{{121}}{{100}}} \right) = x\left( {\dfrac{{11}}{{10}}} \right) + x\)
\(2100\left( {\dfrac{{121}}{{100}}} \right) = \dfrac{{21x}}{{10}}\)
\( \Rightarrow x = 1210\)

Alternative method:
Here, (1 + r) = 1 + $\displaystyle\frac{1}{{10}} = \displaystyle\frac{{11}}{{10}}$
Ratio of principals of two installments = 1 : $\displaystyle\frac{{10}}{{11}}$ = 11 : 10
Sum of ratios = 11 + 10 = 21
Therefore, Principal of first instalment = 2100 × $\displaystyle\frac{{11}}{{21}}$ = Rs. 1100
Therefore, Installment = Principal of first instalment × (1 + r)
= 1100 × $\displaystyle\frac{{11}}{{10}}$ = Rs. 1210

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5. A man borrows Rs. 820 and undertakes to pay back with compound interest @ 5% p.a. in 2 equal yearly installments at the end of first and second year. What is the amount of each installment?
a. 400
b. 420
c. 441
d. 410

Answer: C

Explanation:
The value of 820 after two years = The value of the installment after 1 year + The value of second installment
\( \Rightarrow 820{\left( {1 + \dfrac{5}{{100}}} \right)^2}\) = \(x\left( {1 + \dfrac{5}{{100}}} \right)\) + \(x\)
\( \Rightarrow 820 \times \dfrac{{441}}{{400}} = \dfrac{{21}}{{20}}x + x\)
\( \Rightarrow 820 \times \left( {\dfrac{{441}}{{400}}} \right) = \dfrac{{41}}{{20}}x\)
\( \Rightarrow x = 441\)

Alternative Method:
Here, (1 + r) = 1 + $\displaystyle\frac{1}{{20}} = \displaystyle\frac{{21}}{{20}}$
Ratio of principals of two instalments = 1 : $\displaystyle\frac{{20}}{{21}}$ = 21 : 20
Sum of ratios = 21 + 20 = 41
Therefore, Principal of first installment = $\displaystyle\frac{{21}}{{41}}$ × 820 = Rs. 420
Therefore, Installment = Principal of first installment × (1 + r) = 420 × $\displaystyle\frac{{21}}{{20}}$ = Rs. 441

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6. A man borrows Rs. 1820 and undertakes to pay back with compound interest @ 20% p.a. in 3 equal yearly installments at the end of first, second and third years. What is the amount of each installment?
a. 864
b. 850
c. 820
d. 900

Answer: A

Explanation:
The value of 1820 after three years = The value of the first installment after 2 years + The value of second installment after 1 year + the value of the third installment
\( \Rightarrow 1820{\left( {1 + \dfrac{{20}}{{100}}} \right)^3}\) = \(x{\left( {1 + \dfrac{{20}}{{100}}} \right)^2}\) + \(x\left( {1 + \dfrac{{20}}{{100}}} \right) + x\)
\( \Rightarrow 1820\left( {\dfrac{{216}}{{125}}} \right)\) = \(x\left( {\dfrac{{36}}{{25}}} \right) + x\left( {\dfrac{6}{5}} \right) + x\)
\( \Rightarrow 1820\left( {\dfrac{{216}}{{125}}} \right)\) = \(x\left( {\dfrac{{36 + 30 + 25}}{{25}}} \right)\)
\( \Rightarrow 1820\left( {\dfrac{{216}}{{125}}} \right) = x\left( {\dfrac{{91}}{{25}}} \right)\)
\( \Rightarrow x = 864\)

Alternative Method:
Here, (1 + r) = 1 + $\displaystyle\frac{1}{5} = \displaystyle\frac{6}{5}$
Ratio of principals for three years = 1 : $\displaystyle\frac{5}{6}:\left( {\displaystyle\frac{5}{6}} \right)^2 $
= $6^2 $ : 6 × 5 : $5^2 $ (On multiplying each ratio by $6^2 $)
= 36 : 30 : 25
Sum of the ratios = 36 + 30 + 25 = 91
Therefore, Principal of first installment = $\displaystyle\frac{{36}}{{91}}$ × 1820 = Rs. 720
Therefore, Installment = Principal of first installment × (1 + r) = 720 × $\displaystyle\frac{6}{5}$ = Rs. 864

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7. A sum of Rs.550 was taken a loan. This is to be repaid in two equal annual installments. If the rate of interest be 20% compounded annually, then the value of each installment is :
a. Rs.421
b. Rs.396
c. Rs.360
d. Rs.350

Answer: C

Explanation:
Let the value of each instalment be Rs.x. Then,
$\displaystyle\frac{x}{{\left( {1 + \displaystyle\frac{{20}}{{100}}} \right)}} + \displaystyle\frac{x}{{{{\left( {1 + \displaystyle\frac{{20}}{{100}}} \right)}^2}}} = 550$
or $\displaystyle\frac{{5x}}{6} + \displaystyle\frac{{25x}}{{36}} = 550$ or x = 360

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8. A loan was repaid in two annual instalments of Rs.112 each. If the rate of interest be 10% per annum compounded annually, the sum borrowed was :
a. Rs.200
b. Rs.210
c. Rs.217.80
d. Rs.280

Answer: B

Explanation:
Principal = (Present value of Rs.121 due 1 year hence ) + (Present value of Rs.121 due 2 years hence )
= Rs. $\displaystyle\frac{{121}}{{\left( {1 + \displaystyle\frac{{10}}{{100}}} \right)}} + \displaystyle\frac{{121}}{{{{\left( {1 + \displaystyle\frac{{10}}{{100}}} \right)}^2}}}$ = Rs.210