1. What does Rs. 250 amounts to in 2 years with compound interest at the rate of 4% in the 1^{st} year and 8% in the second year ?

a) Rs. 280

b) Rs. 280.80

c) Rs. 468

d) Rs. 290.80

Explanation:

$$\eqalign{ & {\text{Principal = Rs 250}} \cr & {{\text{R}}_1} = 4\% ,\,\,\,\,\,\,\,\,\,{{\text{R}}_2} = 8\% \cr & {\text{Amount}}\,{\text{after}}{1^{st}}\,{\text{year}} \cr & = 250\left( {1 + \frac{4}{{100}}} \right) = {\text{Rs}}{\text{. }}260 \cr & {\text{Amount after }}{{\text{2}}^{{\text{nd}}}}{\text{ year}} \cr & {\text{ = }}260\left( {1 + \frac{8}{{100}}} \right) \cr & = {\text{Rs}}{\text{. }}280.80 \cr} $$

2. The compound interest on a certain sum of money for 2 years at 5% is Rs. 328, then the sum is =

a) Rs. 3000

b) Rs. 3600

c) Rs. 3200

d) Rs. 3400

Explanation: Go check with option one by one (Go with option (C) and check it.)

Principal is Rs. 3200

3200 of 5% for 1

^{st}year = 160

then, principal = 3200 + 160 = 3360

3360 of 5% for 2

^{nd}year = 168

Interest = 160 + 168 = 328

3. The compound interest on a certain sum of money for 2 years at 5% per annum is Rs 410. The simple interest on the same sum at the same rate and for the same time is =

a) Rs. 400

b) Rs. 300

c) Rs. 350

d) Rs. 405

Explanation:

$$\eqalign{ & {\text{Rate of interest 5}}\% \cr & = \frac{1}{{20}} \cr & {\text{Let principal}} \cr & {\text{ = }}{\left( {20} \right)^2}{\text{ = 400 units}} \cr & \Rightarrow {\text{ Total compound interest }} \cr & {\text{41 Units }} \to {\text{Rs. 410 }} \cr & {\text{1 Units }} \to {\text{Rs. 10 }} \cr & {\text{400 Units }} \to {\text{Rs. 400 }} \cr & {\text{Total simple interest}} \cr & {\text{ = Rs. 400}} \cr} $$

4. A sum of money lent out at compound interest increases in value by 50% in 5 years. A person wants to lend three different sums x, y and z for 10, 15 and 20 years respectively at the above rate in such a way that he gets back equal sums at the end of their respective periods. The ratio x : y : z is =

a) 6 : 9 : 4

b) 9 : 4 : 6

c) 9 : 6 : 4

d) 6 : 4 : 9

Explanation:

$$\eqalign{ & P{\left( {1 + \frac{R}{{100}}} \right)^5} = 150\% \,{\text{of }}P = \frac{3}{2}P \cr & \Rightarrow {\left( {1 + \frac{R}{{100}}} \right)^5} = \frac{3}{2} \cr} $$

$$x{\left( {1 + \frac{R}{{100}}} \right)^{10}} = y{\left( {1 + \frac{R}{{100}}} \right)^{15}} = $$ $$z{\left( {1 + \frac{R}{{100}}} \right)^{20}}$$

$$ \Rightarrow x{\left\{ {{{\left( {1 + \frac{R}{{100}}} \right)}^5}} \right\}^2} = $$ $$y{\left\{ {{{\left( {1 + \frac{R}{{100}}} \right)}^5}} \right\}^3} = $$ $$z{\left\{ {{{\left( {1 + \frac{R}{{100}}} \right)}^5}} \right\}^4}$$

$$\eqalign{ & \Rightarrow x \times {\left( {\frac{3}{2}} \right)^2} = y \times {\left( {\frac{3}{2}} \right)^3} = z \times {\left( {\frac{3}{2}} \right)^4} \cr & \Rightarrow \frac{{9x}}{4} = \frac{{27y}}{8} = \frac{{81z}}{{16}} = k({\text{say}}) \cr & \Rightarrow x = \frac{{4k}}{9},y = \frac{{8k}}{{27}},z = \frac{{16k}}{{81}} \cr & x:y:z = \frac{{4k}}{9}:\frac{{8k}}{{27}}:\frac{{16k}}{{81}} \cr & x:y:z = 36:24:16 \cr & x:y:z = 9:6:4 \cr} $$

5. Under the Rural Housing Scheme, the Delhi Development Authority (DDA) allotted a house to Kamal Raj for Rs. 126100. This payment is to be made in three equal annual instalments. If the money is reckoned at 5% per annum compound interest, then how much is to be paid by Kamal Raj in each instalment ?

a) Rs. 45205

b) Rs. 46305

c) Rs. 47405

d) Rs. 48505

Explanation: Let the value of each instalment be Rs. x

Then, (P.W. of Rs. x due 1 year hence) + (P.W. of Rs. x due 2 year hence) + (P.W. of Rs. x due 3 year hence) = 126100

$$\eqalign{ & \frac{x}{{\left( {1 + \frac{5}{{100}}} \right)}} + \frac{x}{{{{\left( {1 + \frac{5}{{100}}} \right)}^2}}} + \frac{x}{{{{\left( {1 + \frac{5}{{100}}} \right)}^3}}} = 126100 \cr & \frac{{20x}}{{21}} + \frac{{400x}}{{441}} + \frac{{8000x}}{{9261}} = 126100 \cr & \frac{{8820x + 8400x + 8000x}}{{9261}} = 126100 \cr & \frac{{25220x}}{{9261}} = 126100 \cr & x = \left( {\frac{{126100 \times 9261}}{{25220}}} \right) \cr & x = 46305 \cr} $$

6. A sum of Rs 210 was taken as a loan. This is to be paid back in two equal installments. If the rate of interest be 10% compounded annually, then the value of each installment is = ?

a) Rs. 127

b) Rs. 121

c) Rs. 210

d) Rs. 225

Explanation:

$$\eqalign{ & {\text{Rate of interest}} \Rightarrow {\text{ 10% = }}\frac{1}{{10}} \cr & {\text{Each installment of 2 years}} \cr & \frac{{10}}{{11}} \times \frac{{\left( {10 + 11} \right)}}{{11}} \times {\text{ Installment = P}}{\text{.A}} \cr & \frac{{10}}{{11}} \times \frac{{\left( {10 + 11} \right)}}{{11}} \times {\text{ Installment = 210}} \cr & {\text{Installment = 121}} \cr} $$

7. A certain sum will amount to Rs 12100 in 2 years at 10% per annum of compound interest, interest being compounded annually. The sum is = ?

a) Rs. 12000

b) Rs. 6000

c) Rs. 8000

d) Rs. 10000

Explanation: Amount = 12,100; r = 10%, t = 2 yrs

$$\eqalign{ & {\text{Amount}} = P{\left[ {1 + \frac{r}{{100}}} \right]^t} \cr & 12100 = P{\left[ {1 + \frac{{10}}{{100}}} \right]^2} \cr & 12100 = P{\left[ {\frac{{11}}{{10}}} \right]^2} \cr & 12100 = P \times \frac{{11}}{{10}} \times \frac{{11}}{{10}} \cr & P = \frac{{12100 \times 10 \times 10}}{{11 \times 11}} \cr & P = 10000 \cr} $$

8. Find the rate percent per annum if Rs. 2000 amounts to Rs. 2315.25 in one and half years interest being compounded half yearly.

a) 10 %

b) 11.5 %

c) 5 %

d) 20 %

Explanation:

$$\eqalign{ & {\text{compounded half yearly}} \cr & {\text{Rate = }}\frac{{\text{R}}}{2} \cr & {\text{Time = }}\frac{{{\text{2T}}}}{3} \cr & {\text{Amount = P}}{\left( {1 + \frac{{\text{R}}}{{2 \times 100}}} \right)^3} \cr & 2315.25 = 2000{\left( {1 + \frac{{\text{R}}}{{2 \times 100}}} \right)^3} \cr & \frac{{2315.25}}{{2000}} = {\left( {1 + \frac{{\text{R}}}{{2 \times 100}}} \right)^3} \cr & \frac{{231525}}{{200000}} = {\left( {1 + \frac{{\text{R}}}{{2 \times 100}}} \right)^3} \cr & \frac{{9261}}{{8000}} = {\left( {1 + \frac{{\text{R}}}{{2 \times 100}}} \right)^3} \cr & {\left( {\frac{{21}}{{20}}} \right)^3} = {\left( {1 + \frac{{\text{R}}}{{2 \times 100}}} \right)^3} \cr & 1 + \frac{{\text{R}}}{{200}} = \frac{{21}}{{20}} \cr & {\text{R = 10}}\% \cr} $$

9. One can purchase a flat from a house building society for Rs. 55000 cash or on the terms that he should pay Rs. 4275 as cash down payment and get the rest in three equal installments. The society charges interest at the rate of 16% per annum compounded half-yearly. If the flat is purchased under installment plan, find the value of each installment ?

a) Rs. 18756

b) Rs. 19292

c) Rs. 19683

d) Rs. 20285

Explanation: Total cost of the flat = Rs. 55000

Down payment = Rs. 4275

Balance = Rs. (55000 - 4275) = Rs. 50725

Rate of interest = 8% per half year

Let the value of each instalment be Rs. x

P.W. of Rs. x due 6 months hence + P.W. of Rs. x due 1 year hence + P.W. of Rs. x due $$1\frac{1}{2}$$ years hence = 50725

$$ \frac{x}{{\left( {1 + \frac{8}{{100}}} \right)}} + $$ $$\frac{x}{{{{\left( {1 + \frac{8}{{100}}} \right)}^2}}} + $$ $$\frac{x}{{{{\left( {1 + \frac{8}{{100}}} \right)}^3}}} = $$ $$50725$$

$$\eqalign{ & \frac{{25x}}{{27}} + \frac{{625x}}{{729}} + \frac{{15625x}}{{19683}} = 50725 \cr & \frac{{50725x}}{{19683}} = 50725 \cr & x = \left( {\frac{{50725 \times 19683}}{{50725}}} \right) = 19683 \cr} $$

10. The sum of money which when given on compound interest at 18% per annum would fetch Rs 960 more when the interest is payable half-yearly then when it was payable annually for 2 years is =

a) Rs. 60000

b) Rs. 30000

c) Rs. 40000

d) Rs. 50000

Explanation: Rate of interest = 18%

Time = 2 year

When the interest is payable half yearly

Then, rate of interest = 9%

Time = 4 half - years

Let the principal be Rs. x

$$\eqalign{ & {\text{C}}{\text{.I}}{\text{. = }}x\left[ {{{\left( {1 + \frac{R}{{100}}} \right)}^T} - 1} \right]{\text{ }} \cr & = x\left[ {{{\left( {1 + \frac{9}{{100}}} \right)}^4} - 1} \right] \cr & = x\left[ {{{\left( {\frac{{109}}{{100}}} \right)}^4} - 1} \right] \cr & = x\left[ {1.4116 - 1} \right] \cr & = Rs.\,0.4116x \cr & {\text{According to question}} \cr & = x\left[ {{{\left( {1 + \frac{{18}}{{100}}} \right)}^2} - 1} \right] \cr & = x\left[ {{{\left( {\frac{{118}}{{100}}} \right)}^2} - 1} \right] \cr & = x\left[ {{{\left( {1.18} \right)}^2} - 1} \right] \cr & = x\left[ {1.3924 - 1} \right] \cr & = Rs.\,0.3924x \cr & {\text{According to question,}} \cr & 0.4116x - 0.3924x = 960 \cr & x = \frac{{960}}{{0.0192}} \cr & x = \frac{{960 \times 10000}}{{192}} \cr & x = 50000 \cr} $$