Compound-Interest - 01

Explanation:-

Interest for 2 years at the rate of 15% $$\eqalign{ & {\text{ = 15 + 15 + }}\frac{{15 \times 15}}{{100}} \cr & = 32.25\% \cr & {\text{According to question,}} \cr & {\text{32}}{\text{.25}}\% {\text{ = 3225}} \cr & {\text{100}}\% {\text{ = }}\frac{{3225}}{{32.25}} \times 100 \cr & = 100 \times 100 = 10000 \cr}$$

Explanation:-

$$\eqalign{ & {\text{R}}\% {\text{ = }}\frac{{2662 - 2420}}{{2420}} \times 100 \cr & = \frac{{242}}{{2420}} \times 100 \cr & = 10\% \cr & {\text{2 years CI}}\% \cr & {\text{ = 10 + 10 + }}\frac{{10 \times 10}}{{100}} \cr & = 21\% \cr & {\text{So, 121}}\% {\text{ = 2420}} \cr & \Rightarrow {\text{100}}\% {\text{ = 2000}} \cr}$$

Explanation:-

$$\eqalign{ & {\text{Rate of interest = 5}}\% {\text{ p}}{\text{.c}}{\text{.p}}{\text{.a}}{\text{.}} \cr & {\text{If time 3 years than CI}} - {\text{SI}} \cr & {\text{ = }}P\left[ {{{\left( {\frac{R}{{100}}} \right)}^3} + 3{{\left( {\frac{R}{{100}}} \right)}^2}} \right] \cr & \Rightarrow 122 = P\left[ {{{\left( {\frac{5}{{100}}} \right)}^3} + 3{{\left( {\frac{5}{{100}}} \right)}^2}} \right] \cr & \Rightarrow 122 = P\left( {\frac{{125}}{{1000000}} + \frac{{75}}{{10000}}} \right) \cr & \Rightarrow 122 = P\left[ {\frac{{125 + 7500}}{{1000000}}} \right] \cr & \Rightarrow 122 = P\left[ {\frac{{7525}}{{1000000}}} \right] \cr & \Rightarrow P = \frac{{122 \times 1000000}}{{7625}} \cr & \Rightarrow {\text{Rs}}{\text{. 16000}} \cr}$$

Explanation:-

Let the total sum of money invested by Shashi be Rs. x
In scheme A money invested at simple interest for 6 years at a rate of 12% p.a. $$\therefore \frac{2}{3}{\text{of }}x \times \frac{{12 \times 6}}{{100}} = \frac{{48x}}{{100}}....(i)$$ In scheme B money at compound interest for 2 year at a rate of 10% p.a. $$\eqalign{ & \frac{x}{3}{\left( {1 + \frac{{10}}{{100}}} \right)^2} - \frac{x}{3} \cr & \Rightarrow \frac{x}{3}{\left( {1 + \frac{{10}}{{100}}} \right)^2} - \frac{x}{3} = \frac{{7x}}{{100}} \cr}$$ According to given information we get $$\eqalign{ & \Rightarrow \frac{{48x}}{{100}} + \frac{{7x}}{{100}} = 2750 \cr & \Rightarrow 55x = 2750 \times 100 \cr & \Rightarrow x = \frac{{2750 \times 100}}{{55}} \cr & \Rightarrow x = Rs.\,5000 \cr}$$

Explanation:-

Let the sum of money be rs. P
Then, $$\eqalign{ & \left[ {P{{\left( {1 + \frac{R}{{100}}} \right)}^t} - P} \right] = {\text{C}}{\text{.I}}{\text{.}} \cr & \left[ {P{{\left( {1 + \frac{{10}}{{100}}} \right)}^2} - P} \right] = 525 \cr & \Rightarrow P{\left( {\frac{{11}}{{10}}} \right)^2} - 1 = 525 \cr & \Rightarrow P\left( {\frac{{121}}{{100}} - 1} \right) = 525 \cr & \Rightarrow P\left( {\frac{{21}}{{100}}} \right) = 525 \cr & \Rightarrow P = \frac{{525 \times 100}}{{21}} \cr & \Rightarrow P = {\text{Rs}}{\text{.}}\,2500 \cr & \therefore {\text{ Sum of money}} \cr & {\text{ = Rs}}{\text{. 2500}} \cr}$$ Simple interest on the same sum Rs. 2500 for 4 (double the time) years at 5% (half the rate of percent per annum) is
So, $$\eqalign{ & {\text{S}}{\text{.I}}{\text{. = Rs}}{\text{.}}\left( {\frac{{2500 \times 5 \times 4}}{{100}}} \right) \cr & \,\,\,\,\,\,\,\,\,{\text{ = Rs}}{\text{. 500}} \cr}$$

Explanation:-

$$\eqalign{ & {\text{Rate of interest = 5}}\% {\text{ per annum}} \cr & {\text{Time = 2 year}} \cr & {\text{Accroding to question,}} \cr & P\left[ {{{\left( {1 + \frac{r}{{100}}} \right)}^n} - 1} \right] - \frac{{P \times r \times t}}{{100}}{\text{ = 63}} \cr & P\left[ {{{\left( {1 + \frac{5}{{100}}} \right)}^2} - 1} \right] - \frac{{P \times 5 \times 2}}{{100}}{\text{ = 63}} \cr & \Rightarrow P\left[ {{{\left( {1 + \frac{5}{{100}}} \right)}^2} - 1} \right] - \frac{{10P}}{{100}}{\text{ = 63}} \cr & \Rightarrow P\left[ {{{\left( {\frac{{105}}{{100}}} \right)}^2} - 1} \right] - \frac{{10P}}{{100}}{\text{ = 63}} \cr & \Rightarrow P\left( {\frac{{11025 - 10000}}{{10000}}} \right) - \frac{{10P}}{{100}} = 63 \cr & \Rightarrow \frac{{1025P}}{{10000}} - \frac{{10P}}{{100}} = 63 \cr & \Rightarrow \frac{{1025P - 1000P}}{{10000}} = 63 \cr & \Rightarrow 65P = Rs.630000 \cr & \Rightarrow P = \frac{{630000}}{{25}} \cr & \Rightarrow P = Rs25200 \cr & {\text{Hence}},\,{\text{sum Rs}}{\text{. 25200}} \cr}$$

Explanation:-

$$\eqalign{ & {\text{According to the question}} \cr & {\text{Principal = Rs S}} \cr & {\text{Rate }}\% {\text{ = 2r}}\,\% {\text{ p}}{\text{.a}}{\text{.}} \cr & {\text{Time = 3 years}} \cr & \therefore {\text{A = P}}{\left( {1 + \frac{{\text{r}}}{{100}}} \right)^T} \cr & \Leftrightarrow {\text{A = S}}{\left( {1 + \frac{{{\text{2r}}}}{{100}}} \right)^3} \cr & \Leftrightarrow {\text{A = S}}{\left( {1 + \frac{{\text{r}}}{{50}}} \right)^3} \cr}$$

Explanation:-

$$\eqalign{ & {\text{According to the question}} \cr & {\text{Amount}} = {\text{ }}{\left( {1 + \frac{{\text{R}}}{{2 \times 100}}} \right)^{2 \times {\text{t}}}} \cr & \Rightarrow 68921 = 64000{\left( {1 + \frac{5}{{2 \times 100}}} \right)^{2 \times {\text{t}}}} \cr & \Rightarrow \frac{{68921}}{{64000}} = {\left( {1 + \frac{1}{{40}}} \right)^{2 \times {\text{t}}}} \cr & \Rightarrow {\left( {\frac{{41}}{{40}}} \right)^3} = {\left( {\frac{{41}}{{40}}} \right)^{2 \times {\text{t}}}} \cr & \Rightarrow 2{\text{t = 3}} \cr & \Rightarrow {\text{t = }}\frac{3}{2} \cr & \Rightarrow {\text{t = 1}}\frac{1}{2}{\text{ years}} \cr}$$

Explanation:-

Total cost of the flat = Rs. 55000.
Down payment = Rs. 4275
Balance = Rs. (55000 - 4275) = Rs. 50725
Rate of interest = 85 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 1¹/₂ years hence = 50725 $$\eqalign{ & \Rightarrow \frac{x}{{\left( {1 + \frac{8}{{100}}} \right)}} + \frac{x}{{{{\left( {1 + \frac{8}{{100}}} \right)}^2}}} + \frac{x}{{{{\left( {1 + \frac{8}{{100}}} \right)}^3}}} = 50725 \cr & \Rightarrow \frac{{25x}}{{27}} + \frac{{625x}}{{729}} + \frac{{15625x}}{{19683}} = 50725 \cr & \Rightarrow \frac{{50725x}}{{19683}} = 50725 \cr & \Rightarrow x = \left( {\frac{{50725 \times 19683}}{{507250}}} \right) = 19683 \cr}$$

Explanation:-

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