Compound Interest - Study Mode

[#111] 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 = ?
Correct Answer

(B) Rs. 121

Explanation

Solution: $$eqalign{
& { ext{Rate of interest}} Rightarrow { ext{ 10% = }}frac{1}{{10}} cr
& { ext{Each installment of 2 years}} cr
& Rightarrow frac{{10}}{{11}} imes frac{{left( {10 + 11}
ight)}}{{11}} imes { ext{ Installment = P}}{ ext{.A}} cr
& Rightarrow frac{{10}}{{11}} imes frac{{left( {10 + 11}
ight)}}{{11}} imes { ext{ Installment = 210}} cr
& Rightarrow { ext{Installment = 121}} cr} $$

[#112] Find the rate percent per annum if Rs. 2000 amounts to Rs. 2315.25 in one and half years interest being compounded half yearly.
Correct Answer

(A) 10%

Explanation

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

[#113] 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 ?
Correct Answer

(C) Rs. 19683

Explanation

Solution: 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 $$1frac{1}{2}$$ years hence = 50725 $$ Rightarrow frac{x}{{left( {1 + frac{8}{{100}}}
ight)}} + $$ xa0xa0 $$frac{x}{{{{left( {1 + frac{8}{{100}}}
ight)}^2}}} + $$ xa0 $$frac{x}{{{{left( {1 + frac{8}{{100}}}
ight)}^3}}} = $$ xa0xa0 $$50725$$ $$eqalign{
& Rightarrow frac{{25x}}{{27}} + frac{{625x}}{{729}} + frac{{15625x}}{{19683}} = 50725 cr
& Rightarrow frac{{50725x}}{{19683}} = 50725 cr
& Rightarrow x = left( {frac{{50725 imes 19683}}{{50725}}}
ight) = 19683 cr} $$

[#114] 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 = ?
Correct Answer

(D) Rs. 50000

Explanation

Solution: 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 $$eqalign{
& { ext{C}}{ ext{.I}}{ ext{. = }}xleft[ {{{left( {1 + frac{R}{{100}}}
ight)}^T} - 1}
ight]{ ext{ }} cr
& = xleft[ {{{left( {1 + frac{9}{{100}}}
ight)}^4} - 1}
ight] cr
& = xleft[ {{{left( {frac{{109}}{{100}}}
ight)}^4} - 1}
ight] cr
& = xleft[ {1.4116 - 1}
ight] cr
& = Rs.,0.4116x cr
& { ext{According to question}} cr
& = xleft[ {{{left( {1 + frac{{18}}{{100}}}
ight)}^2} - 1}
ight] cr
& = xleft[ {{{left( {frac{{118}}{{100}}}
ight)}^2} - 1}
ight] cr
& = xleft[ {{{left( {1.18}
ight)}^2} - 1}
ight] cr
& = xleft[ {1.3924 - 1}
ight] cr
& = Rs.,0.3924x cr
& { ext{According to question,}} cr
& 0.4116x - 0.3924x = 960 cr
& Rightarrow x = frac{{960}}{{0.0192}} cr
& Rightarrow x = frac{{960 imes 10000}}{{192}} cr
& Rightarrow x = 50000 cr
& { ext{Hence, sum of money}} cr
& { ext{ = Rs. 50000}} cr} $$

[#115] In what time will Rs 64000 amounts to Rs 68921 at 5% per annum interest being compounded half yearly ?
Correct Answer

(A) $$1frac{1}{2}$$ years

Explanation

Solution: $$eqalign{
& { ext{According to the question,}} cr
& { ext{Amount}} = { ext{ }}{left( {1 + frac{{ ext{R}}}{{2 imes 100}}}
ight)^{2 imes { ext{t}}}} cr
& Rightarrow 68921 = 64000{left( {1 + frac{5}{{2 imes 100}}}
ight)^{2 imes { ext{t}}}} cr
& Rightarrow frac{{68921}}{{64000}} = {left( {1 + frac{1}{{40}}}
ight)^{2 imes { ext{t}}}} cr
& Rightarrow {left( {frac{{41}}{{40}}}
ight)^3} = {left( {frac{{41}}{{40}}}
ight)^{2 imes { ext{t}}}} cr
& Rightarrow 2{ ext{t = 3}} cr
& Rightarrow { ext{t = }}frac{3}{2} cr
& Rightarrow { ext{t = 1}}frac{1}{2}{ ext{ years}} cr} $$