# Simple and Compound Interest - Excercise 1

## Questions

1.   A sum of money lent at compound interest for 2 years at 20% per annum would fetch Rs.482 more, if the interest was payable half yearly than if it was payable annually . The sum is

• 10000
• 20000
• 40000
• 50000

Sum = Rs x
C.I. when compounded half yearly = [$x{(1+\frac{10}{100})^4}$] - $x$= ${\frac{4641}{10000}}x$
C.I. when compounded annually = [$x{(1+\frac{10}{100})^2}$] - $x$ = ${\frac{11}{25}}x$
Difference => ${\frac{4641}{10000}}x$ - ${\frac{11}{25}}x$ = 482
$x$ = 20000

2.   The population of a town was 3600 three years back. It is 4800 right now. What will be the population three years down the line, if the rate of growth of population has been constant over the years and has been compounding annually?

• Rs.6000
• Rs.6400
• Rs.6500
• Rs.6600

The population grew from 3600 to 4800 in 3 years.
That is a growth of 1200 on 3600 during three year span.
Therefore, the rate of growth for three years has been constant.
The rate of growth during the next three years will also be the same.
Therefore, the population will grow from 4800 by = ${4800\times\frac{1}{3}}$ => 1600
Hence, the population three years from now will be = 4800 + 1600 = 6400

3.   A sum of money amounts to Rs.6690 after 3 years and to Rs.10,035 after 6 years on compound interest. Find the sum ?

• 4360
• 4460
• 4560
• 4660
P(${1+\frac{R}{100})^3}$ = 6690--(i) and P(${1+\frac{R}{100})^6}$ = 10035--(ii)
we get (${1+\frac{R}{100})^3}$ = ${\frac{10025}{6690}}$ => ${\frac{3}{2}}$
Substituting this value in (i), we get: ${P\times\frac{3}{2}}$ = 6690
or P = ${6690\times\frac{2}{3}}$ = 4460
4.   A chartered bank offers a five-year Escalator Guaranteed Investment Certificate. In successive years it pays annual interest rates of 4%, 4.5%, 5%, 5.5%, and 6%, respectively, compounded at the end of each year. The bank also offers regular five-year GICs paying a fixed rate of 5% compounded annually. Calculate and compare the maturity values of $1000 invested in each type of GIC • 1276.28 • 1234 • 1278 • 1256 The Correct answer is :1276.28 FV = 1000(1.04)(1.045)(1.05)(1.055)(1.06) = 1276.14 The maturity value of the regular GIC is FV = 1000${(1.05)^5}$= 1276.28 5. The present worth of Rs.169 due in 2 years at 4% per annum compound interest is • Rs.156.25 • Rs.150 • Rs.140 • Rs125.25 The Correct answer is :Rs.156.25 Present Worth = Rs.${\frac{P}{\left(1+\frac{R}{100}\right)^{n}}}$= Rs.${\frac{169}{\left(1+\frac{4}{100}\right)^{2}}}$= Rs.${\frac{169\times25\times25}{26\times26}}$= Rs.156.25 6. If Rs. 500 amounts to Rs. 583.20 in two years compounded annually, find the rate of interest per annum. • 6.00% • 7.00% • 8.00% • 9.00% The Correct answer is :8.00% Principal = Rs. 500; Amount = Rs. 583.20; Time = 2 years. Let the rate be R% per annum Then, [500(1+${\frac{R}{100})^2}$] = 583.20 or [ 1+(${\frac{R}{100})]^2}$=${ \frac{5832}{5000}}$=>${\frac{11664}{10000}}$[ 1+(${\frac{R}{100})]^2}$=${(\frac{108}{100})^2}$or 1 +${\frac{R}{100}}$=${\frac{108}{100}}$or R = 8 So, rate = 8% p.a. 7. A certain sum amounts to Rs.7350 in 2 years and to Rs.8575 in 3 years. Find the sum and rate percent • 3400 • 4400 • 5400 • 6400 The Correct answer is :5400 Interest for 1 year is the same whether it's simple interest or the compound interest. Now interest of third year = ( 8575 - 7350) = 1225; Means principal for this interest is 7350 The compound interest taken If 7350 is the principal interest = 1225 If 100 is the principal interest =${\frac{1225}{7350}\times}$100 =${\frac{50}{3}}$% When a thing increases for two successive times the overall increase on initial amount =${(a+b) + \frac{a\times b}{100}}$Therefore overall interest for two years =${\frac{50}{3}+\frac{50}{3}+[(\frac{50}{3}\times\frac{50}{3}\times100)]}$=${\frac{325}{9}}$% Therefore amount after 2 years = 100+${\frac{325}{9}}$=${\frac{1225}{9}}$If 1225/9 is the amount, principal = 100 if 7350 is the amount, then principal =${\frac{900}{1225}\times7350}$=5400 So sum = 5400; Rate =${\frac{50}{3}}$% 8. Rs. 5887 is divided between Shyam and Ram, such that Shyam's share at the end of 9 years is equal to Ram's share at the end of 11 years, compounded annually at the rate of 5%. Find the share of Shyam. • 3567 • 3452 • 3087 • 3544 The Correct answer is :3087 Shyam's share${(1+0.05)^{9}}$= Ram's share${(1+0.05)^{11}}$Shyam's share / Ram's share =${\frac{(1+0.5)^{11}}{(1+0.5)^9}}$=>${(1+0.05)^{2}}$=${\frac{441}{400}}$rnTherefore Shyam's share =${\frac{441}{841}\times5887}$= 3087 9. In what time will Rs. 1000 become Rs. 1331 at 10% per annum compounded annually • 1 year • 2 years • 3 years • 4 years The Correct answer is :3 years Principal = Rs. 1000; Amount = Rs. 1331; Rate = 10% p.a. Let the time be n years. Then,n[ 1000 (1+ (${1+\frac{10}{100})^n}$] = 1331 or (${\frac{11}{10})^n}$=${\frac{1331}{1000}}$= (${\frac{11}{10})^3}$= 3 years 10. The difference between compound interest and simple interest on a sum for two years at 10% per annum, where the interest is compounded annually is Rs.16. if the interest were compounded half yearly , the difference in two interests would be • Rs.24.81 • Rs.26.90 • Rs.31.61 • Rs.32.40 The Correct answer is :Rs.24.81 For 1st year S.I = C.I. Rs.16 is the S.I. on S.I. for 1 year Rs.10 is S.I on Rs.100 Rs.16 is S.I on Rs.(${\frac{100}{10}\times16}$) = Rs.160 So, S.I on principal for 1 year at 10% is Rs.160 Principal = Rs.${\frac{(100\times160)}{(10\times1)}}$= Rs. 1600 Amount for 2 years compounded half yearly = Rs.[${1600\times(1+(\frac{5}{100})^4)}$] = Rs. 1944.81 C.I= Rs.(1944.81 - 1600) = Rs.344.81 S.I.= Rs${\frac{(1600\times10\times2)}{100}}$= Rs.320 Rs.(344.81 - 320) = Rs.24.81 11. What will Rs.1500 amount to in three years if it is invested in 20% p.a. compound interest, interest being compounded annually • 2592 • 2492 • 2352 • 2334 The Correct answer is :2592 The usual way to find the compound interest is given by the formula A =${P(1+\frac{R}{100})^n}$In this formula, A is the amount at the end of the period of investment P is the principal that is invested r is the rate of interest in % p.a And n is the number of years for which the principal has been invested. In this case, it would turn out to be A =${1500(1+\frac{20}{100})^3}$= 2592. 12. A man invests Rs.5000 for 3 years at 5% p.a. compound interest reckoned yearly. Income tax at the rate of 20% on the interest earned is deducted at the end of each year. Find the amount at the end of the third year • Rs.5624.32 • Rs.5423 • Rs.5634 • Rs.5976 The Correct answer is :Rs.5624.32 5% is the rate of interest. 20% of the interest amount is paid as tax. i.e 80% of the interest amount stays back if we compute the rate of interest as 80% of 5% = 4% p.a., we will get the same value The interest accrued for 3 years in compound interest = 3 x simple interest on principal + 3 x interest on simple interest + 1 x interest on interest on interest. = 3 x 200 + 3 x 8 + 1 x 0.32 = 624.32 The amount at the end of 3 years = 5000 + 624.32 = 5624.32 13. Shawn invested one half of his savings in a bond that paid simple interest for 2 years and received Rs.550 as interest. He invested the remaining in a bond that paid compound interest, interest being compounded annually, for the same 2 years at the same rate of interest and received Rs.605 as interest. What was the value of his total savings before investing in these two bonds • Rs.5500 • Rs.11000 • Rs.22000 • Rs.2750 The Correct answer is :Rs.2750 Shawn received an extra amount of (Rs.605 - Rs.550) =>Rs.55 On his compound interest paying bond as the interest that he received in the first year also earned interest in the second year. The extra interest earned on the compound interest bond = Rs.55 The interest for the first year =${\frac{550}{2}}$= Rs.275 Therefore, the rate of interest =${\frac{55}{275}\times}$100 = 20% p.a. 20% interest means that Shawn received 20% of the amount he invested in the bonds as interest. If 20% of his investment in one of the bonds = Rs.275, then his total investment in each of the bonds = =${ \frac{275}{20}\times}$100 = 1375. As he invested equal sums in both the bonds, his total savings before investing =${2\times1375}$= Rs.2750. 14. The effective annual rate of interest corresponding to a nominal rate of 6% per annum payable half-yearly is • 6.06% • 6.07% • 6.08% • 6.09% The Correct answer is :6.09% Amount of Rs. 100 for 1 year when compounded half-yearly = Rs.100${\times(1+\frac{3}{100})^2}$= Rs.106.09 Effective rate = (106.09 -100)% = 6.09% 15. The difference between compound interest and simple interest on an amount of Rs.15,000 for 2 years is Rs. 96. What is the rate of interest per annum? • 8 • 10 • 12 • 14 The Correct answer is :8 [15000${\times (1+\frac{R}{100})^{2}}$- 15000] -${\frac{15000\times R\times2}{100}}\$ = 96