Question

A sum of money lent out at simple interest amount ₹2200 in one year and ₹2800 in 4 years. Find the sum of money and the rate of interest.

Answer 1

amount1 = ₹2200 and and time = 1yrs

Find the interest1 :
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S.I1 = (p × r × t )/100

S.I1 = (p × r × 1 / 100 ) = pr / 100

solve for 'p'
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amount1 = principal + S.I1

₹2200 = p + ( pr /100 )

2200 × 100 = 100p + pr

220000 = p ( 100 + r )

p = 220000 / ( 100 + r ) -----(1)

Find the interest2:
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amount2 = ₹2800 , and time = 4yrs

S.I2 = ( p × r × t )/ 100 = (p × r × 4/100 )

S.I2 = pr / 25

solve for 'p' :
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amount2 = principal + S.I2

2800 = p + ( pr/ 25 )

25p + pr = 2800×25

p ( 25 + r ) = 70000

p = 70000/ ( 25 + r ) ---------(2)

equalize (1) and (2) , we get

220000/ ( 100 + r ) = 70000/ ( 25 + r )

220000 / 70000 = (100 + r ) / ( 25 + r )

22 / 7 = ( 100 + r ) / ( 25 + r )

22 × ( 25 + r ) = 7 × ( 100 + r )

550 + 22r = 700 + 7r

15r = 150 => r = 10 %

put value of 'r' in (2) , we get

p = 70000 / ( 25 + r )

p = 70000 / ( 25 + 10 )

p = 70000 ÷ 35 = ₹2000

therefore the required sum =₹2000

rate of interest = 10 %

Answer : sum = ₹2000 , rate = 10% p.a

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Answer 2

Let  the  principle be  P  and rate  of  interest  r.

For  one  year,

Time (t) = 1 year

rate  of interest = r

principle  = P

Simple  interest (I₁) = A - P = 2200 - P

Amount  = Rs. 2200

We know that,

$S.I.=\frac{p\times r\times t}{100}\\\;\\I_1=\frac{P\times r\times1}{100}\\\;\\A-P=\frac{Pr}{100}\\\;\\2200-P=\frac{Pr}{100}\;\;\;................i)$

For  four   year,

Time (t) = 4 year

rate  of interest = r

principle  = P

Simple  interest (I₂) = A - P = 2800 - P

Amount  =  Rs.2800

We know that,

$S.I.=\frac{p\times r\times t}{100}\\\;\\I_2=\frac{P\times r\times4}{100}\\\;\\A-P=\frac{4Pr}{100}\;\;........................ii)$

Comparing  Eq. i) and  ii)

$2200-P=\frac{2800-P}{4}\\\;\\2200-P=\frac{2800}{4}-\frac{P}{4}\\\;\\2200-P=700-\frac{P}{4}\\\;\\2200-700=P-\frac{P}{4}\\\;\\1500=\frac{4P-P}{4}\\\;\\6000=3P\\\;\\P=2000\\\;\\\;\\\text{Putting P=2000 in eq. ii)}\\\;\\\frac{2800-P}{4}=\frac{Pr}{100}\\\;\\\frac{2800-2000}{4}=\frac{2000\times r}{100}\\\;\\\frac{800}{4}\\\;\\200=20r\\\;\\r=10\%$