Question

Mr. Raju lent out a certain sum on simple interest and the same sum on compound interest at a certain rate
of interest per annum. If the ratio between the difference of compound interest and simple interest of 2
years and that of 3 years is 10 : 31, what is the rate of interest per annum?​

Answer 1

Given :-

• Mr. Raju lent out a certain sum on simple interest and the same sum on compound interest at a certain rate
• of interest per annum.
• The ratio between the difference of compound interest and simple interest of 2 years and that of 3 years is 10 : 31 .

To Find :-

• what is the rate of interest per annum ?

Solution :-

we know that,

• Difference between the compound interest and simple interest for 2 years at same Principal and same rate of interest is = { Principal * (rate)² } / (100)²
• Difference between the compound interest and simple interest for 3 years at same Principal and same rate of interest is = [ {principal * (rate)² * (300 + rate) } / (100)³ ]

Let us assume that, certain sum is Rs.P and rate of interest is R % per annum.

given that,

→ (Difference between CI and SI for years) / (Difference between CI and SI for years) = 10 / 31

Putting all values in above told formula we get,

→ [{P * (R)²} / (100)²] / [ {P * (R)² * (300 + R) } / (100)³ ] = 10/31

→ {(PR²)/(100)²} * [(100)³/{PR²(300+R)}] = 10/31

→ 100 / (300 + R) = 10/31

cross - Multiply,

→ 10(300 + R) = 3100

dividing both sides by 10,

→ 300 + R = 310

→ R = 310 - 300

→ R = 10% (Ans.)

Hence, Rate of interest per annum is 10% .

See how to prove formula of Difference between the compound interest and simple interest for 3 years at same Principal and same rate of interest :-

Difference between si and ci for 3 years formula

https://brainly.in/question/16133467 .

Answer 2

Given : Mr. Raju lent out a certain sum on simple interest and the same sum on compound interest at a certain rate  of interest per annum.

the ratio between the difference of compound interest and simple interest of 2  years and that of 3 years is 10 : 31

To Find  : what is the rate of interest per annum

Solution:

Let say Amount = 100P

Simple interest in 2Years  = 100P *  R * 2 / 100  =  2PR

Compound interest in 2 Years = 100P(1 + R/100)²  - 100P

= 100P ( R²/100²  + 2R/100)

= 2PR   +   P R²/100

Difference in Interest in 2 Years = 2PR   + P R²/100  -  2PR

=   P R²/100

Simple interest in 3Years  = 100P *  R * 3 / 100  =  3PR

Compound interest in 3 Years = 100P(1 + R/100)³  - 100P

= 100P ( 1 + R³/100³ + 3R²/100² + 3R/100)  - 100P

= PR³/100² + 3PR²/100 + 3PR

Difference on interest in 3 Years =  PR³/100² + 3PR²/100 + 3PR  -   3PR

= PR³/100² + 3PR²/100

= PR²/100( R/100 + 3)

 (P R²/100 ) / (P R²/100)( 3 + R /100)   = 10/31

=> 31 =  10 ( 3 + R/100)

=> 31 = 30  + R/10

=>  1 = R/10

=>  R =  10

rate of interest per annum = 10 %

Taking P = 100

Year           SI                   CI                   Difference

1                 10                    10                  0

2                20                   21                   1    

3                30                 33.1                 3.1    

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