Question

Compound interest on a sum at some rate of interest after two years is rs. 23441.25 and compound interest on rs1500 more sum at 20% less rate of interest for two years is rs. 19152.what is the sum?

Answer 1

Answer:

Let the Principle = P and rate of interest = r in decimal. So, we have the first equation as

P * (1+r)^2 - P = 23441.25

Or, P [(1+r^2+2r) -1] = 23441.25

Or, P * r * (r+2) = 23441.25 ………… (1)

And for the second equation, Principle = P + 1500, rate = 20% less rate = r* (1 - 0.2) = 0.8r

We have the second equation

(P+1500) * (1+0.8r)^2 - (P+1500) = 19152

Or, (P+1500) * [(1+0.64r^2+1.6r) - 1] = 19152

Or, (P+1500) * 0.8r * (0.8r+2) = 19152 ……….. (2)

Dividing equation (2) by (1), we get,

[(P+1500)/P] * 0.8 * [(0.8r+2)/(r+2)] = 19152/23441.25 = 0.8170212766

Or, (P+1500)/P = 0.8170212766/0.8 * (r+2)/(0.8r+2)

Or, 1+1500/P =1.0213 * (r+2)/(0.8r+2)

Or, 1500/P = [1.0213 * (r+2)/(0.8r+2)] - 1

Or, P = 1500 / [{1.0213 * (r+2)/(0.8r+2)} - 1] ……. (3)

From equation (3), there are two unknowns parameters - P and r. This can be solved by very cumbersome trial & error method. There are value sets for P and r but only one set will satisfy equation (1) and (2).

Let us start the trial :

Assume r = 0.05 ( I.e., 5%), P from eqn (3) = 57020.4

Assume, r = 0.1, P = 48200.22

Assume r = 0.15, P = 41955.27

Assume r = 0.2, P = 35761.59

Putting these value sets to left hand side of equation (1) and (2) to see the values if they are equal to right hand side,

For r = 0.05 and P = 57020.4

From eqn (1), LHS = 57020.4 * 0.5 * 2.05 = 5844.59 not matching

Try with r = 0.1 and P = 48200.22,

Eqn(1) LHS = 10122 not matching

Try with r = 0.2, P = 35761.59, Eqn(1) LHS = 15735 not matching

From eqn (3), let r = 0.4, P = 26540.57

From eqn(2) LHS = 26540.57 * 0.4 * 2.4 =25978.85 slightly more.

Try with r = 0.35, P from eqn (3) = 28486.94

From eqn(1) LHS = 28586.94 * 0.35 * 2.35 = 23430.51 - nearly matching

Like this, you have to proceed.

So, r = 0.35 I.e., 35%, Now, from eqn(1),

P = 23441.25/(0.35 *2.35) = Rs. 28500

From eqn (2), LHS = (28500+1500) * (0.8*0.35) * (2+0.35 * 0.8) = 19152, So matching both equations

So, we have the following answers :

1. Principle = Rs.28500 and Rate = 35% Ans