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Q#1

The Simple interest of some amount is $4000 for two years, while Compound interest is $4225. Find rate of interest.

Q#2

Jame's two sons are aged 9 and 13 years when he dies. He leaves $40,000 in a fund which earns 5% annually to be so divided that each shall receive the same amount at age 21. What part of the $40,000 does each receive?

Q#3

The rate of simple interest in two banks X and Y are 2:3 A person wants to deposit his savings in these two banks such that he receives equal quarterly interest from both. what percentage of amount that he should deposit in two banks X and Y?

Ayaz,

1. The simple interest is given by the formula Fs = P(1+r)^n, where P is the initial principal, r is the interest rate (componded annually) and n is the number of years P is invested. The formula for compound interest requires a parameter for the number of times per year the principal is compounded. Since you don't provide this in your question, I'll use the formula for continuous compounding, Fc = Pe^nr. We want to find r given an unknown initial amount P and the 2 differnet interest calculatons. To get rid of P, take the ratio to get(for 2 years)

4225/4000 = 1.056 = e2r/(1+r)^2.

This is not an easy equation to solve for r (its transcendental), so we need to try approximate methods. Analytically, we can expand the exponential to get the ratio (take square root fist to simplify)

sqrt(1.056) ≈ (1+r+r^2/2 + ...))/(1+r)^2. Rearranging (and letting the dust settle), I end up with the quadratic for r

r^2-2zr-2z = 0 where z = sqrt(1.056)-1 = 0.0276. Using the familiar quadratic formula to solve for r, get r = (0.0276)±().237). This seemed a little high, but if you plot R(r) = Fc(r)/Fs(r) vs. r, it looks about right (i.e., R(0.24) ≈ 1.056). See Excel image.

2. We need to figure out the initial fractions of $40,000, f1 and f2, for each son. Son 1 turns 21 in 12 years and son 2 in 8. The principals for each son are P1 = f1(40000) and P2 = f2(40000). Note tht f1+f2 = 1. Using the formula fro simple interest, take the ratio of the investment amount when each turns 21 (for 5% interest)

(40000-P2)/P2 = 1.05^12/1.05^8 = 1.05^4. Again, rearranging (40000/P2 - 1)(1.05)^4. This gives P2 = 19878 which means that P1 = 20122.

3. We need to figure out what fraction, fx and fy, of an initial amount to put in each bank if the ratio of their interest rates is 3rx = 2ry (rx = rate for bank x, etc). Note that fx+fy=1. If the investor want the same return after a quarter, then

Fx = fxP(1+rx/4) = fyP(1+ry/4). Again the Ps cancel out. Manipulation gives

fx = (8+3rx)/(16+5x). so that the investor can take the rate from bank x and figure out the ratio.

Hope this helps

Randy

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