What is the easiest way to solve this question?

Question

A sum of 1440 $ is lent out in three parts in such a way that the interest on first part at 2% for 3 years, second part at 3% for 4 years and third part at 4% for 5 years are equal. Find the difference between the largest and the smallest sum.

Answer 1

I would do absolutely nothing and hire someone to do all of that for me, because well, I would be way to afraid to mess up and lose money.

Answer 2

let x,y,z be the amount invested

we have therefore   x+y+z  =  1440

also we have

x*2*3/100   = y * 3 *4/  100  = z*4*5/100

(interest are equal) ie

6x/100  = 12y/100  = 20z/100   = k(let it be equal to k)

so we have

  x  = 100k/6

  y  = 100k/12

  z = 100k/20

but we know x+y+z =1440

subsitute in terms of k

    100k(1/6  +  1/12  +  1/20)  = 1440

    100k(10+5+3)/60  = 1440         60 is LCM

   k  =   (1440 * 60)/(100 * 18)

    k  = 48

we get   x = 48*100/6   =  800

            y  =  48 * 100/12  =  400

            z = 48 * 100/20    = 240

diff between largest and smallest  = 800 - 240 =560

Answer 3

You don't say if interest is compounded or not.

I will assume that it is, since there is a good answer out there using straight interest.

compound interest @ 2% for 3 years:

  X (1.02)^3 - X = 0.061208 X

compound interest @ 3% for 4 years:

Y (1.03)^4 - Y = 0.12550881 Y

compound interest @ 4% for 5 years:

Z (1.04)^5 - Z = 0.2166529024 Z

Note that I have subtracted X, Y, Z from the calculations above.

That is because calculating X(1.02)^3 gives us the value of the interest AND principal. To get value of interest alone, I then subtract principal from this calculation. A previous answerer used the same approach, but failed to subtract the principal.

Now the 3 interests are the same. Therefore

0.061208 X = 0.125508881 Y = 0.2166529024 Z

Y = 0.061208 X / 0.12220881  =  0.4877 X

Z = 0.061208 X / 0.2166529024  =  0.2825 X

Now X + Y + Z = 1440

So  X + 0.4877 X  +  0.2825 X = 1.7702 X = 1440

X = 813.47

Y = 0.4877 (813.47) = 396.72

Z = 0.2825 (813.47) = 229.81

DIFFERENCE = 813.47 - 229.81 = 583.66

Calculate interest:

813.47 (1.02)^3 - 813.47 = 863.29 - 813.47 = 49.79

396.72 (1.03)^4 - 396.72 = 446.51 - 396.72 = 49.79

229.81 (1.04)^5 - 229.81 = 279.60 - 229.81 = 49.79

Answer 4

Let the three amounts be x, y, and z. The simple interest on each of the parts is the (amount * rate * years).

x + y + z = 1440 (all parts add to 1440)

0.06x = 0.12y = 0.2z (the interest is the same on each part)

x + 0.5x + 0.3x = 1440 (substitute for y and z in first equation)

x = 800

y = 0.5x = 400

z = 0.3x = 240

Compound interest is the same idea, but a little more arithmetic:

x + y + z = 1440 (all parts add to 1440)

[(1.02)^3-1]x = [(1.03)^4-1]y = [(1.04)^5-1]z (same interest on each)

0.061208x = 0.12550881y = 0.216652902z

1.7702x = 1440

x=813.47

y=396.71

z=229.82

Answers: (800-240) = 560 simple interest

(813.47-229.82) = 583.65 compound interest

Answer 5

2% for 3 years as a multiple (if compounded interest) = 1.06 approx

3% for 4 years as a multiple (if compounded interest) = 1.13 approx

4% for 5 years as a multiple (if compounded interest) = 1.22 approx

so for the three sums, A+B+C = 1440 and 1.06A = 1.13B = 1.22C

so B = 0.94A, C = 0.87A, so 2.81A = 1440

A = 512, C = 445, so the difference is 67.

there may be a little rounding error, to avoid, do the ratios between the different interest rates to greater number of sig figs. And verify that I am correct that they want you to consider the interest for each sum as compounded annually.