Question

A total of ₹ 8,500 was invested in three interest-earning accounts. The interest rates were 2%, 3% and 6% if the total simple interest for one year was ₹ 380 and the amount invested at 6% was equal to the sum of the amounts in the other two accounts, then how much was invested in each account? (use Cramer’s rule)

Answer 1

Let the amount invested at 2%, 3%, and 6% are x, y, and z respectively.

x + y + z = 8500  ........(1)

`(x xx 2/100) + (y xx 3/100) + (z xx 6/100)` = 380

`(2x)/100 + (3y)/100 + (6z)/100` = 380

2x + 3y + 6z = 38000  ........(2)

x + y = z

x + y – z = 0  ........(3) 

Here `Delta = |(1, 1, 1),(2, 3, 6),(1, 1, -1)|`

= 1(– 3 – 6) – 1(– 2 – 6) + 1(2 – 3)

= 1(– 9) -1(– 8) + 1(–1)

= – 9 + 8 – 1

= – 2 ≠ 0

∴ We can apply Cramer’s Rule

`Delta_x = |(8500, 1, 1),(38000, 3, 6),(0, 1, -1)|`

= 8500(– 3 – 6) – 1(– 38000 – 0) + 1(38000 – 0)

= 8500(– 9) + 38000 + 38000

= – 76500 + 76000

= – 500

`Delta_y = |(1, 8500, 1),(2, 38000, 6),(1, 0, -1)|`

= 1(– 38000 – 0) – 8500(– 2 – 6) + 1(0 – 38000)

= – 38000 + 68000 – 38000

= 68000 – 76000

= – 8000

`Delta_z = |(1, 1, 8500),(2, 3, 38000),(1, 1, 0)|`

= (0 – 38000) – 1(0 – 38000) + 8500(2 – 3)

= – 38000 + 38000 – 8500

= – 8500

∴ By Cramer’s Rule

x = `Delta_x/Delta = (- 500)/(- 2)` = 250

x = `Delta_y/Delta = (- 8000)/(- 2)` = 4000

z = `Delta_z/Delta = (- 8500)/(- 2)` = 4250

∴ Amount invested at 2% is ₹ 250

∴ Amount invested at 3% is ₹ 4,000

∴ Amount invested at 6% is ₹ 4,250