Question

Answer the following question:

An amount of ₹ 5000 is put into three investments at the rate of interest of 6%, 7% and 8% per annum respectively. The total annual income is ₹ 350. If the combined income from the first two investments is ₹ 70 more than the income from the third. Find the amount of each investment.

Answer 1

Let the amount of each investment be ₹ x, ₹ y and ₹ z.

According to the given conditions,

x + y + z = 5000,

6%x + 7%y + 8%z = 350

∴ `6/100 x + 7/100y + 8/100z` = 350

∴ 6x + 7y + 8z = 35000

and 6%x + 7%y = 8%z + 70

∴ `6/100x + 7/100y = 8/100z + 70`

∴ 6x + 7y = 8z + 7000

∴ 6x + 7y – 8z = 7000

∴ D = `|(1, 1, 1),(6, 7, 8),(6, 7, -8)|`

= 1(–56 – 56) –1(–48 – 48) + 1(42 – 42)

=  –112 + 96 + 0

= –16 ≠ 0

D_x = `|(5000, 1, 1),(35000, 7, 8),(7000, 7, -8)|`

Taking 1000 common from C₁, we get

D_x = `1000|(5, 1, 1),(35, 7, 8),(7, 7, -8)|`

Applying C₁ → C₁ – 5C₃ and C₂ → C₂ – C₃, we get

D_x = `1000|(0, 0, 1),(-5, -1, 8),(47, 15, -8)|`

= 1000 [0 – 0 + 1(–75 + 47)]

= 1000 x (–28)

= – 28000 

D_y = `|(1, 5000, 1),(6, 35000, 8),(6, 7000, -8)|`

Taking 1000 common from C₂, we get

D_y = `1000|(1, 5, 1),(6, 35, 8),(6, 7, -8)|`

Applying C₁ → C₁ – C₃ and C₂ → C₂ – 5C₃, we get

D_y = `1000|(0, 0, 1),(-2, -5, 8),(14, 47, -8)|`

= 1000 [0 – 0 + 1(–94 + 70)

= 1000 (–24)

= –24000

D_z = `|(1, 1, 5000),(6, 7, 35000),(6, 7, 7000)|`

Taking 1000 common from C₃, we get

D_z = `1000|(1, 1, 5),(6, 7, 35),(6, 7, 7)|`

Applying C₁ → C₁ – C₂ and C₃ → C₃ – 5C₂, we get

D_z = `1000|(0, 1, 0),(-1, 7, 0),(-1, 7, -28)|`

= 1000[0 – 1(28 – 0) + 0]

= 1000 x (–28)

= –28000

By Cramer’s Rule,

x = `"D"_x/"D" = (-28000)/(-16)` = 1750,

y = `"D"_y/"D" = (-24000)/(-16)` = 1500,

z = `"D"_z/"D" = (-28000)/(-16)` = 1750,

∴ The amounts of investments are ₹ 1750, ₹ 1500 and ₹ 1750.