Question

An amount of ₹ 5,000 is invested in three plans at rates 6%, 7% and 8% per annum respectively. The total annual income from these investments is ₹ 350. If the total annual income from first two investments is ₹ 70 more than the income from the third, find the amount invested in each plan by using Cramer’s Rule.

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

∴ `6/100x + 7/100y + 8/100z` = 350

∴ 6x + 7y + 8z = 35000
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

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

Taking 1000 common from C₁, we get

D_x = `|(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|(0, 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

∴ Amounts of investments are ₹ 1750, ₹ 1500 and ₹ 1750.