Question

An amount of ₹ 5000 is invested in three types of investments, at interest rates 6%, 7%, 8% per annum respectively. The total annual income from these investimest is ₹ 350. If the total annual income from first two investment is ₹ 70 more than the income from the third, find the amount of each investment using matrix method.

Answer 1

Let the amounts in three investments by ₹ x, ₹ y and ₹ z respectively. 

Then,

x + y + z = 5000 

Since the rate of interest in these investments are 6%, 7% and 8% respectively, the annual income of the three investments are \(\frac{6x}{100}\),\(\frac{7y}{100}\)and \(\frac{8z}{100}\)respectively.

According to the given conditions,

\(\frac{6x}{100}\) + \(\frac{7y}{100}\) + \(\frac{8z}{100}\) = 350

i.e. 6x + 7y + 8z = 35000

Also,

\(\frac{6x}{100}\) + \(\frac{7y}{100}\)  = \(\frac{8z}{100}\) + 70

i.e. 6x + 7y – 8z = 7000 

Hence,

The system of linear equation is :

x + y + z = 5000 

6x + 7y + 8z = 35000 

6x + 7y – 8z = 7000

These equations can be written in matrix form as :

By equality of matrices,

x + y + z = 5000 …(1) 

y + 2z = 5000 …(2) 

-16z = -28000 ….(3) 

From (3), 

z = 1750

Substituting z = 1750 in (2), we get, 

y + 2(1750) = 5000 

∴ y = 5000 – 3500 = 1500 

Substituting y = 1500, z = 1750 in (1), we get, 

x + 1500 + 1750 = 5000 

∴ x = 5000 – 3250 = 1750 

∴ x = 1750, y = 1500, z = 1750 

Hence, 

The amounts of the three investments are ₹ 1750, ₹ 1500 and ₹ 1750 respectively.