Question

A sum of Rs. 20000 invested in two
parts at 8% per annum and at 12% per
annum. If the rate of interest on the
whole investment is 9% per annum,
then find the sum invested at 8% per
annum.​

Answer 1

$\large\underline{\sf{Solution-}}$

Let assume that

Sum invested at 8 % per annum = Rs x

and

Sum invested at 12 % per annum = Rs y

According to statement,

Total sum = Rs 20000

$\red{\rm :\longmapsto\:x + y = 20000 - - - (1)}$

Case :- 1

Sum invested, p = Rs x

Time, n = 1 year

Rate of interest, r = 8 % per annum

We know,

Simple Interest on a certain sum of money Rs p invested at the rate of r % per annum for n years is

$\green{\boxed{ \bf \: SI \: = \: \dfrac{p \: \times \: r \: \times \: t}{100} }}$

Thus,

Simple Interest in this case is

$\rm :\longmapsto\:SI_1 = \dfrac{x \times 8 \times 1}{100}$

$\green{\rm :\longmapsto\:SI_1 = \dfrac{8x}{100} - - - (1)}$

Case :- 2

Sum invested, p = Rs y

Time, n = 1 year

Rate of interest, r = 12 % per annum

We know,

Simple Interest on a certain sum of money Rs p invested at the rate of r % per annum for n years is

$\green{\boxed{ \bf \: SI \: = \: \dfrac{p \: \times \: r \: \times \: t}{100} }}$

Thus,

Simple Interest in this case is

$\rm :\longmapsto\:SI_1 = \dfrac{y \times 12 \times 1}{100}$

$\green{\rm :\longmapsto\:SI_2 = \dfrac{12y}{100} - - - (2)}$

Case :- 3

Sum invested, p = Rs x + y

Time, n = 1 year

Rate of interest, r = 9 % per annum

We know,

Simple Interest on a certain sum of money Rs p invested at the rate of r % per annum for n years is

$\green{\boxed{ \bf \: SI \: = \: \dfrac{p \: \times \: r \: \times \: t}{100} }}$

Thus,

Simple Interest in this case is

$\rm :\longmapsto\:SI_1 = \dfrac{(x + y) \times 9 \times 1}{100}$

$\green{\rm :\longmapsto\:SI_3 = \dfrac{9(x + y)}{100} - - - (3)}$

According to statement,

$\rm :\longmapsto\:SI_1 + SI_2 = SI_3$

$\rm :\longmapsto\:\dfrac{8x}{100} + \dfrac{12y}{100} = \dfrac{9(x + y)}{100}$

$\rm :\longmapsto\:8x + 12y = 9x + 9y$

$\rm :\longmapsto\:x - 3y = 0$

$\bf\implies \:x = 3y - - - (2)$

On substituting equation (2) in equation (1), we get

$\rm :\longmapsto\:3y + y = 12000$

$\rm :\longmapsto\:4y= 12000$

$\bf\implies \:y = 3000$

On substituting the value of y in equation (2), we get

$\bf\implies \:x = 9000$

Thus,

• Sum invested at 8 % per annum = Rs 9000

and

• Sum invested at 12 % per annum = Rs 3000