Question

Rs.4615 is divided into 3 parts and each part is lent at the rate of 15 percent per annum on simple interest for 4 6 and 8 years respectively.The interest received after the respective time periods on each part is equal.How much money is lent for 8 years on simple interest?​

Answer 1

$\begin{gathered}\begin{gathered}\bf \:Let - \begin{cases} &\sf{first \: part \: be \: Rs \: x} \\ &\sf{second \: part \: be \: Rs \: y}\\ &\sf{third \: part \: be \: Rs \: z} \end{cases}\end{gathered}\end{gathered}$

So,

$\bf\implies \:x \: + \: y \: + \: z \: = \: 4615 - - - (1)$

Now,

$\begin{gathered}\begin{gathered}\bf \:Case \: 1 - \begin{cases} &\sf{Principal \: Rs \: x} \\ &\sf{Time, \: {t_{1}} = 4 \: years}\\ &\sf{Rate, \: {r_{1}} = 15\%} \end{cases}\end{gathered}\end{gathered}$

We know,

$\: \: \: {\sf{\ Simple \: interest \: = \: \dfrac{P \times R \times T}{100}}}$

where,

• P represents Principal

• R represents Rate of interest

• T represents Time

So,

Simple interest in this case is

$\rm :\longmapsto\:S.I._{1} \: = \: \dfrac{x \times 15 \times 4}{100} = \dfrac{60x}{100} - - - (2)$

$\begin{gathered}\begin{gathered}\bf \:Case \: 2 - \begin{cases} &\sf{Principal \: Rs \: y} \\ &\sf{Time, \: {t_{2}} = 6 \: years}\\ &\sf{Rate, \: {r_{2}} = 15\%} \end{cases}\end{gathered}\end{gathered}$

So, Simple interest in this case is

$\rm :\longmapsto\:S.I._{2} \: = \: \dfrac{y \times 15 \times 6}{100} = \dfrac{90y}{100} - - - (3)$

$\begin{gathered}\begin{gathered}\bf \:Case \: 3 - \begin{cases} &\sf{Principal \: Rs \: z} \\ &\sf{Time, \: {t_{3}} = 8 \: years}\\ &\sf{Rate, \: {r_{3}} = 15\%} \end{cases}\end{gathered}\end{gathered}$

So, Simple interest in this case is

$\rm :\longmapsto\:S.I._{3} \: = \: \dfrac{z \times 15 \times 8}{100} = \dfrac{120z}{100} - - - (4)$

According to statement,

It is given that interest received after the respective time period on each part is same,

$\bf\implies \:S.I._{1} = S.I._{2} = S.I._{3}$

$\rm :\implies\:\dfrac{60x}{100} = \dfrac{90y}{100} = \dfrac{120z}{100}$

$\: \: \: \: (\tt \: on \: dividing \: each \: term \: by \: \dfrac{30}{100} , \: we \: get)$

$\rm :\longmapsto\:2x = 3y = 4z$

$\sf \: Let \: \: 2x \: = \: 3y \: = \: 4z \: = \: k$

$\begin{gathered}\begin{gathered}\bf \:So, \: we \: have - \begin{cases} &\sf{x = \dfrac{k}{2} - - (5)} \\ &\sf{y = \dfrac{k}{3} - - (6) }\\ &\sf{z = \dfrac{k}{4} - - (7) } \end{cases}\end{gathered}\end{gathered}$

On substituting, the values of x, y, z in equation (1), we get

$\rm :\longmapsto\:\dfrac{k}{2} + \dfrac{k}{3} + \dfrac{k}{4} = 4615$

$\rm :\longmapsto\:\dfrac{6k + 4k + 3k}{12} = 4615$

$\rm :\longmapsto\:\dfrac{13k}{12} = 4615$

$\bf\implies \:k = \dfrac{4615 \times 12}{13} = 4260 - - (8)$

$\begin{gathered}\begin{gathered}\bf \:So, \: value \: of \: each \: part - \begin{cases} &\sf{x = \dfrac{4260}{2} = Rs \: 2130} \\ &\sf{y = \dfrac{4260}{3} = Rs \: 1420 }\\ &\sf{z = \dfrac{4260}{4} = Rs \: 1065} \end{cases}\end{gathered}\end{gathered}$

So, it implies,

Rs 1065 is lent for 8 years on Simple Interest.

Additional Information :-

Principal: The principal is the amount that initially borrowed from the bank or invested. The principal is denoted by P.

Rate: Rate is the rate of interest at which the principal amount is given to someone for a certain time. The rate of interest is denoted by R.

Time: Time is the duration for which the principal amount is given to someone. Time is denoted by T.

Amount: When a person takes a loan from a bank, he/she has to return the principal borrowed plus the interest amount, and this total returned is called Amount.

Amount = Principal + Simple Interest