Question

Sandeep borrowed Rs.3000 for 4 years and Rs.8000 for $$2{\displaystyle \frac{1}{2}}$$ year from Sonia at the same rate of interest. The total interest earned by Sonia is Rs.2880. Find the rate of interest.

Answer 1

Hint: Here we will find the rate of interest by using the Simple Interest method. First, we will find the interest incurred on the principal amount for Rs 3000 and the interest incurred on the principal amount for Rs 8000 using the formula. Then we will add the simple interest obtained in both the cases and equate it to total interest earned. We will simplify it further to get the required rate of interest.

Formula Used:
 $$SI={\displaystyle \frac{P\times R\times T}{100}}$$, where, $$P$$ is the principal, $$R$$ is the rate of interest,$$T$$ is the time period and $$SI$$ is the simple interest.

Complete step-by-step answer:
It is given that Sandeep borrowed Rs.3000 for 4 years.
Let rate of interest, $$R=r$$.
Substituting $$P=3000$$, $$T=4$$ and $$R=r$$ in the equation $$SI={\displaystyle \frac{P\times R\times T}{100}}$$ we get,$$SI={\displaystyle \frac{3000\times r\times 4}{100}}$$………………$$\left(1\right)$$
Now, Sandeep borrowed Rs.8000 for $$2{\displaystyle \frac{1}{2}}$$ years.
Substituting $$P=8000$$, $$T=2{\displaystyle \frac{1}{2}}={\displaystyle \frac{5}{2}}$$ and $$R=r$$ in the equation $$SI={\displaystyle \frac{P\times R\times T}{100}}$$ we get,
$$SI={\displaystyle \frac{8000\times {\displaystyle \frac{5}{2}}\times r}{100}}$$………………$$\left(2\right)$$
As given that the total Interest earned by Sonia is Rs.2880. So, sum of both the Interest will be equal to Rs.2880.
Therefore, on adding equation $$\left(1\right)$$ and $$\left(2\right)$$ and equating 2880, we get
$${\displaystyle \frac{3000\times r\times 4}{100}}+{\displaystyle \frac{8000\times {\displaystyle \frac{5}{2}}\times r}{100}}=2880$$
Simplifying the expression, we get
$$\Rightarrow 120r+200r=2880$$
Adding the terms, we get
$$\Rightarrow 320r=2880$$
Dividing both sides by 320, we get
$$\Rightarrow r=9\mathrm{\%}$$
Therefore, the interest rate at which Sonia lent Sandeep money is $$9\mathrm{\%}$$.

Note: Simple interest is a term used in banking, the financial sector etc. It can be calculated on a monthly yearly or daily basis. Simple Interest is calculated only on the principal amount or the original amount. We need to keep in mind that the interest on a sum of money and the amount paid with interest are two different things. Compound interest is another type of interest that is calculated on both principal and interest incurred during the period.