Question

A sum of money lent at simple interest amounts to 4745 in 3 years and to 5475 in
5 years. Find the sum and the rate per cent per annum.​

Answer 1

Given :-

• Amount after 5 years = ₹5475
• Amount after 3 years = ₹4745

To find :-

• Principal
• Rate %
• Sum after 1 year

Formula to use :-

$\dfrac{simple \: interest \times 100}{ principal \times time} = rate$

Important points to note :-

• Amount = Principal + Simple Interest
• Principal = Amount - Simple Interest
• Simple interest = Amount - Principal

Let principal be x and rate be y.

Simple interest for 2 years :-

(Simple interest for 5 years) - (Simple Interest for 3 years)

As we already know that Simple Interest = Amount - Principal,

substituting the values,

Simple interest (2years) = (5475-x) - (4745-x)

By removing the brackets,

Simple interest = 5475 - x - 4745 + x

Simple interest = ₹730 for 2 years.

Simple interest is the same for each year. Hence if for 2 years it's ₹730, then for 1 year it will be ½ of ₹730

Simple interest for 1 year :-

$\dfrac{730}{2}$

$= 365$

Principal = Amount - Simple Interest

x = 4745 - (3×365) [As the simple interest for 3 years will be 365 + 365 + 365]

x = ₹3650

(or)

x = 5475 - (5×365)

x = ₹3650

Now that we know the principal, by using the formula, let us calculate the rate.

$\dfrac{365 \times 100}{3650} = y$

$\dfrac{36500}{3650} = y$

Reducing to the lowest terms,

$10\% = y$

Hence the rate is 10% per annum

Amount after 1 year = 3650 + 365

Amount = ₹4015

Some more formulas :-

$simple \: interest = \dfrac{principal \times time \times rate}{100}$

$time = \dfrac{simple \: interest \times 100}{principal \times time}$

$principal = \dfrac{simple \: interest \times 100}{time \times \: rate}$