If the sum of Rs. 2400 grows by 1/5 of itself every year, in how many years and months will it amount to Rs. 3240?
Answer (2)
Apply the compound interest formula: A = P ( 1 + r ) t .
Substitute the given values: 3240 = 2400 ( 1.2 ) t .
Solve for t : t = l n ( 1.2 ) l n ( 1.35 ) ≈ 1.646 years.
Convert to years and months: 1 year and 8 months. The final answer is 1 year and 8 months .
Explanation
Understanding the Problem We are given that an initial sum of Rs. 2400 grows by 1/5 of itself every year and we want to find out how long it will take for it to amount to Rs. 3240. The growth is compounded annually.
Stating the Formula Let P be the initial amount, A be the final amount, r be the rate of growth per year, and t be the number of years. The formula for compound interest is: A = P ( 1 + r ) t
Substituting the Values We are given P = 2400 , A = 3240 , and r = 5 1 = 0.2 . Substituting these values into the formula, we get: 3240 = 2400 ( 1 + 0.2 ) t
Simplifying the Equation Simplifying the equation, we have: 3240 = 2400 ( 1.2 ) t
Isolating the Exponential Term Dividing both sides by 2400, we get: 2400 3240 = ( 1.2 ) t
Further Simplification Simplifying the fraction, we have: 20 27 = ( 1.2 ) t
Decimal Form Which is: 1.35 = ( 1.2 ) t
Taking the Natural Logarithm To solve for t , we take the natural logarithm of both sides: ln ( 1.35 ) = t ln ( 1.2 ) t = ln ( 1.2 ) ln ( 1.35 )
Calculating t Calculating the value of t , we find: t ≈ 1.646 years.
Converting to Years and Months Since we need to find the answer in years and months, we separate the integer part and the fractional part of t . The integer part is 1 year. The fractional part is 0.646 . To find the number of months, we multiply the fractional part by 12: 0.646 × 12 ≈ 7.752 Rounding to the nearest whole number, we get 8 months.
Final Answer Therefore, it will take 1 year and 8 months for Rs. 2400 to amount to Rs. 3240.
Examples
Compound interest is a powerful tool in finance. For example, if you invest money in a savings account or a certificate of deposit, the interest earned is often compounded, meaning that you earn interest not only on your initial investment but also on the accumulated interest from previous periods. Understanding how compound interest works can help you make informed decisions about your investments and savings, allowing you to estimate how your money will grow over time. This concept is also applicable to loans, where interest is charged on the outstanding balance, affecting the total amount you repay.
It takes approximately 1 year and 8 months for Rs. 2400 to grow to Rs. 3240 at a rate of 1/5 per year. This was calculated using the compound growth formula by substituting the given values and solving for the time variable t . The final result indicates the duration needed for the initial amount to reach the specified final amount based on the determined growth rate.
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