the original amount at the beginning o
the first year?
किसी राशि पर वार्षिक संयोजित चक्रवृद्धि ब्याज दूसं
वर्ष के लिए ₹ 1,320 है और तीसरे वर्ष के लिए
₹ 1,452 है। पहले वर्ष की शुरूआत में मूल राशि कितन
होगी?

$₹ ₹$

Ask by Parry Coles. in India

Mar 24,2025

Question

the original amount at the beginning o
the first year?
किसी राशि पर वार्षिक संयोजित चक्रवृद्धि ब्याज दूसं
वर्ष के लिए ₹ 1,320 है और तीसरे वर्ष के लिए
₹ 1,452 है। पहले वर्ष की शुरूआत में मूल राशि कितन
होगी?

$₹ ₹$

Ask by Parry Coles. in India

Mar 24,2025

UpStudy AI · Accepted Answer

Let the principal amount at the beginning be $$ P $$ and the annual interest rate be $$ r $$ (in decimal form).

1. After one year the amount becomes $$ P(1+r) $$.
2. After two years the amount is $$ P(1+r)^2 $$. Hence, the interest earned in the second year is
   $$
   P(1+r)^2 - P(1+r) = P(1+r)\left[(1+r) - 1\right] = P(1+r)r.
   $$
   It is given that this interest is ₹1,320. Therefore,
   $$
   P(1+r)r = 1320. \tag{1}
   $$

3. After three years the amount is $$ P(1+r)^3 $$. Thus, the interest earned in the third year is
   $$
   P(1+r)^3 - P(1+r)^2 = P(1+r)^2\left[(1+r) - 1\right] = P(1+r)^2r.
   $$
   It is given that this interest is ₹1,452. So,
   $$
   P(1+r)^2r = 1452. \tag{2}
   $$

4. Dividing equation (2) by equation (1) gives:
   $$
   \frac{P(1+r)^2r}{P(1+r)r} = \frac{1452}{1320},
   $$
   which simplifies to:
   $$
   1+r = \frac{1452}{1320}.
   $$
   Simplify the fraction by dividing numerator and denominator by 12:
   $$
   1+r = \frac{1452 \div 12}{1320 \div 12} = \frac{121}{110}.
   $$
   Thus,
   $$
   1+r = \frac{121}{110} \quad \Longrightarrow \quad r = \frac{121}{110} - 1 = \frac{121-110}{110} = \frac{11}{110} = 0.1.
   $$

5. Now substitute $$ r = 0.1 $$ into equation (1):
   $$
   P(1+r)r = P(1.1)(0.1) = 0.11P = 1320.
   $$
   Solving for $$ P $$:
   $$
   P = \frac{1320}{0.11} = 12000.
   $$

Thus, the original amount at the beginning of the first year is $$ \boxed{₹12,000} $$.
The original amount at the beginning of the first year is ₹12,000.

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