The question is:
Rajpal borrowed a certain amount at 10% per annum simple interest for 1 \frac{1}{2} years. He gave the same amount as loan to a friend and charged compound interest (compounded semi-annually) at the same rate for the same period. In this way, he earned Rs.305. Find the sum of the money.
The options provided are:
* Rs.30,500
* Rs.40,000
* Rs.40,305
* Rs.30,000
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Rajpal borrowed a certain amount from a bank at 10% per annum simple interest for 1.5 years. He then lent the same amount to a friend at 10% per annum compound interest, compounded semi-annually, for the same period. Through this transaction, he earned a profit of Rs.305. We need to find the principal amount.
Let's calculate the simple interest that Rajpal pays on his borrowed amount. Using the simple interest formula SI equals P times R times T divided by 100, where P is the unknown principal, R is 10 percent per annum, and T is 1.5 years. Substituting these values, we get SI equals P times 10 times 1.5 divided by 100, which simplifies to 0.15P. This means Rajpal pays simple interest equal to 15% of the principal amount.
Now let's set up the compound interest calculation for Rajpal's lending. The compound interest formula is A equals P times 1 plus r over n to the power of n times t. For semi-annual compounding, we have P as the principal, r equals 0.10 as the annual rate, n equals 2 for semi-annual compounding, and t equals 1.5 years. Substituting these values, we get A equals P times 1 plus 0.10 over 2 to the power of 2 times 1.5, which simplifies to P times 1.05 to the power of 3. This represents 3 compounding periods of 6 months each.
Now let's complete the compound interest calculation. We need to calculate 1.05 to the power of 3. First, 1.05 squared equals 1.1025. Then, 1.05 cubed equals 1.1025 times 1.05, which gives us 1.157625. Therefore, the amount A equals P times 1.157625. The compound interest CI equals A minus P, which is P times 1.157625 minus 1, giving us 0.157625P. Notice how compound interest grows faster than simple interest due to the compounding effect.
Now let's solve for the principal amount using the profit equation. Rajpal's profit equals compound interest minus simple interest, which is 305 rupees. Substituting our calculated values: 0.157625P minus 0.15P equals 305. This simplifies to 0.007625P equals 305. Dividing both sides by 0.007625, we get P equals 40,000 rupees. Let's verify: Simple interest is 40,000 times 0.15 equals 6,000. Compound interest is 40,000 times 0.157625 equals 6,305. The profit is 6,305 minus 6,000 equals 305 rupees, which matches our given condition. Therefore, the answer is Rs.40,000.