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6 copies of a book can be bought for a c...

6 copies of a book can be bought for a certain sum payable at the end of a year, and 7 copies of the same book can be bought for the same sum in cash money. What is the rate per annum of simple interest (correct to one decimal place)?

A

0.109

B

0.155

C

0.167

D

0.304

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem step by step, we can follow these calculations: ### Step 1: Define the Variables Let the cost price of one book be \( x \). ### Step 2: Calculate the Total Cost for 6 Copies The total cost for 6 copies of the book, payable at the end of the year, is: \[ \text{Total cost for 6 copies} = 6x \] ### Step 3: Calculate the Total Cost for 7 Copies The total cost for 7 copies of the book, paid in cash, is: \[ \text{Total cost for 7 copies} = 7x \] ### Step 4: Understand the Interest Concept Since the total cost for 6 copies is paid at the end of the year, it includes the principal amount plus the interest for one year. The total cost for 7 copies is paid immediately in cash, so it does not include any interest. ### Step 5: Set Up the Equation The amount paid for 6 copies at the end of the year can be expressed as: \[ 6x = 7x + \text{Interest} \] The interest can be calculated using the formula for simple interest: \[ \text{Interest} = \text{Principal} \times \text{Rate} \times \text{Time} \] Here, the principal is \( 7x \), the rate is \( r \) (in decimal), and the time is 1 year. Therefore, we can write: \[ \text{Interest} = 7x \cdot r \cdot 1 = 7xr \] ### Step 6: Substitute the Interest into the Equation Substituting the expression for interest into the equation gives: \[ 6x = 7x + 7xr \] ### Step 7: Rearranging the Equation Rearranging the equation, we get: \[ 6x - 7x = 7xr \] \[ -x = 7xr \] ### Step 8: Solve for the Rate \( r \) Dividing both sides by \( x \) (assuming \( x \neq 0 \)): \[ -1 = 7r \] Thus, \[ r = -\frac{1}{7} \] ### Step 9: Convert to Percentage To express the rate as a percentage, we multiply by 100: \[ \text{Rate} = -\frac{1}{7} \times 100 \approx -14.2857\% \] ### Step 10: Correct to One Decimal Place Since we are looking for the rate per annum of simple interest, we take the absolute value and round it to one decimal place: \[ \text{Rate} \approx 14.3\% \] ### Final Answer The rate per annum of simple interest is approximately **14.3%**. ---
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