Question

The average number of copies of a book sold per day by a shopkeeper is 60 in the initial seven days and 63 in the initial eight days, after the book launch. On the ninth day, she sells 11 copies less than the eighth day, and the average number of copies sold per day from the second day to the ninth day becomes 66. The number of copies sold on the first day of the book launch is:

Hint:

When dealing with averages over overlapping time intervals, convert each average to a total sum. Then, use differences of these sums to find individual day values and set up equations to solve for the unknowns.

Answer 1

Correct Answer: 59

Let $x_i$ denote the number of copies sold on day $i$, and let $S_n$ be the total copies sold in the first $n$ days. Step 1: Use the given averages for the first $7$ and $8$ days. Average for first $7$ days is $60$: \[ S_7 = 7 \times 60 = 420. \] Average for first $8$ days is $63$: \[ S_8 = 8 \times 63 = 504. \]
Step 2: Find the sales on the $8$th day. \[ x_8 = S_8 - S_7 = 504 - 420 = 84. \]
Step 3: Find the sales on the $9$th day. On the $9$th day, she sells $11$ copies less than on the $8$th day: \[ x_9 = x_8 - 11 = 84 - 11 = 73. \]
Step 4: Use the average from day $2$ to day $9$. From day $2$ to day $9$ there are $8$ days, and the average is $66$: \[ S_{2\text{--}9} = 8 \times 66 = 528. \]
Step 5: Relate totals to find $x_1$. Total sales in the first $9$ days: \[ S_9 = S_8 + x_9 = 504 + 73 = 577. \] But $S_9$ is also \[ S_9 = x_1 + S_{2\text{--}9} = x_1 + 528. \] So, \[ x_1 + 528 = 577 \;\Rightarrow\; x_1 = 577 - 528 = 49. \] Therefore, the number of copies sold on the first day is \(49\).