### CAT 1996 Question Paper Question 25

Instructions

Answer the questions based on the following information. A salesman enters the quantity sold and the price into the computer. Both the numbers are two-digit numbers. But, by mistake, both the numbers were entered with their digits interchanged.Although, the total sales value remained the same, i.e. Rs. 1,148, the total inventory sold got reduced by 54.

Question 25

# What is the actual quantity sold?

Solution

Total sales value $$= 1148 = 4\cdot7\cdot41$$
Let AB be the actual price of the product and CD be the actual quantity.

Since the quantity/inventory increased by 54 upon reversing
$$CD-DC\ =\ (10\cdot C+D)-\left(10\cdot D+C\right)=9\cdot\left(C-D\right)$$
wkt, $$9\cdot\left(C-D\right)=54\ \ \longrightarrow\ \ C-D=6$$

Now, wkt, $$AB\cdot CD=1148\ \ \&\ \ BA\cdot DC=1148$$
The possible values of CD are $$93, 82$$ & $$71$$
Since only 82 divides 1148, $$CD=82$$, Therefore, $$AB=14$$
Actual price and Actual quantity are 14 and 82 respectively.

Alternatively,
Since final value $$= 1148 = 4\cdot7\cdot41$$ remains the same,
1148 can be represented as product of interchangeable numbers i.e. $$41 \times 28$$ and $$82 \times 14$$
As inventory decreased by 54, so, the quantity sold should be 82 and actual price will be 14.