## Sets and Unions

**A shop sells three type of products i.e. Pen, Pencil & Notebook. On a survey for checking the sales of each product of the shop he found that the no. of people who bought only pen, only pencil, & only notebook are in A.P. in no particular order. Similarly, the number of people who bought exactly two of the three products are in A.P. too.**

It was also found that the no. of people who bought all the products is 1/20th of the number of people who bought pencil only which in turn is equal to half of the number of people who bought notebook only. The number of people that bought both pen & pencil is 15, whereas that of those who bought pencil & notebook is 19. The number of people who bought notebook are 120, which is more than the no. of people who bought pen (which is a 2 digit no above 50).

1) What is the total no people that visits the shop?

a) 220 b) 231 c) 233 d) 240

2) How many people bought both pen and notebook?

a) 15 b) 17 c) 19 d) 21

*Detailed* Solution

Let the no of people that bought all the three products be ‘x’, and no of people that bought both pen and notebook be y .

Then no of people that bought pencil only = 20 x

And those who bought notebook only = 40 x

Venn diagram according to question can be drawn as:

Thus from the figure we get 40x + (19 -x) + x + y = 120

=) 40x + y = 101

Now if we take a close look at the equation we can conclude that the value x can be either ‘1’ or ‘2’

Putting x = 1 in the equation we get y =61. But we have to also meet the additional condition that 15- x, 19-x, y are in A.P. which is not possible in this case. Hence it is rejected.

Thus the value of x = 2, hence we will get y = 21, which also satisfies the additional condition.

Now if we see the figure again we can see that only pen can either be 0, 60, and 120 to satisfy the additional condition. And as it cannot be less than 50 (Specified in the question), the only value only pencil can take = 60

Thus 1) Total no of people that visited the shop = 60 + 13 + 2 + 40 + 17 + 80 + 21 = 233

2) The no people that bought both pen and notebook = 21

**Correct Answer:** 233, 21

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