Question

A shop sells bags in three sizes: small, medium and large. A large bag costs Rs.1000, a medium bag costs Rs.200, and a small bag costs Rs.50. Three buyers, Ashish, Banti and Chintu, independently buy some numbers of these types of bags. The respective amounts spent by Ashish, Banti and Chintu are equal. Put together, the shop sells 1 large bag, 15 small bags and some medium bags to these three buyers. What is the minimum number of medium bags that the shop sells to them?

• A. 5
• B. 9
• C. 4
• D. 10
• E. 7

Answer 1

Answer: (E)

Given that the total amounts spent by Ashish, Banti and Chintu are equal, we denote this common amount as \( x \). The shop sells 1 large bag, 15 small bags, and some medium bags. The cost of a large bag is Rs.1000, a medium bag is Rs.200, and a small bag is Rs.50. We express the total cost in terms of these variables: 

Let \( M \) be the total number of medium bags sold. The total cost of the bags sold is expressed as:

x3=(1*1000)+(M*200)+(15*50)

Simplifying, we have:

x3=1000+200M+750

x3=1750+200M

Since the amount spent by each buyer, \( x \), is equal, we divide the total cost by 3:

x=1750+200M3

\( x \) must be an integer as the amount spent is the same for each buyer. For this quotient to be an integer, \( 1750 + 200M \) must be divisible by 3.

Now, calculating \( 1750 \mod 3 \):

1750=1749+1

1749 \mod 3 = 0, so \( 1750 \equiv 1 \mod 3 \).

Now, let \( 200M \equiv 2 \mod 3 \). Since \( 200 \equiv 2 \mod 3 \), it implies \( 200M \equiv 2 \cdot M \mod 3 \).

For \( 200M \equiv 2 \mod 3 \), \( 2M \equiv 2 \mod 3 \), further simplifies to \( M \equiv 1 \mod 3 \).

M	Check
1	No
2	No
3	No
4	Yes

The smallest value of \( M \) that satisfies \( M \equiv 1 \mod 3 \) and allows for divisors of 3 is \( M = 7 \).

Hence, the minimum number of medium bags is \( 7 \).