Math Problem Statement

In a market of food stalls there were three food stalls of Ajay, Raj and Mohit who were selling different fast foods and it is found at the end of the day that total 600 customers have visited market. 50% customers purchased from Ajay; 55% customers purchased from Raj; 45% customers purchased from Mohit. 20% of customers who purchased from Ajay also purchased from other two stalls. Customers who purchased from any of two stalls are 190. Customers of only Ajay are 40 more than that of only Raj. Customers who purchased only from Ajay and Raj are 80.

Solution

Let's break down the information and set up equations based on the data provided.

Define:

  • Let AA, RR, and MM be the sets of customers who bought from Ajay, Raj, and Mohit, respectively.

Given information:

  1. Total customers in the market: ARM=600|A \cup R \cup M| = 600.
  2. Customers who bought from Ajay: A=50%×600=300|A| = 50\% \times 600 = 300.
  3. Customers who bought from Raj: R=55%×600=330|R| = 55\% \times 600 = 330.
  4. Customers who bought from Mohit: M=45%×600=270|M| = 45\% \times 600 = 270.
  5. Customers who bought from all three stalls: ARM=20%×300=60|A \cap R \cap M| = 20\% \times 300 = 60.
  6. Customers who bought from any two stalls: AR+RM+MA2ARM=190|A \cap R| + |R \cap M| + |M \cap A| - 2 |A \cap R \cap M| = 190.
  7. Customers who only bought from Ajay are 40 more than those who only bought from Raj.
  8. Customers who bought only from Ajay and Raj: ARARM=80|A \cap R| - |A \cap R \cap M| = 80.

Let's solve step-by-step.

Step 1: Set up the information in equations

Let:

  • AR=x|A \cap R| = x
  • RM=y|R \cap M| = y
  • MA=z|M \cap A| = z

From information given:

  1. ARM=60|A \cap R \cap M| = 60.
  2. x+y+z2×60=190x + y + z - 2 \times 60 = 190.
    • Simplifying, x+y+z=310x + y + z = 310.
  3. x60=80x - 60 = 80, so x=140x = 140.

Step 2: Substitute values

Using x=140x = 140:

  1. 140+y+z=310140 + y + z = 310 gives y+z=170y + z = 170.

Step 3: Use the remaining information

Let:

  • Customers who only bought from Ajay be aa,
  • Customers who only bought from Raj be rr,
  • Customers who only bought from Mohit be mm.

We know:

  1. a=r+40a = r + 40.
  2. From the inclusion-exclusion principle: ARM=a+r+m+x+y+z+60=600|A \cup R \cup M| = a + r + m + x + y + z + 60 = 600 Substituting known values gives: a+r+m+140+170+60=600a + r + m + 140 + 170 + 60 = 600 Simplifying, a+r+m=230a + r + m = 230

Step 4: Solve for individual values of aa, rr, and mm

Using a=r+40a = r + 40:

  1. Substitute a=r+40a = r + 40 in a+r+m=230a + r + m = 230: (r+40)+r+m=230(r + 40) + r + m = 230 2r+m=1902r + m = 190

Now we need further equations or assumptions to solve fully for each

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Math Problem Analysis

Mathematical Concepts

Set Theory
Inclusion-Exclusion Principle
Percentage
Systems of Equations

Formulas

Inclusion-Exclusion Principle: |A ∪ B ∪ C| = |A| + |B| + |C| - |A ∩ B| - |B ∩ C| - |C ∩ A| + |A ∩ B ∩ C|
Percentage calculation: Part = (Percentage / 100) * Total
System of equations for solving unknowns

Theorems

Inclusion-Exclusion Principle

Suitable Grade Level

Grades 10-12