Question

Jasmine is reviewing recent orders at her deli to determine which meats she should order. She found that of 1,000 orders, 550 customers ordered turkey, 305 customers ordered ham, and 225 customers ordered neither turkey nor ham. Based on this data, how many of the next 1,000 customers will order both turkey and ham? Show your work, and use complete sentences.

Answer 1

Let x denote the number who order turkey but not ham. On the diagram, it's the non-intersecting part of one of the circles. 

Let y denote the number who order ham but not turkey. On the diagram, it's the non-intersecting part of the other circle. 

The intersection of the circles by definition represents the number who order both turkey and ham. It is denoted on the diagram by x ∩ y. Let's call it z. 

The rectangle denotes all of the 1000 orders; the part of the rectangle outside of the circles is the number of customers who order neither turkey nor ham. Let's denote this region as w. 

Using the given info, we have the following equations to solve: 

x + y + z + w = 1000 
x + z = 550 
y + z = 305 
w = 225 

So we immediately know the value of w. Put it into the first equation: 

x + y + z + w = 1000 
x + y + z = 1000 - 225 = 775 

so now we have the system 

x + y + z = 775 
x + z = 550 
y + z = 305 

Add the second and third equations, and subtract the first equation from the result: 

x + y + 2z = 855 
x + y + 1z = 775 
------------------------ 
. . . . . . z = 80 

So we can expect 80 of the next 1000 customers to order both ham and turkey. That's the question that was asked. 

You can find the values of the others as follows: 

Put 80 for z in the second and third equations and solve for x and y: 

x + 80 = 550 
x = 470 

y + 80 = 305 
y = 225 

So the full solution is x = 470, y = 225, z = 80, w = 225.

The final answer is 80 will order both turkey and ham.