Q:  Superburger sells hamburgers with the choice of ketchup, mustard and relish. One day they sold 256 hamburgers; 140 had mustard, 140 had ketchup, 84 had ketchup and relish, 62 had mustard but no relish, 68 had ketchup and mustard, 38 had all three condiments and 20 had none.

(a) The number sold with relish only is?

(b)The number sold with no relish is?

A:  This problem would be best solved with a Venn Diagram.  You should be drawing it yourself as you read along…

Draw 3 circles that all over-lap:

Circle K:  Burgers with Ketchup

Circle R:  Burgers with Relish

Circle M:  Burgers with Mustard

Like so:

Now, we always want to fill Venn Diagrams with the most complicated information first.

38 had all three condiments.  Put that number in like so:

Now fill in these pieces (I am going out of order on purpose to make sure we get it right):

84 had ketchup and relish; 68 had ketchup and mustard; 20 had none.

Note:  84 had ketchup and relish.  The total number in the section that overlaps K and R should be 84.  38 is already in that circle, so 84 – 38 = 46 should be in the remaining piece.

68 had ketchup and mustard… The K and M overlap should total to 68… So, 68 – 38 = 30 should go in the remaining.  Look:

The 20 goes on the outside because it represents the burgers with nothing.

Now let’s add these pieces of info:

140 had ketchup; 62 had mustard but no relish.

The K circle should total up to 140… So 140 – 30 – 38 – 46 = 26 is the number that should go in the remaining K spot.

62 should be the total in the M but no R section…  There is already 30 in that section, so 62 – 30 = 32 should go in the M but no R section… Look:

Now, use this piece of info:

140 had mustard

In the remaining Mustard slot should go 140 – 32 – 38 – 30 = 40.

Now, for the final space…. We know there are 256 total hamburgers… So, we need to take 256 subtract all of the numbers to see how may are left for that last space….

256 – 20 – 40 – 46 – 38 – 30 – 32 – 26 = 24

Fill it in!  Almost done:

The Venn Diagram is complete.  Now we can answer the questions:

(a) The number sold with relish only is?

The number that is in the R circle but no other circle:  24.

(b)The number sold with no relish is?

Every number not in the R circle:  20 + 32 + 30 + 26 = 108.