Venn diagram
Dave's sister baked 3 dozen pies of which half contained chocolate, two thirds contained marshmallows, three-fourths contained cayenne, and one-sixth contained salted soy nuts. What is the largest possible number of pies that had none of these ingredients?
If 27 contained all three, then 9 could contain none.
To find the largest possible number of pies that had none of the mentioned ingredients, we need to consider the overlapping conditions of the ingredients.
Let's start by finding the number of pies that contained each individual ingredient:
- Half contained chocolate: (1/2) * 3 dozen = 1.5 dozen
- Two-thirds contained marshmallows: (2/3) * 3 dozen = 2 dozen
- Three-fourths contained cayenne: (3/4) * 3 dozen = 2.25 dozen
- One-sixth contained salted soy nuts: (1/6) * 3 dozen = 0.5 dozen
To find the largest possible number of pies with none of these ingredients, we need to subtract the total number of pies with any of the mentioned ingredients from the total number of pies:
Total number of pies = 3 dozen = 3
Total number of pies with any of the mentioned ingredients = 1.5 dozen + 2 dozen + 2.25 dozen + 0.5 dozen = 6.25 dozen
Number of pies with none of these ingredients = Total number of pies - Total number of pies with any of the mentioned ingredients
= 3 dozen - 6.25 dozen
Since we can't have a negative number of pies, the result is 0.
Therefore, the largest possible number of pies that had none of these ingredients is 0.
To find the largest possible number of pies that had none of these ingredients, we can use a Venn diagram.
Step 1: Draw three overlapping circles to represent chocolate, marshmallows, and cayenne ingredients.
Step 2: Label the regions of the Venn diagram as follows:
- Let "C" represent the region for chocolate pies.
- Let "M" represent the region for marshmallow pies.
- Let "CA" represent the region for cayenne pies.
Step 3: Fill in the given information inside the appropriate regions:
- Half contained chocolate, so shade the "C" region to show that it covers 50% of the total pies.
- Two-thirds contained marshmallows, so shade the "M" region to show that it covers 66.67% of the total pies.
- Three-fourths contained cayenne, so shade the "CA" region to show that it covers 75% of the total pies.
- There is no specific information about salted soy nuts, so you can leave that region blank.
Step 4: To find the largest possible number of pies that had none of the ingredients, find the region that is not shaded in any of the circles. This region represents pies that don't contain any of the mentioned ingredients.
Step 5: Count the number of pies in the region that doesn't have any shading. This will give you the largest possible number of pies that had none of the ingredients.
Note: The question doesn't mention the total number of pies baked by Dave's sister. Therefore, you won't be able to determine the exact number of pies that had none of the ingredients, but you can find the largest possible number based on the given information and the Venn diagram.