After a genetics experiment on 67 pea plants, the number of plants having certain characteristics was tallied, with the results given below. Answer parts (a) through (c).
22 were tall
35 had green peas
42 had smooth peas
12 were tall and had green peas
20 had green peas and smooth peas
6 had all 3 characteristics
7 had none of the characteristics
A- find the number of plants that were tall and had smooth peas
B- How many plants were tall and had peas that were neither smooth nor green
C- how many plants were not tall but had peas that were smooth and green?
I know Im supposed to use a venn diagram set up but Im stuck on how to find the other numbers.
To solve this problem, we can use a Venn diagram to visualize the relationships between the characteristics of the pea plants. Let's start by creating a Venn diagram with three intersecting circles: one for tall plants, one for plants with green peas, and one for plants with smooth peas.
Based on the given information, we can fill in the following values in the Venn diagram:
- The number of plants that are tall is 22.
- The number of plants with green peas is 35.
- The number of plants with smooth peas is 42.
- The number of plants that are tall and have green peas is 12.
- The number of plants that have green peas and smooth peas is 20.
- The number of plants that have all three characteristics is 6.
- The number of plants that have none of the characteristics is 7.
Now let's solve for parts (a) through (c) of the question:
A) To find the number of plants that are tall and have smooth peas, we need to find the overlapping region between the tall plants and plants with smooth peas. Looking at the Venn diagram, this corresponds to the region where the circles for tall plants and smooth peas intersect.
To determine this value, we can subtract the number of plants that have all three characteristics from the number of plants that have green peas and smooth peas: 20 - 6 = 14. Therefore, there are 14 plants that are tall and have smooth peas.
B) To find the number of plants that are tall and have peas that are neither smooth nor green, we need to exclude the overlapping regions with green peas and smooth peas from the total number of tall plants. In other words, we need to subtract the number of plants that have all three characteristics and the number of plants with green peas and smooth peas from the total number of tall plants.
To determine this value, we can subtract the sum of the number of plants with all three characteristics and the number of plants with green peas and smooth peas from the total number of tall plants: 22 - (6 + 12) = 4. Therefore, there are 4 plants that are tall and have peas that are neither smooth nor green.
C) To find the number of plants that are not tall but have peas that are smooth and green, we need to subtract the overlapping region with tall plants from the number of plants that have green peas and smooth peas. Looking at the Venn diagram, this corresponds to the region where the circles for green peas and smooth peas intersect but exclude the tall plants region.
To determine this value, we can subtract the number of plants that have all three characteristics from the number of plants with green peas and smooth peas: 20 - 6 = 14. Therefore, there are 14 plants that are not tall but have peas that are smooth and green.
Using the Venn diagram and the calculations above, we can answer parts (a) through (c) of the question.
To find the other numbers, we can use the information given and set up a Venn diagram. Let's go step-by-step:
Step 1: Draw a Venn diagram with three intersecting circles representing tall plants, green peas, and smooth peas. Label the regions as: tall (T), green peas (G), and smooth peas (S).
Step 2: Begin by filling in the values you already know:
- From the information given, we know that the total number of pea plants observed is 67.
- We also know that 7 of them had none of the characteristics, so we can place this value outside of all circles.
Step 3: Use the information given to fill in the overlapping regions, working from the center to the outer regions:
- We know that 6 plants had all three characteristics, so place 6 in the region where all three circles intersect (TGS).
- We know that 20 plants had green and smooth peas but were not necessarily tall. Place this value in the intersection of the green peas circle (G) and the smooth peas circle (S), but outside the tall plant circle (T).
- Additionally, we know that 12 plants were tall and had green peas. Since we have already accounted for the plants with all three characteristics, we need to put 6 in the intersection of the tall plant circle (T) and the green peas circle (G) that does not intersect with the smooth peas circle (S).
Step 4: Fill in the remaining regions:
- Since we have 35 plants with green peas, and we have placed 6 in the intersection of the tall plant circle (T) and the green peas circle (G) without the smooth peas circle (S), we are left with 35 - 6 = 29 plants that have green peas and smooth peas but are not necessarily tall. Place this value in the intersection of the green peas circle (G) and the smooth peas circle (S) without the tall plant circle (T).
- We know that 42 plants had smooth peas, and we have already placed 6 in the intersection of the tall plant circle (T), green peas circle (G), and smooth peas circle (S). This means there are 42 - 6 = 36 plants that have smooth peas but are not necessarily tall or have green peas. Place this value in the region of the smooth peas circle (S) that does not intersect with the tall plant circle (T) or the green peas circle (G).
Step 5: Calculate the missing values using the information provided:
(a) To find the number of plants that were tall and had smooth peas, we need to add up the counts in the intersection of the tall plant circle (T) and the smooth peas circle (S). From the Venn diagram, we can see that this value is 6.
(b) To find the number of plants that were tall and had peas that were neither smooth nor green, we need to subtract the counts in the intersections of the tall plant circle (T) and the green peas circle (G), and the tall plant circle (T) and the smooth peas circle (S), from the count of tall plants (22). From the Venn diagram, we can see that this value is 22 - 6 - 6 = 10.
(c) To find the number of plants that were not tall, but had peas that were smooth and green, we need to add up the counts in the intersection of the green peas circle (G) and the smooth peas circle (S). From the Venn diagram, we can see that this value is 29.
So, the answers are:
(a) The number of plants that were tall and had smooth peas is 6.
(b) The number of plants that were tall and had peas that were neither smooth nor green is 10.
(c) The number of plants that were not tall, but had peas that were smooth and green is 29.