a survvey of vegetable gardeners showed the following:
69 grew tomatoes
25 grew cucumbers
30 grew zucchini
18 grew both tomatoes and cucumbers
21 grew both tomatoes and zucchini
14 grew both cucumber and zucchini
12 grew all 3
12 had none of these
1. how many grew only cucumbers
2. how many grew both cucumbers and tomatoes but not zucchini
3. how many grew only tomatoes
4. how many grew none of these 3 or only zucchini
5. how vegetable gardeners were surveyed
Use processes indicated in your later post.
To answer these questions, we'll need to use a method called the Principle of Inclusion-Exclusion, which allows us to calculate the number of elements in different sets when we have overlapping information.
First, let's define the symbols we'll use:
- T: Number of gardeners who grew tomatoes
- C: Number of gardeners who grew cucumbers
- Z: Number of gardeners who grew zucchini
Now, let's solve each question step-by-step using the given information:
1. How many grew only cucumbers?
To find the number of gardeners who grew only cucumbers, we need to subtract the number of gardeners who grew cucumbers and other vegetables from the total number of gardeners who grew cucumbers.
First, we know that 25 gardeners grew cucumbers (C). We can subtract the gardeners who grew both cucumbers and tomatoes (18) and the gardeners who grew all three (12).
So, the number of gardeners who grew only cucumbers is: C - (C ∩ T) - (C ∩ Z) - (C ∩ T ∩ Z)
Substituting the given values, we get: 25 - 18 - 14 - 12 = 25 - 44 = -19
However, a negative value for gardeners is not possible. This implies that there was an error in the given information or that we made a mistake in our calculations.
2. How many grew both cucumbers and tomatoes but not zucchini?
To find the number of gardeners who grew both cucumbers and tomatoes but not zucchini, we need to subtract the number of gardeners who grew all three from the number of gardeners who grew both cucumbers and tomatoes.
We are given that 12 gardeners grew all three (C ∩ T ∩ Z). Therefore, the number of gardeners who grew both cucumbers and tomatoes is 18 (C ∩ T) - 12 (C ∩ T ∩ Z) = 6.
3. How many grew only tomatoes?
To find the number of gardeners who grew only tomatoes, we need to subtract the gardeners who grew tomatoes and other vegetables from the total number of gardeners who grew tomatoes.
First, we know that 69 gardeners grew tomatoes (T). We can subtract the gardeners who grew both tomatoes and cucumbers (18) and the gardeners who grew both tomatoes and zucchini (21).
So, the number of gardeners who grew only tomatoes is: T - (T ∩ C) - (T ∩ Z) - (T ∩ C ∩ Z)
Substituting the given values, we get: 69 - 18 - 21 - 12 = 69 - 51 = 18.
4. How many grew none of these three or only zucchini?
To find the number of gardeners who grew none of these three vegetables or only zucchini, we need to subtract the gardeners who grew either of the other two vegetables or all three from the total number of gardeners.
We know that 12 gardeners grew all three (C ∩ T ∩ Z). Additionally, we have 12 gardeners who grew none of these three.
So, the number of gardeners who grew none of these three or only zucchini is: 12 + (Z - (C ∩ Z) - (T ∩ Z) - (C ∩ T ∩ Z))
Substituting the given values, we get: 12 + (30 - 14 - 21 - 12) = 12 + 30 - 47 = -5.
Similar to the first question, we obtained a negative value which is not possible, indicating an error in the given information or in our calculations.
5. How were the vegetable gardeners surveyed?
The provided information does not specify how the vegetable gardeners were surveyed. If the survey method is relevant to answering the questions or if additional information is required, it would be necessary to obtain that information.