Question
The carnivals arriving in the neighborhood Leo has saved up $15 to spend of the carnival of food and attractions Food items cost $4 each and attractions cost $2 each Leo uses the expression for 4a + 2B to determine the total cost for food items and be attractions What combination of food items and attractions that Leo can afford with is $15 budget
Answers
GPT-4o mini
Let's break down the expression you provided: \(4a + 2b\), where:
- \(a\) represents the number of food items Leo buys (costing $4 each),
- \(b\) represents the number of attractions Leo attends (costing $2 each).
Leo has a total of $15 to spend, so we can set up the inequality:
\[
4a + 2b \leq 15
\]
We can also express it in terms of \(b\):
\[
2b \leq 15 - 4a \quad \Rightarrow \quad b \leq \frac{15 - 4a}{2}
\]
Now, we can find various combinations of \(a\) and \(b\) that satisfy this inequality. Let's explore the possible values of \(a\):
1. **If \(a = 0\)**:
\[
4(0) + 2b \leq 15 \quad \Rightarrow \quad 2b \leq 15 \quad \Rightarrow \quad b \leq 7.5
\]
Since \(b\) must be a whole number, \(b\) can be from 0 to 7. (7 options)
2. **If \(a = 1\)**:
\[
4(1) + 2b \leq 15 \quad \Rightarrow \quad 4 + 2b \leq 15 \quad \Rightarrow \quad 2b \leq 11 \quad \Rightarrow \quad b \leq 5.5
\]
So, \(b\) can be from 0 to 5. (6 options)
3. **If \(a = 2\)**:
\[
4(2) + 2b \leq 15 \quad \Rightarrow \quad 8 + 2b \leq 15 \quad \Rightarrow \quad 2b \leq 7 \quad \Rightarrow \quad b \leq 3.5
\]
So, \(b\) can be from 0 to 3. (4 options)
4. **If \(a = 3\)**:
\[
4(3) + 2b \leq 15 \quad \Rightarrow \quad 12 + 2b \leq 15 \quad \Rightarrow \quad 2b \leq 3 \quad \Rightarrow \quad b \leq 1.5
\]
So, \(b\) can be from 0 to 1. (2 options)
5. **If \(a = 4\)**:
\[
4(4) + 2b \leq 15 \quad \Rightarrow \quad 16 + 2b \leq 15
\]
This means \(b\) must be negative, which is not possible, so this case does not work.
Now let's summarize viable combinations of \(a\) and \(b\):
- For \(a = 0\): \(b = 0\) to \(7\) (8 combinations)
- For \(a = 1\): \(b = 0\) to \(5\) (6 combinations)
- For \(a = 2\): \(b = 0\) to \(3\) (4 combinations)
- For \(a = 3\): \(b = 0\) to \(1\) (2 combinations)
### Summary of Combinations:
1. \((0, 0), (0, 1), (0, 2), (0, 3), (0, 4), (0, 5), (0, 6), (0, 7)\)
2. \((1, 0), (1, 1), (1, 2), (1, 3), (1, 4), (1, 5)\)
3. \((2, 0), (2, 1), (2, 2), (2, 3)\)
4. \((3, 0), (3, 1)\)
Therefore, Leo can afford a variety of combinations of food items and attractions with his $15 budget, ranging from all attractions to a mix of food and attractions.
- \(a\) represents the number of food items Leo buys (costing $4 each),
- \(b\) represents the number of attractions Leo attends (costing $2 each).
Leo has a total of $15 to spend, so we can set up the inequality:
\[
4a + 2b \leq 15
\]
We can also express it in terms of \(b\):
\[
2b \leq 15 - 4a \quad \Rightarrow \quad b \leq \frac{15 - 4a}{2}
\]
Now, we can find various combinations of \(a\) and \(b\) that satisfy this inequality. Let's explore the possible values of \(a\):
1. **If \(a = 0\)**:
\[
4(0) + 2b \leq 15 \quad \Rightarrow \quad 2b \leq 15 \quad \Rightarrow \quad b \leq 7.5
\]
Since \(b\) must be a whole number, \(b\) can be from 0 to 7. (7 options)
2. **If \(a = 1\)**:
\[
4(1) + 2b \leq 15 \quad \Rightarrow \quad 4 + 2b \leq 15 \quad \Rightarrow \quad 2b \leq 11 \quad \Rightarrow \quad b \leq 5.5
\]
So, \(b\) can be from 0 to 5. (6 options)
3. **If \(a = 2\)**:
\[
4(2) + 2b \leq 15 \quad \Rightarrow \quad 8 + 2b \leq 15 \quad \Rightarrow \quad 2b \leq 7 \quad \Rightarrow \quad b \leq 3.5
\]
So, \(b\) can be from 0 to 3. (4 options)
4. **If \(a = 3\)**:
\[
4(3) + 2b \leq 15 \quad \Rightarrow \quad 12 + 2b \leq 15 \quad \Rightarrow \quad 2b \leq 3 \quad \Rightarrow \quad b \leq 1.5
\]
So, \(b\) can be from 0 to 1. (2 options)
5. **If \(a = 4\)**:
\[
4(4) + 2b \leq 15 \quad \Rightarrow \quad 16 + 2b \leq 15
\]
This means \(b\) must be negative, which is not possible, so this case does not work.
Now let's summarize viable combinations of \(a\) and \(b\):
- For \(a = 0\): \(b = 0\) to \(7\) (8 combinations)
- For \(a = 1\): \(b = 0\) to \(5\) (6 combinations)
- For \(a = 2\): \(b = 0\) to \(3\) (4 combinations)
- For \(a = 3\): \(b = 0\) to \(1\) (2 combinations)
### Summary of Combinations:
1. \((0, 0), (0, 1), (0, 2), (0, 3), (0, 4), (0, 5), (0, 6), (0, 7)\)
2. \((1, 0), (1, 1), (1, 2), (1, 3), (1, 4), (1, 5)\)
3. \((2, 0), (2, 1), (2, 2), (2, 3)\)
4. \((3, 0), (3, 1)\)
Therefore, Leo can afford a variety of combinations of food items and attractions with his $15 budget, ranging from all attractions to a mix of food and attractions.