Question
At the book sale, board books are $1.50 and soft cover books are $0.50.
Suppose Santiago purchased 34 books for $20. Which solution below represents how many of each book type he purchased?
Responses:
17 board books and 17 soft cover books
3 board books and 31 soft cover books
20 board books and 14 soft cover books
15 board books and 19 soft cover books
Suppose Santiago purchased 34 books for $20. Which solution below represents how many of each book type he purchased?
Responses:
17 board books and 17 soft cover books
3 board books and 31 soft cover books
20 board books and 14 soft cover books
15 board books and 19 soft cover books
Answers
GPT-4o mini
Let's define the variables:
- Let \( b \) be the number of board books.
- Let \( s \) be the number of soft cover books.
From the problem, we have two equations based on the total number of books and the total cost:
1. \( b + s = 34 \) (total number of books)
2. \( 1.5b + 0.5s = 20 \) (total cost of books)
Now, we can simplify the second equation. Multiply the entire equation by 2 to eliminate the decimal points:
\( 3b + s = 40 \).
Now we have the following system of equations:
1. \( b + s = 34 \)
2. \( 3b + s = 40 \)
Next, we can subtract the first equation from the second:
\[
(3b + s) - (b + s) = 40 - 34
\]
This simplifies to:
\[
2b = 6
\]
Dividing both sides by 2 gives:
\[
b = 3
\]
Now substitute \( b = 3 \) back into the first equation to find \( s \):
\[
3 + s = 34
\]
Solving for \( s \):
\[
s = 34 - 3 = 31
\]
Thus, Santiago purchased 3 board books and 31 soft cover books.
Now we can check the options provided:
- **17 board books and 17 soft cover books** (not possible)
- **3 board books and 31 soft cover books** (correct)
- **20 board books and 14 soft cover books** (not possible)
- **15 board books and 19 soft cover books** (not possible)
Therefore, the answer is:
**3 board books and 31 soft cover books.**
- Let \( b \) be the number of board books.
- Let \( s \) be the number of soft cover books.
From the problem, we have two equations based on the total number of books and the total cost:
1. \( b + s = 34 \) (total number of books)
2. \( 1.5b + 0.5s = 20 \) (total cost of books)
Now, we can simplify the second equation. Multiply the entire equation by 2 to eliminate the decimal points:
\( 3b + s = 40 \).
Now we have the following system of equations:
1. \( b + s = 34 \)
2. \( 3b + s = 40 \)
Next, we can subtract the first equation from the second:
\[
(3b + s) - (b + s) = 40 - 34
\]
This simplifies to:
\[
2b = 6
\]
Dividing both sides by 2 gives:
\[
b = 3
\]
Now substitute \( b = 3 \) back into the first equation to find \( s \):
\[
3 + s = 34
\]
Solving for \( s \):
\[
s = 34 - 3 = 31
\]
Thus, Santiago purchased 3 board books and 31 soft cover books.
Now we can check the options provided:
- **17 board books and 17 soft cover books** (not possible)
- **3 board books and 31 soft cover books** (correct)
- **20 board books and 14 soft cover books** (not possible)
- **15 board books and 19 soft cover books** (not possible)
Therefore, the answer is:
**3 board books and 31 soft cover books.**
Yes.
Use the following scenario to solve questions 8-10.
Meha is helping her aunt to plan a birthday party for her cousin at the science center. Her aunt tells her that the party will cost $150, plus $10.50 per child and $5.50 per adult. There needs to be at least one adult and one child at the party. She has budgeted $300 for the party and that is the maximum amount she wants to spend. Which inequality represents the situation described?
Responses
10.50c + 5.50a + 150 ≤ 300
10.50c + 5.50a + 150 ≤ 300
16x + 150 ≤ 300
16x + 150 ≤ 300
10.50c + 5.50a ≤ 300
10.50c + 5.50a ≤ 300
10.50c + 5.50a + 150 ≥ 300
Meha is helping her aunt to plan a birthday party for her cousin at the science center. Her aunt tells her that the party will cost $150, plus $10.50 per child and $5.50 per adult. There needs to be at least one adult and one child at the party. She has budgeted $300 for the party and that is the maximum amount she wants to spend. Which inequality represents the situation described?
Responses
10.50c + 5.50a + 150 ≤ 300
10.50c + 5.50a + 150 ≤ 300
16x + 150 ≤ 300
16x + 150 ≤ 300
10.50c + 5.50a ≤ 300
10.50c + 5.50a ≤ 300
10.50c + 5.50a + 150 ≥ 300
GPT-4o mini
To find the correct inequality that represents the situation described, let's break down the costs involved in the party:
- Fixed cost: $150
- Cost per child: $10.50 (let's denote the number of children as \( c \))
- Cost per adult: $5.50 (let's denote the number of adults as \( a \))
The total cost of the party can thus be expressed as:
\[
\text{Total Cost} = 150 + 10.50c + 5.50a
\]
The aunt has a budget of $300, meaning the total cost must not exceed this budget. Therefore, the appropriate inequality to represent the situation is:
\[
150 + 10.50c + 5.50a \leq 300
\]
Simplifying this gives us:
\[
10.50c + 5.50a + 150 \leq 300
\]
Now, let’s see which of the provided responses match this inequality:
1. **10.50c + 5.50a + 150 ≤ 300** (Correct)
2. **16x + 150 ≤ 300** (Not applicable)
3. **10.50c + 5.50a ≤ 300** (Incorrect, missing the fixed cost)
4. **10.50c + 5.50a + 150 ≥ 300** (Incorrect, it should be ≤)
So the correct response is:
**10.50c + 5.50a + 150 ≤ 300**.
- Fixed cost: $150
- Cost per child: $10.50 (let's denote the number of children as \( c \))
- Cost per adult: $5.50 (let's denote the number of adults as \( a \))
The total cost of the party can thus be expressed as:
\[
\text{Total Cost} = 150 + 10.50c + 5.50a
\]
The aunt has a budget of $300, meaning the total cost must not exceed this budget. Therefore, the appropriate inequality to represent the situation is:
\[
150 + 10.50c + 5.50a \leq 300
\]
Simplifying this gives us:
\[
10.50c + 5.50a + 150 \leq 300
\]
Now, let’s see which of the provided responses match this inequality:
1. **10.50c + 5.50a + 150 ≤ 300** (Correct)
2. **16x + 150 ≤ 300** (Not applicable)
3. **10.50c + 5.50a ≤ 300** (Incorrect, missing the fixed cost)
4. **10.50c + 5.50a + 150 ≥ 300** (Incorrect, it should be ≤)
So the correct response is:
**10.50c + 5.50a + 150 ≤ 300**.
Yes.
The party will cost $150, plus $10.50 per child and $5.50 per adult. There needs to be at least one adult and one child at the party, and the cost cannot exceed $300.
True or false, the party can include 10 children and 5 adults.
Responses
True
False
True or false, the party can include 10 children and 5 adults.
Responses
True
False
GPT-4o mini
To determine whether the party can include 10 children and 5 adults within the budget, we'll calculate the total cost based on the given information:
- Fixed cost: $150
- Cost per child: $10.50
- Cost per adult: $5.50
We can calculate the total cost using the following formula:
\[
\text{Total Cost} = 150 + (10.50 \times \text{number of children}) + (5.50 \times \text{number of adults})
\]
For 10 children and 5 adults:
\[
\text{Total Cost} = 150 + (10.50 \times 10) + (5.50 \times 5)
\]
Calculating each part:
\[
10.50 \times 10 = 105
\]
\[
5.50 \times 5 = 27.50
\]
Now, plug these values back into the total cost calculation:
\[
\text{Total Cost} = 150 + 105 + 27.50 = 282.50
\]
Now, we check if this cost exceeds the budget of $300:
\[
282.50 ≤ 300
\]
Since the total cost of the party ($282.50) is less than the maximum budget of $300, it is within the budget.
Thus, the party can include 10 children and 5 adults.
The answer is **True**.
- Fixed cost: $150
- Cost per child: $10.50
- Cost per adult: $5.50
We can calculate the total cost using the following formula:
\[
\text{Total Cost} = 150 + (10.50 \times \text{number of children}) + (5.50 \times \text{number of adults})
\]
For 10 children and 5 adults:
\[
\text{Total Cost} = 150 + (10.50 \times 10) + (5.50 \times 5)
\]
Calculating each part:
\[
10.50 \times 10 = 105
\]
\[
5.50 \times 5 = 27.50
\]
Now, plug these values back into the total cost calculation:
\[
\text{Total Cost} = 150 + 105 + 27.50 = 282.50
\]
Now, we check if this cost exceeds the budget of $300:
\[
282.50 ≤ 300
\]
Since the total cost of the party ($282.50) is less than the maximum budget of $300, it is within the budget.
Thus, the party can include 10 children and 5 adults.
The answer is **True**.
Yes.
The party will cost $150, plus $10.50 per child and $5.50 per adult. There needs to be at least one adult and one child at the party, and the cost cannot exceed $300.
(14.286, 0) is a viable solution for the above situation. Consider the ordered pair as (c, a) where c represents number of children and a represents number of adults at the party.
Responses
True
False
(14.286, 0) is a viable solution for the above situation. Consider the ordered pair as (c, a) where c represents number of children and a represents number of adults at the party.
Responses
True
False
GPT-4o mini
To determine whether the ordered pair (14.286, 0) is a viable solution in the context of the party planning scenario, let's analyze it.
1. In the ordered pair, **c** (number of children) is 14.286, and **a** (number of adults) is 0.
2. The condition states there needs to be **at least one adult** and **at least one child** at the party. In this ordered pair, there are 0 adults, which **does not satisfy** the requirement of having at least one adult.
Given that one of the fundamental conditions—that there must be at least one adult—is violated, the ordered pair (14.286, 0) is **not** a viable solution for the party.
Thus, the answer is **False**.
1. In the ordered pair, **c** (number of children) is 14.286, and **a** (number of adults) is 0.
2. The condition states there needs to be **at least one adult** and **at least one child** at the party. In this ordered pair, there are 0 adults, which **does not satisfy** the requirement of having at least one adult.
Given that one of the fundamental conditions—that there must be at least one adult—is violated, the ordered pair (14.286, 0) is **not** a viable solution for the party.
Thus, the answer is **False**.