At the book sale, board books are $1.50 and soft cover books are $0.50.

Question

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

Answer 1

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.

Answer 2

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

Answer 3

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.

Answer 4

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

Answer 5

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.

Answer 6

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

Answer 7

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.