Question

Rational numbers m and n are plotted on the number line below.

The number line showing numbers from -4 to 4 each divided into 4 segments has point n halfway between 0 and the first segment to the left of 0 and point m halfway between third and the fourth segment to the right of 0.

Based on the number line, which statement is true?
A. n - m
The value of
is positive.

B. n + m
The value of
is positive.

C. n + m
The value of
is negative.

D. -(n - m)
The value of
is negative.

Answer 1

What is the result of this calculation?

1/2 times 4/3 + 5/6 divided by 7/8 =




A. 1 27/48


B. 1 13/21


C. 1 33/42


D. 1 19/48

Answer 2

Rashad purchased movie tickets for himself and two friends. After buying the tickets, he has $17 left over to buy snacks for the movie. Snack prices (including tax) are shown below:

Candy $1.50
Popcorn $4.00
Drink $2.75

Which of the following combinations of snacks could Rashad purchase?
A.
2 drinks and 3 popcorns

B.
3 popcorns and 3 candies

C.
4 popcorns and 1 candy

D.
4 drinks and 2 popcorns

Answer 3

Simplify:
8x + 3 (x + 5) - 5 (x - 4)
A.
6x + 35

B.
6x + 1

C.
6x - 5
D.
43x

Answer 4

To analyze the positions of m and n on the number line, let's identify their exact values based on the described locations.

1. The point n is halfway between 0 and the first segment to the left of 0. Since the number line goes from -4 to 4 and is divided into 4 segments, the positions of the segments are as follows:
- -4
- -3
- -2
- -1
- 0
- 1
- 2
- 3
- 4

The first segment to the left of 0 is -1. Therefore, halfway between 0 and -1 is:
\[
n = \frac{0 + (-1)}{2} = \frac{-1}{2} = -0.5
\]

2. The point m is halfway between the third and fourth segments to the right of 0. The segments to the right of 0 are:
- 1
- 2
- 3
- 4

The third segment is 3, and the fourth is 4. Thus, halfway between 3 and 4 is:
\[
m = \frac{3 + 4}{2} = \frac{7}{2} = 3.5
\]

Now, we have the values:
- \( n = -0.5 \)
- \( m = 3.5 \)

Now let's evaluate each option:

A. \( n - m \):
\[
n - m = -0.5 - 3.5 = -4
\]
The value is negative.

B. \( n + m \):
\[
n + m = -0.5 + 3.5 = 3
\]
The value is positive.

C. \( n + m \):
The same as above, the value is positive.

D. \( -(n - m) \):
Since we calculated \( n - m = -4 \):
\[
-(n - m) = -(-4) = 4
\]
The value is positive.

The only true statement is **B**: \( n + m \) is positive. Thus the answer is:

**B. The value of \( n + m \) is positive.**

Answer 5

To solve the expression \( \frac{1}{2} \times \frac{4}{3} + \frac{5}{6} \div \frac{7}{8} \), we need to follow the order of operations, which means we will perform the multiplication and division first, then the addition.

### Step 1: Calculate \( \frac{1}{2} \times \frac{4}{3} \)
\[
\frac{1}{2} \times \frac{4}{3} = \frac{1 \times 4}{2 \times 3} = \frac{4}{6} = \frac{2}{3}
\]

### Step 2: Calculate \( \frac{5}{6} \div \frac{7}{8} \)
Dividing by a fraction is the same as multiplying by its reciprocal.
\[
\frac{5}{6} \div \frac{7}{8} = \frac{5}{6} \times \frac{8}{7} = \frac{5 \times 8}{6 \times 7} = \frac{40}{42} = \frac{20}{21}
\]

### Step 3: Add \( \frac{2}{3} + \frac{20}{21} \)
To add these fractions, we need a common denominator. The least common multiple of the denominators (3 and 21) is 21.

Convert \( \frac{2}{3} \) to a fraction with a denominator of 21:
\[
\frac{2}{3} = \frac{2 \times 7}{3 \times 7} = \frac{14}{21}
\]

Now we can add the fractions:
\[
\frac{14}{21} + \frac{20}{21} = \frac{14 + 20}{21} = \frac{34}{21}
\]

### Step 4: Convert \( \frac{34}{21} \) to a mixed number
To convert \( \frac{34}{21} \):
- \( 34 \div 21 = 1 \) with a remainder of \( 13 \).

So, \( \frac{34}{21} = 1 \frac{13}{21} \).

Thus, the final result is:
\[
1 \frac{13}{21}
\]

The correct answer is **B: 1 13/21**.

Answer 6

To determine which combination of snacks Rashad could purchase with his $17 leftover after buying movie tickets, we need to calculate the total cost of each option:

**Snack Prices:**
- Candy: $1.50
- Popcorn: $4.00
- Drink: $2.75

Now let's calculate the total cost for each of the options given.

### Option A: 2 drinks and 3 popcorns
- Cost of 2 drinks: \( 2 \times 2.75 = 5.50 \)
- Cost of 3 popcorns: \( 3 \times 4.00 = 12.00 \)
- Total cost: \( 5.50 + 12.00 = 17.50 \)

### Option B: 3 popcorns and 3 candies
- Cost of 3 popcorns: \( 3 \times 4.00 = 12.00 \)
- Cost of 3 candies: \( 3 \times 1.50 = 4.50 \)
- Total cost: \( 12.00 + 4.50 = 16.50 \)

### Option C: 4 popcorns and 1 candy
- Cost of 4 popcorns: \( 4 \times 4.00 = 16.00 \)
- Cost of 1 candy: \( 1 \times 1.50 = 1.50 \)
- Total cost: \( 16.00 + 1.50 = 17.50 \)

### Option D: 4 drinks and 2 popcorns
- Cost of 4 drinks: \( 4 \times 2.75 = 11.00 \)
- Cost of 2 popcorns: \( 2 \times 4.00 = 8.00 \)
- Total cost: \( 11.00 + 8.00 = 19.00 \)

### Summary of Total Costs:
- Option A: $17.50 (too high)
- Option B: $16.50 (within budget, $0.50 left)
- Option C: $17.50 (too high)
- Option D: $19.00 (too high)

Based on the calculations above, the only combination of snacks that Rashad could purchase with his $17 is:

**B. 3 popcorns and 3 candies.**

Answer 7

To simplify the expression \( 8x + 3(x + 5) - 5(x - 4) \), we follow these steps:

### Step 1: Distribute the terms

Distributing \( 3 \) in \( 3(x + 5) \):
\[
3(x + 5) = 3x + 15
\]

Distributing \( -5 \) in \( -5(x - 4) \):
\[
-5(x - 4) = -5x + 20
\]

### Step 2: Substitute back into the expression

Now we can rewrite the original expression:
\[
8x + 3x + 15 - 5x + 20
\]

### Step 3: Combine like terms

Now combine the \( x \) terms and the constant terms:
- Combine \( 8x \), \( 3x \), and \( -5x \):
\[
8x + 3x - 5x = 6x
\]

- Combine the constants \( 15 \) and \( 20 \):
\[
15 + 20 = 35
\]

### Step 4: Write the simplified expression

Putting it all together, we have:
\[
6x + 35
\]

Thus, the simplified expression is:

**A. \( 6x + 35 \)**.