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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.

7 answers

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.

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

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.

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

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.

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

B.
6x + 1

C.
6x - 5
D.
43x

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. \( 6x + 35 \).